Calculation of the convexity adjustment to the forward rate in the Vasicek model  for the forward exotic contracts

Куликов Александр В.; Малых Николай О.; Постевой Иван С.

doi:10.31857/S042473880021701-0

Главная>Номер 3>Calculation of the convexity adjustment to the forward rate in the Vasicek model for the forward exotic contracts

Calculation of the convexity adjustment to the forward rate in the Vasicek model for the forward exotic contracts

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Calculation of the convexity adjustment to the forward rate in the Vasicek model for the forward exotic contracts

Аннотация

Код статьи

S042473880021701-0-1

DOI

10.31857/S042473880021701-0

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Куликов Александр Владимирович Связаться с автором

Аффилиация: Физтех-школа прикладной математики и информатики, Московский физико-технический институт
Адрес: Долгопрудный, Российская Федерация

Малых Николай Олегович

Аффилиация: Физтех-школа прикладной математики и информатики, Московский физико-технический институт
Адрес: Российская Федерация

Постевой Иван Сергеевич

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Том 58 Номер 3

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115-128

Аннотация

In the following article, we consider forward contracts, which are financial instruments used to buy or sell some assets at a certain point moment in the future, and at the fixed price. Such contracts are customizable and traded over-the-counter, unlike futures, which are standardized contracts traded at exchanges. Particularly, we focus on in-arrears interest rate forward contracts (in-arrears FRA). The difference from the vanilla FRA: floating rate is immediately paid after it is fixed. We calculate the convexity adjustment to the forward simple interest rate in the single-factor Vasicek stochastic model for such contracts with different payment dates. With the help of the no-arbitrage market condition it is shown that such adjustments should be non-negative when payments occur before the end of accrual period and should be negative when payments occur after accrual period. We also studied in-arrears forward and option contracts, where fixed interest rate and principal, on which this rate is accrued, are denominated in different currencies (so called quanto in-arrears FRA and quanto in-arrears options). We checked that quanto in-arrears FRA equals in-arrears FRA in case when rates and principal are from the same currency market, and that quanto in-arrears option contract prices are greater than those of vanilla options.

Ключевые слова

convexity adjustment; forward rate agreement (FRA); Vasicek model; no-arbitrage market; in-arrears FRA; iFRA; quanto FRA; LIBOR; MOSPRIME

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17.04.2022

Дата публикации

22.09.2022

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ГОСТ	Куликов А. В. , Малых Н. О. , Постевой И. С. Calculation of the convexity adjustment to the forward rate in the Vasicek model for the forward exotic contracts // Экономика и математические методы. – 2022. – T. 58. – Номер 3 C. 115-128 . URL: https://emmras.ru/calculation-of-the-convexity-adjustment-to-the-forward-rate-in-the-vasicek-model-for-the-forward-exotic-contracts/?version_id=96382. DOI: 10.31857/S042473880021701-0
MLA	Kulikov, Aleksandr, Malykh, Nickolay, Postevoy, Ivan "Calculation of the convexity adjustment to the forward rate in the Vasicek model for the forward exotic contracts." Economics and the Mathematical Methods. 58.3 (2022).:115-128. DOI: 10.31857/S042473880021701-0
APA	Kulikov A., Malykh N., Postevoy I. (2022). Calculation of the convexity adjustment to the forward rate in the Vasicek model for the forward exotic contracts. Economics and the Mathematical Methods. vol. 58, no. 3, pp.115-128 DOI: 10.31857/S042473880021701-0


1	1. Introduction	1. Introduction 1. Introduction

2	Forward contracts are widely used financial instruments used for purchase/sell of some asset at a certain date in future at the specified fixed price.	Forward contracts are widely used financial instruments used for purchase/sell of some asset at a certain date in future at the specified fixed price. Forward contracts are widely used financial instruments used for purchase/sell of some asset at a certain date in future at the specified fixed price.

3	An example of forward contract is a forward rate agreement (FRA) on interest rate as an underlying asset, which we define in the next section.	An example of forward contract is a forward rate agreement (FRA) on interest rate as an underlying asset, which we define in the next section. An example of forward contract is a forward rate agreement (FRA) on interest rate as an underlying asset, which we define in the next section.

4	FRA is a cash settled contract with the payment based on the net difference between the floating interest rate and the fixed rate (Hull, 2017). Fixed rate makes the initial price of the FRA being equal to 0 is called forward rate.	FRA is a cash settled contract with the payment based on the net difference between the floating interest rate and the fixed rate (Hull, 2017). Fixed rate makes the initial price of the FRA being equal to 0 is called <em>forward rate</em>. FRA is a cash settled contract with the payment based on the net difference between the floating interest rate and the fixed rate (Hull, 2017). Fixed rate makes the initial price of the FRA being equal to 0 is called <em>forward rate</em>.

5	There is an exotic in-arrears contract which is settled at the beginning of the forward period — not at the end. The forward rate of an in-arrears contract is greater than the forward rate of a vanilla contract and the difference between these two rates depends on stochastic model used to simulate financial processes and is called convexity adjustment.	There is an exotic <em>in-arrears contract</em> which is settled at the beginning of the forward period — not at the end. The forward rate of an in-arrears contract is greater than the forward rate of a vanilla contract and the difference between these two rates depends on stochastic model used to simulate financial processes and is called convexity adjustment. There is an exotic <em>in-arrears contract</em> which is settled at the beginning of the forward period — not at the end. The forward rate of an in-arrears contract is greater than the forward rate of a vanilla contract and the difference between these two rates depends on stochastic model used to simulate financial processes and is called convexity adjustment.

6	Studies on this topic may be found in (Mcinerney, Zastawniak, 2015), where LIBOR in-arrears rate was considered. The adjustment was calculated using the replication strategy and solving stochastic differential equation in the LIBOR market model. Another approach using the change of measure was studied in (Palsser, 2003), where simple lognormal stochastic model was chosen to calculate an in-arrears forward LIBOR rate. In (Gaminha, Gaspar, Oliveira, 2015), authors explored the Vasicek and Cox–Ingersoll–Ross models within LIBOR in-arrears rate. The authors obtained the adjustment from stochastic differential equation (SDE) numerical solution of convexity term SDE and found the partial closed-form solution for Vasicek model. There are also researches on in-arrears options – caps and floors (Hagan, 2003) where prices of options were found using the replication strategy for option-like pay-off. Finally, in the previous paper, written by two authors of this article, (Malykh, Postevoy, 2019), pricing of in-arrears FRA and in-arrears interest rate options using change of measure were considered.	Studies on this topic may be found in (Mcinerney, Zastawniak, 2015), where LIBOR in-arrears rate was considered. The adjustment was calculated using the replication strategy and solving stochastic differential equation in the LIBOR market model. Another approach using the change of measure was studied in (Palsser, 2003), where simple lognormal stochastic model was chosen to calculate an in-arrears forward LIBOR rate. In (Gaminha, Gaspar, Oliveira, 2015), authors explored the Vasicek and Cox–Ingersoll–Ross models within LIBOR in-arrears rate. The authors obtained the adjustment from stochastic differential equation (SDE) numerical solution of convexity term SDE and found the partial closed-form solution for Vasicek model. There are also researches on in-arrears options – caps and floors (Hagan, 2003) where prices of options were found using the replication strategy for option-like pay-off. Finally, in the previous paper, written by two authors of this article, (Malykh, Postevoy, 2019), pricing of in-arrears FRA and in-arrears interest rate options using change of measure were considered. Studies on this topic may be found in (Mcinerney, Zastawniak, 2015), where LIBOR in-arrears rate was considered. The adjustment was calculated using the replication strategy and solving stochastic differential equation in the LIBOR market model. Another approach using the change of measure was studied in (Palsser, 2003), where simple lognormal stochastic model was chosen to calculate an in-arrears forward LIBOR rate. In (Gaminha, Gaspar, Oliveira, 2015), authors explored the Vasicek and Cox–Ingersoll–Ross models within LIBOR in-arrears rate. The authors obtained the adjustment from stochastic differential equation (SDE) numerical solution of convexity term SDE and found the partial closed-form solution for Vasicek model. There are also researches on in-arrears options – caps and floors (Hagan, 2003) where prices of options were found using the replication strategy for option-like pay-off. Finally, in the previous paper, written by two authors of this article, (Malykh, Postevoy, 2019), pricing of in-arrears FRA and in-arrears interest rate options using change of measure were considered.

7	There is also another kind of exotic forward contracts – quanto FRA, in which the notional principal amount is denominated in a currency other than the currency in which the interest rate is settled.	There is also another kind of exotic forward contracts – quanto FRA, in which the notional principal amount is denominated in a currency other than the currency in which the interest rate is settled. There is also another kind of exotic forward contracts – quanto FRA, in which the notional principal amount is denominated in a currency other than the currency in which the interest rate is settled.

8	Such contracts were studied in (Lin, 2012), where author used forward measure pricing methodology to derive the valuation formulas within the Heath–Jarrow–Morton interest rate model. Research on quanto interest rate options may be found in (Hsieh, Chou, Chen, 2015), where authors also adopted martingale probability measure to obtain options pricing in the cross-currency LIBOR market model.	Such contracts were studied in (Lin, 2012), where author used forward measure pricing methodology to derive the valuation formulas within the Heath–Jarrow–Morton interest rate model. Research on quanto interest rate options may be found in (Hsieh, Chou, Chen, 2015), where authors also adopted martingale probability measure to obtain options pricing in the cross-currency LIBOR market model. Such contracts were studied in (Lin, 2012), where author used forward measure pricing methodology to derive the valuation formulas within the Heath–Jarrow–Morton interest rate model. Research on quanto interest rate options may be found in (Hsieh, Chou, Chen, 2015), where authors also adopted martingale probability measure to obtain options pricing in the cross-currency LIBOR market model.

9	In this article we are going to continue our previous work and expand change of measure method in a single-factor Vasicek stochastic model (Vasicek, 1977) to consider cases, when the payment in FRA occurs in other dates, — not only at the beginning or at the end of the forward period. We prove that the convexity adjustment is negative when the settlement date takes place after the forward period. We also apply it to explore quanto FRA. Moreover, we combine it with the in-arrears FRA and come to the in-arrears quanto FRA. At the end, in-arrears quanto options are briefly considered.	In this article we are going to continue our previous work and expand <em>change of measure method</em> in a single-factor Vasicek stochastic model (Vasicek,<strong> </strong>1977) to consider cases, when the payment in FRA occurs in other dates, — not only at the beginning or at the end of the forward period. We prove that the convexity adjustment is negative when the settlement date takes place after the forward period. We also apply it to explore quanto FRA. Moreover, we combine it with the in-arrears FRA and come to the in-arrears quanto FRA. At the end, in-arrears quanto options are briefly considered. In this article we are going to continue our previous work and expand <em>change of measure method</em> in a single-factor Vasicek stochastic model (Vasicek,<strong> </strong>1977) to consider cases, when the payment in FRA occurs in other dates, — not only at the beginning or at the end of the forward period. We prove that the convexity adjustment is negative when the settlement date takes place after the forward period. We also apply it to explore quanto FRA. Moreover, we combine it with the in-arrears FRA and come to the in-arrears quanto FRA. At the end, in-arrears quanto options are briefly considered.

10	2. Definitions	2. Definitions 2. Definitions

11	Let us introduce definitions which we use further in this paper.	Let us introduce definitions which we use further in this paper. Let us introduce definitions which we use further in this paper.

12	Definition 1. Zero-coupon bond (ZCB) with maturity T is a security which promises to pay owner 1currency unit at T. We denote ZCB price at the moment t by $P (t, T)$ , where $P (t, T)$ is an $F_{t}$ -measurable function and $P (T, T) = 1$ .	<strong>Definition 1.</strong> Zero-coupon bond (ZCB) with maturity <em>T</em> is a security which promises to pay owner 1currency unit at <em>T</em>. We denote ZCB price at the moment <em>t</em> by <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></math> , where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></math> is an <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></math> -measurable function and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>P</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mn>1</mn></math> . <strong>Definition 1.</strong> Zero-coupon bond (ZCB) with maturity <em>T</em> is a security which promises to pay owner 1currency unit at <em>T</em>. We denote ZCB price at the moment <em>t</em> by <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></math> , where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></math> is an <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></math> -measurable function and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>P</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mn>1</mn></math> .

13	LIBOR is the indicative rate on which banks are willing to lend money each other, LIBID is the indicative rate on which banks are willing to borrow money. We assume equivalence of LIBID and LIBOR. MOSPRIME is a Russian analogue of the LIBOR rate, i.e. MOSPRIME is the indicative rate on which banks are willing to lend money to each other in rubles. We also make standard “Black–Sholes–Merton model” assumptions: no transaction costs; no default risk; no funding risk; no liquidity risk.	LIBOR is the indicative rate on which banks are willing to lend money each other, LIBID is the indicative rate on which banks are willing to borrow money. We assume equivalence of LIBID and LIBOR. MOSPRIME is a Russian analogue of the LIBOR rate, i.e. MOSPRIME is the indicative rate on which banks are willing to lend money to each other in rubles. We also make standard “Black–Sholes–Merton model” assumptions: no transaction costs; no default risk; no funding risk; no liquidity risk. LIBOR is the indicative rate on which banks are willing to lend money each other, LIBID is the indicative rate on which banks are willing to borrow money. We assume equivalence of LIBID and LIBOR. MOSPRIME is a Russian analogue of the LIBOR rate, i.e. MOSPRIME is the indicative rate on which banks are willing to lend money to each other in rubles. We also make standard “Black–Sholes–Merton model” assumptions: no transaction costs; no default risk; no funding risk; no liquidity risk.

14	Now we define LIBOR rate and forward rate agreement more precisely.	Now we define LIBOR rate and forward rate agreement more precisely. Now we define LIBOR rate and forward rate agreement more precisely.

15	Definition 2. We denote LIBOR spot rate at the moment t for a time period $α > 0$ by $L (t, t, t + α)$ . Bank can lend (or borrow) N currency units at the time t for a period $α$ and get (return) $N (1 + α L (t, t, t + α))$ currency units at the moment $t + α$ . Technically, MOPSRIME rate definition is similar to the LIBOR one, i.e. it is a spot rate with simple compounding. We use the LIBOR and MOSPRIME terms interchangeably through the article.	<strong>Definition 2.</strong> We denote LIBOR spot rate at the moment <em>t</em> for a time period <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>α</mi><mo>></mo><mn>0</mn></math> by <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mi>α</mi><mo>)</mo></math> . Bank can lend (or borrow) <em>N</em> currency units at the time <em>t</em> for a period <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>α</mi></math> and get (return) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>α</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mi>α</mi><mo>)</mo><mo>)</mo></math> currency units at the moment <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>+</mo><mi>α</mi></math> . Technically, MOPSRIME rate definition is similar to the LIBOR one, i.e. it is a spot rate with simple compounding. We use the LIBOR and MOSPRIME terms interchangeably through the article. <strong>Definition 2.</strong> We denote LIBOR spot rate at the moment <em>t</em> for a time period <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>α</mi><mo>></mo><mn>0</mn></math> by <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mi>α</mi><mo>)</mo></math> . Bank can lend (or borrow) <em>N</em> currency units at the time <em>t</em> for a period <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>α</mi></math> and get (return) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>α</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mi>α</mi><mo>)</mo><mo>)</mo></math> currency units at the moment <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>+</mo><mi>α</mi></math> . Technically, MOPSRIME rate definition is similar to the LIBOR one, i.e. it is a spot rate with simple compounding. We use the LIBOR and MOSPRIME terms interchangeably through the article.

16	Definition 3. Forward rate agreement (FRA) is an over-the-counter contract for the exchange of two cash flows at a certain date in future. Floating reference rate is fixed at $T_{1}$ . Buyer of this contract at $t \leq T_{1}$ with maturity $T_{2}$ , fixed rate K and principal N, agrees on following obligation between counterparties at $T_{2}$ :	<strong>Definition 3. </strong>Forward rate agreement (FRA) is an over-the-counter contract for the exchange of two cash flows at a certain date in future. Floating reference rate is fixed at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> . Buyer of this contract at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> , fixed rate <em>K</em> and principal <em>N</em>, agrees on following obligation between counterparties at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> : <strong>Definition 3. </strong>Forward rate agreement (FRA) is an over-the-counter contract for the exchange of two cash flows at a certain date in future. Floating reference rate is fixed at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> . Buyer of this contract at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> , fixed rate <em>K</em> and principal <em>N</em>, agrees on following obligation between counterparties at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> :

17	pay $(T_{2} - T_{1}) K N$ currency units to contract counterparty, receive $(T_{2} - T_{1}) L (T_{1}, T_{1}, T_{2}) N$ currency units from contract counterparty.	<ol> <li><em>pay</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi><mi>N</mi></math> currency units to contract counterparty, </li> <li><em>receive</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>N</mi></math> currency units from contract counterparty. </li></ol> <ol> <li><em>pay</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi><mi>N</mi></math> currency units to contract counterparty, </li> <li><em>receive</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>N</mi></math> currency units from contract counterparty. </li></ol>

18	The price of the FRA at $T_{2}$ is equal to $(T_{2} - T_{1}) (L (T_{1}, T_{1}, T_{2}) - K) N$ .	The price of the FRA at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> is equal to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi><mo>)</mo><mi>N</mi></math> . The price of the FRA at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> is equal to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi><mo>)</mo><mi>N</mi></math> .

19	For simplicity, we assume that principal amount $N = 1$ .	For simplicity, we assume that principal amount <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi><mo>=</mo><mn>1</mn></math> . For simplicity, we assume that principal amount <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi><mo>=</mo><mn>1</mn></math> .

20	Definition 4. Forward rate $L (t, T_{1}, T_{2})$ is the fixed rate K which makes price of the FRA contract at t equal to 0 for $t \leq T_{1} \leq T_{2}$ .	<strong>Definition 4.</strong> Forward rate <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> is the fixed rate <em>K</em> which makes price of the FRA contract at <em>t</em> equal to 0 for <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> . <strong>Definition 4.</strong> Forward rate <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> is the fixed rate <em>K</em> which makes price of the FRA contract at <em>t</em> equal to 0 for <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> .

21	It can be shown (Hull, 2017), that $L (t, T_{1}, T_{2}) = \frac{P (t, T_{1}) - P (t, T_{2})}{(T_{2} - T_{1}) P (t, T_{2})}$ .	It can be shown (Hull, 2017), that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></math> . It can be shown (Hull, 2017), that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></math> .

22	Now, we consider exotic in-arrears FRA: this contract is settled at time $T_{1}$ .	Now, we consider exotic in-arrears FRA: this contract is settled at time <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> . Now, we consider exotic in-arrears FRA: this contract is settled at time <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> .

23	Definition 5. In-arrears FRA (iFRA) is an over-the-counter contract for the exchange of two cash flows at a certain date. Floating reference rate is fixed at $T_{1}$ . Buyer of this contract at $t \leq T_{1}$ with maturity $T_{1}$ , fixed rate K and principal N, agrees on following obligation between counterparties at $T_{1}$ (not $T_{2}$ ):	<strong>Definition </strong><strong>5</strong><strong>.</strong> In-arrears FRA (iFRA) is an over-the-counter contract for the exchange of two cash flows at a certain date. Floating reference rate is fixed at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> . Buyer of this contract at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> , fixed rate <em>K</em> and principal <em>N</em>, agrees on following obligation between counterparties at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> (not <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> ): <strong>Definition </strong><strong>5</strong><strong>.</strong> In-arrears FRA (iFRA) is an over-the-counter contract for the exchange of two cash flows at a certain date. Floating reference rate is fixed at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> . Buyer of this contract at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> , fixed rate <em>K</em> and principal <em>N</em>, agrees on following obligation between counterparties at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> (not <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> ):

24	pay $(T_{2} - T_{1}) K N$ currency units to counterparty, receive $(T_{2} - T_{1}) L (T_{1}, T_{1}, T_{2}) N$ currency units from counterparty.	<ol> <li><em>pay</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi><mi>N</mi></math> currency units to counterparty, </li> <li><em>receive</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>N</mi></math> currency units from counterparty. </li></ol> <ol> <li><em>pay</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi><mi>N</mi></math> currency units to counterparty, </li> <li><em>receive</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>N</mi></math> currency units from counterparty. </li></ol>

25	The price of the iFRA at $T_{1}$ is equal to $(T_{2} - T_{1}) (L (T_{1}, T_{1}, T_{2}) - K) N$ .	The price of the iFRA at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> is equal to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi><mo>)</mo><mi>N</mi></math> . The price of the iFRA at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> is equal to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi><mo>)</mo><mi>N</mi></math> .

26	We denote K which gives iFRA a 0 (zero) price at t by $i L (t, T_{1}, T_{2})$ .	We denote <em>K</em> which gives iFRA a 0 (zero) price at <em>t</em> by <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> . We denote <em>K</em> which gives iFRA a 0 (zero) price at <em>t</em> by <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> .

27	A portfolio of assets is called self-financed if its value changes only due to changes in the asset prices.	A portfolio of assets is called self-financed if its value changes only due to changes in the asset prices. A portfolio of assets is called self-financed if its value changes only due to changes in the asset prices.

28	Definition 6. Self-financed potrfolio A is called an arbitrage portfolio on some probability space $(Ω, F, P)$ if its price (value) at the time t is $V^{A} (t) \leq 0$ and $\exists T > t :$ $P (V^{A} (T) \geq 0) = 1$ and $P (V^{A} (T) > 0) > 0$ .	<strong>Definition </strong><strong>6</strong><strong>.</strong> Self-financed potrfolio <em>A</em> is called an arbitrage portfolio on some probability space <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="script">F</mi><mo>,</mo><mi mathvariant="double-struck">P</mi><mo>)</mo></math> if its price (value) at the time <em>t</em> is <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>V</mi></mrow><mrow><mi>A</mi></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>≤</mo><mn>0</mn></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>∃</mo><mi>T</mi><mo>></mo><mi>t</mi><mo>:</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="double-struck">P</mi><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mi>A</mi></mrow></msup><mo>(</mo><mi>T</mi><mo>)</mo><mo>≥</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>1</mn></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="double-struck">P</mi><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mi>A</mi></mrow></msup><mo>(</mo><mi>T</mi><mo>)</mo><mo>></mo><mn>0</mn><mo>)</mo><mo>></mo><mn>0</mn></math> . <strong>Definition </strong><strong>6</strong><strong>.</strong> Self-financed potrfolio <em>A</em> is called an arbitrage portfolio on some probability space <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi mathvariant="script">F</mi><mo>,</mo><mi mathvariant="double-struck">P</mi><mo>)</mo></math> if its price (value) at the time <em>t</em> is <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>V</mi></mrow><mrow><mi>A</mi></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>≤</mo><mn>0</mn></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>∃</mo><mi>T</mi><mo>></mo><mi>t</mi><mo>:</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="double-struck">P</mi><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mi>A</mi></mrow></msup><mo>(</mo><mi>T</mi><mo>)</mo><mo>≥</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>1</mn></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="double-struck">P</mi><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mi>A</mi></mrow></msup><mo>(</mo><mi>T</mi><mo>)</mo><mo>></mo><mn>0</mn><mo>)</mo><mo>></mo><mn>0</mn></math> .

29	We use the assumption of absence of any arbitrage portfolio on the market.	We use the assumption of absence of any arbitrage portfolio on the market. We use the assumption of absence of any arbitrage portfolio on the market.

30	3. In-arrears FRA	3. In-arrears FRA 3. In-arrears FRA

31	It was shown (Malykh, Postevoy, 2019) the that convexity adjustment (CA) for in-arrears FRA under single-factor Vasicek model is:	It was shown (Malykh, Postevoy, 2019) the that convexity adjustment (CA) for in-arrears FRA under single-factor Vasicek model is: It was shown (Malykh, Postevoy, 2019) the that convexity adjustment (CA) for in-arrears FRA under single-factor Vasicek model is:

32	$C A (t, T_{1}, T_{2}) = \frac{1}{T_{2} - T_{1}} \times \frac{P (t, T_{1})}{P (t, T_{2})} (e^{I} - 1),$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>C</mi><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>×</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>(</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mn>1</mn><mo>)</mo><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>C</mi><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>×</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>(</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mn>1</mn><mo>)</mo><mo>,</mo></math>

33	where	where where

34	$I = \frac{σ^{2}}{a^{2}} (\frac{1}{2 a} - \frac{1}{2 a} e^{- 2 a (T_{1} - t)} - \frac{1}{a} e^{- a (T_{2} - T_{1})} + \frac{1}{2 a} e^{- 2 a (T_{2} - T_{1})} + \frac{1}{a} e^{- a (T_{1} + T_{2}) + 2 a t} - \frac{1}{2 a} e^{- 2 a (T_{2} - t)}),$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>I</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mfenced close="" separators="\|"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mn>2</mn><mi>a</mi><mi>t</mi></mrow></msup><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced><mo>,</mo></mrow></mfenced></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>I</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mfenced close="" separators="\|"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mn>2</mn><mi>a</mi><mi>t</mi></mrow></msup><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced><mo>,</mo></mrow></mfenced></math>

35	$P (t, T) = A (t, T) e^{- B (t, T) r (t)},$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>B</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mi>r</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></msup><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>B</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mi>r</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></msup><mo>,</mo></math>

36	$B (t, T) = (1 - e^{- a (T - t)}) / a,$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>B</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mfenced separators="\|"><mrow><mn>1</mn><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><mi>T</mi><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced><mo>/</mo><mi>a</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>B</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mfenced separators="\|"><mrow><mn>1</mn><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><mi>T</mi><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced><mo>/</mo><mi>a</mi><mo>,</mo></math>

37	$A (t, T) = e x p ((B (t, T) - (T - t)) (\frac{θ}{a} - \frac{σ^{2}}{2 a^{2}}) - \frac{σ^{2} B (t, T)^{2}}{4 a}),$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced separators="\|"><mrow><mfenced separators="\|"><mrow><mi>B</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>-</mo><mo>(</mo><mi>T</mi><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mfenced separators="\|"><mrow><mfrac><mrow><mi>θ</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>-</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>B</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn><mi>a</mi></mrow></mfrac></mrow></mfenced><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced separators="\|"><mrow><mfenced separators="\|"><mrow><mi>B</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>-</mo><mo>(</mo><mi>T</mi><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mfenced separators="\|"><mrow><mfrac><mrow><mi>θ</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>-</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>B</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn><mi>a</mi></mrow></mfrac></mrow></mfenced><mo>,</mo></math>

38	where $θ$ and $a$ are constant parameters in this model, which is given by the following SDE for instantaneous interest spot-rate: $d r (t) = (θ - a r (t)) d t + σ d W (t) .$ Now we are going to study other exotic FRAs in this model.	where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>θ</mi></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>a</mi></math> are constant parameters in this model, which is given by the following SDE for instantaneous interest spot-rate: <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><mi>r</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>θ</mi><mo>-</mo><mi>a</mi><mi>r</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><mi>σ</mi><mi>d</mi><mi>W</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>.</mo></math> Now we are going to study other exotic FRAs in this model. where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>θ</mi></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>a</mi></math> are constant parameters in this model, which is given by the following SDE for instantaneous interest spot-rate: <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><mi>r</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>θ</mi><mo>-</mo><mi>a</mi><mi>r</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><mi>σ</mi><mi>d</mi><mi>W</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>.</mo></math> Now we are going to study other exotic FRAs in this model.

39	4. Exotic FRA with different payment time options	4. Exotic FRA with different payment time options 4. Exotic FRA with different payment time options

40	Along with the in-arrears contracts we can construct a FRA with payment date $T_{p a y}$ such as $t < T_{p a y} < T_{1}$ , $T_{1} < T_{p a y} < T_{2}$ , or $T_{2} < T_{p a y}$ . We consider each of these contracts using the same change of measure technique described in (Geman, Karoui, Rochet, 1995).	Along with the in-arrears contracts we can construct a FRA with payment date <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> such as <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> , or <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> . We consider each of these contracts using the same change of measure technique described in (Geman, Karoui, Rochet, 1995). Along with the in-arrears contracts we can construct a FRA with payment date <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> such as <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> , or <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> . We consider each of these contracts using the same change of measure technique described in (Geman, Karoui, Rochet, 1995).

41	Let us denote exotic forward LIBOR rate by $i L$ . Forward rate is the expected value of the future rate under appropriate forward measure (Privault, 2012). Then	Let us denote exotic forward LIBOR rate by <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi></math> . Forward rate is the expected value of the future rate under appropriate forward measure (Privault, 2012). Then Let us denote exotic forward LIBOR rate by <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi></math> . Forward rate is the expected value of the future rate under appropriate forward measure (Privault, 2012). Then

42	$i L (t, T_{p a y}, T_{1}, T_{2}) = E_{Q_{T_{p a y}}} [L (T_{1}, T_{1}, T_{2}) \| F_{t}]$ (1)	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></math> (1) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></math> (1)

43	( $E_{Q_{T_{p a y}}}$ — conditional expectation value).	( <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub></mrow></msub></math> — conditional expectation value). ( <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub></mrow></msub></math> — conditional expectation value).

44	Theorem 1. In a single-factor Vasicek model we have	<strong>Theorem 1.</strong> <em>In a single-factor Vasicek model we have </em> <strong>Theorem 1.</strong> <em>In a single-factor Vasicek model we have </em>

45	$\begin{array}{l} i L (t, T_{p a y}, T_{1}, T_{2}) = L (t, T_{1}, T_{2}) + L (t, T_{1}, T_{2}) \frac{P (t, T_{2}) 1_{T_{p a y} \leq T_{2}} - P (t, T_{p a y})}{P (t, T_{p a y})} + \\ + \frac{P (t, T_{2})}{P (t, T_{p a y}) τ} (\frac{P (t, T_{1})}{P (t, T_{2})} \frac{P (t, T_{p a y})}{P (t, T_{2})} e^{I} - (\frac{P (t, T_{1})}{P (t, T_{2})} - 1) 1_{T_{p a y} \leq T_{2}} - \frac{P (t, T_{p a y})}{P (t, T_{2})}), \end{array}$ (2)	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>-</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>τ</mi></mrow></mfrac><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mn>1</mn></mrow></mfenced><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> (2) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>-</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>τ</mi></mrow></mfrac><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mn>1</mn></mrow></mfenced><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> (2)

46	where	<em>where</em> <em>where</em>

47	$\begin{matrix} I = \frac{σ}{2 a^{3}} (e^{- a \| T_{1} - T_{p a y} \|} - e^{- a (T_{1} + T_{p a y} - 2 t)} - e^{- a (T_{2} - m i n (T_{1}, T_{p a y}))} + e^{- a (T_{2} - 2 t + m i n (T_{1}, T_{p a y}))} - e^{- a (T_{2} + m a x (T_{1} - 2 T_{p a y}, T_{p a y} - 2 T_{1}))} + \\ + e^{- a (T_{2} - 2 t + m a x (T_{1}, T_{p a y}))} + e^{- 2 a (T_{2} - m i n (T_{1}, T_{p a y}))} - e^{- 2 a (T_{2} - t)}) \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>I</mi><mo>=</mo><mfrac><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mfenced close="" separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>\|</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>\|</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow></msup><mo>+</mo></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mn>2</mn><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>)</mo></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow></msup><mfenced open="" separators="\|"><mrow><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>I</mi><mo>=</mo><mfrac><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mfenced close="" separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>\|</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>\|</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow></msup><mo>+</mo></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mn>2</mn><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>)</mo></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow></msup><mfenced open="" separators="\|"><mrow><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced></mtd></mtr></mtable></math>

48	( $1_{T_{p a y} \leq T_{2}}$ — indicator function).	( <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> — indicator function). ( <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> — indicator function).

49	The case $T_{p a y} = T_{1}$ is considered in (Malykh, Postevoy, 2019). Now we consider other cases.	The case <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> is considered in (Malykh, Postevoy, 2019). Now we consider other cases. The case <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> is considered in (Malykh, Postevoy, 2019). Now we consider other cases.

50	4.1. $t < T_{p a y} < T_{1}$	4.1. <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="bold-italic">t</mi><mo><</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">p</mi><mi mathvariant="bold-italic">a</mi><mi mathvariant="bold-italic">y</mi></mrow></msub><mo><</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> 4.1. <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="bold-italic">t</mi><mo><</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">p</mi><mi mathvariant="bold-italic">a</mi><mi mathvariant="bold-italic">y</mi></mrow></msub><mo><</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn>1</mn></mrow></msub></math>

51	Using results from (Privault, 2012), we change the measure to $Q_{T_{2}}$ in (1).	Using results from (Privault, 2012), we change the measure to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> in (1). Using results from (Privault, 2012), we change the measure to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> in (1).

52	$\begin{array}{l} i L (t, T_{p a y}, T_{1}, T_{2}) = \frac{P (t, T_{2})}{P (t, T_{p a y})} E_{Q_{T_{2}}} [L (T_{1}, T_{1}, T_{2}) \frac{1}{P (T_{p a y}, T_{2})} \|F_{t}] = \\ = \frac{P (t, T_{2})}{P (t, T_{p a y})} E_{Q_{T_{2}}} [L (T_{1}, T_{1}, T_{2}) (1 + (T_{2} - T_{p a y}) L (T_{p a y}, T_{p a y}, T_{2})) \| F_{t}] = \\ = \frac{P (t, T_{2})}{P (t, T_{p a y})} (L (t, T_{1}, T_{2}) + (T_{2} - T_{p a y}) E_{Q_{T_{2}}} [L (T_{1}, T_{1}, T_{2}) L (T_{p a y}, T_{p a y}, T_{2}) \| F_{t}]) . \end{array}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced open="\|" close="" separators="\|"><mrow><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfenced separators="\|"><mrow><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced open="\|" close="" separators="\|"><mrow><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfenced separators="\|"><mrow><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>

53	Using the tower property of conditional expectation:	Using the tower property of conditional expectation: Using the tower property of conditional expectation:

54	$\begin{matrix} E_{Q_{T_{2}}} [L (T_{1}, T_{1}, T_{2}) L (T_{p a y}, T_{p a y}, T_{2}) \| F_{t}] = E_{Q_{T_{2}}} [E_{Q_{T_{2}}} [L (T_{1}, T_{1}, T_{2}) L (T_{p a y}, T_{p a y}, T_{2}) \| F_{T_{p a y}}] \| F_{t}] = \\ = E_{Q_{T_{2}}} [L (T_{p a y}, T_{1}, T_{2}) L (T_{p a y}, T_{p a y}, T_{2}) \| F_{t}] . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>

55	Next we find dynamic of the following process under $Q_{T_{2}}$ -measure:	Next we find dynamic of the following process under <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> -measure: Next we find dynamic of the following process under <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> -measure:

56	$\begin{array}{l} L (T_{p a y}, T_{1}, T_{2}) L (T_{p a y}, T_{p a y}, T_{2}) = \\ = \frac{1}{(T_{2} - T_{1}) (T_{2} - T_{p a y})} (\frac{P (T_{p a y}, T_{1}) P (T_{p a y}, T_{p a y})}{P (T_{p a y}, T_{2}) P (T_{p a y}, T_{2})} - \frac{P (T_{p a y}, T_{1})}{P (T_{p a y}, T_{2})} - \frac{P (T_{p a y}, T_{p a y})}{P (T_{p a y}, T_{2})} + 1) . \end{array}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mfenced close="" separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>.</mo></mrow></mfenced></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mfenced close="" separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>.</mo></mrow></mfenced></mtd></mtr></mtable></math>

57	The 2nd and the 3rd terms are the martingales under $Q_{T_{2}}$ -measure. We need to know dynamic of the 1st term.	The 2nd and the 3rd terms are the martingales under <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> -measure. We need to know dynamic of the 1st term. The 2nd and the 3rd terms are the martingales under <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> -measure. We need to know dynamic of the 1st term.

58	$\begin{array}{l} d (\frac{P (t, T_{1}) P (t, T_{p a y})}{P (t, T_{2}) P (t, T_{2})}) = \\ = \frac{P (t, T_{1}) P (t, T_{p a y})}{P (t, T_{2}) P (t, T_{2})} ((ζ^{T_{p a y}} (t) + ζ^{T_{1}} (t) - 2 ζ^{T_{2}} (t)) d W^{T_{2}} (t) + (ζ^{T_{p a y}} (t) - ζ^{T_{2}} (t)) (ζ^{T_{1}} (t) - ζ^{T_{2}} (t)) d t), \end{array}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>d</mi><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced close="" separators="\|"><mrow><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><mn>2</mn><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mi>d</mi><msup><mrow><mi>W</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mfenced open="" separators="\|"><mrow><mo>+</mo><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>t</mi></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>d</mi><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced close="" separators="\|"><mrow><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><mn>2</mn><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mi>d</mi><msup><mrow><mi>W</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mfenced open="" separators="\|"><mrow><mo>+</mo><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>t</mi></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math>

59	where $ζ^{T_{i}} (t) = σ B (t, T_{i})$ . So,	where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>σ</mi><mi>B</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math> . So, where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>σ</mi><mi>B</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math> . So,

60	$\begin{matrix} \frac{P (T_{p a y}, T_{1}) P (T_{p a y}, T_{p a y})}{P (T_{p a y}, T_{2}) P (T_{p a y}, T_{2})} = \frac{P (t, T_{1}) P (t, T_{p a y})}{P (t, T_{2}) P (t, T_{2})} \frac{}{} e x p [\int_{t}^{T_{p a y}} ‍ ((ζ^{T_{p a y}} (t) + ζ^{T_{1}} (t) - 2 ζ^{T_{2}} (t)) d W^{T_{2}} (t) + \\ + (ζ^{T_{p a y}} (t) - ζ^{T_{2}} (t)) (ζ^{T_{1}} (t) - ζ^{T_{2}} (t)) - 0.5 (ζ^{T_{p a y}} (t) + ζ^{T_{1}} (t) - 2 ζ^{T_{2}} (t))^{2} d t)] . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfrac><mrow/><mrow/></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="[" close="" separators="\|"><mrow><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mfenced close="" separators="\|"><mrow><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><mfenced open="" separators="\|"><mrow><mn>2</mn><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mi>d</mi><msup><mrow><mi>W</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo></mrow></mfenced></mrow></mfenced></mtd></mtr><mtr><mtd><mo>+</mo><mfenced separators="\|"><mrow><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mfenced separators="\|"><mrow><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mo>-</mo><mn>0.5</mn><mfenced open="" close="]" separators="\|"><mrow><mfenced separators="\|"><mrow><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><mn>2</mn><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfrac><mrow/><mrow/></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="[" close="" separators="\|"><mrow><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mfenced close="" separators="\|"><mrow><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><mfenced open="" separators="\|"><mrow><mn>2</mn><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mi>d</mi><msup><mrow><mi>W</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo></mrow></mfenced></mrow></mfenced></mtd></mtr><mtr><mtd><mo>+</mo><mfenced separators="\|"><mrow><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mfenced separators="\|"><mrow><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mo>-</mo><mn>0.5</mn><mfenced open="" close="]" separators="\|"><mrow><mfenced separators="\|"><mrow><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><mn>2</mn><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>

61	Now we find expectation under $Q_{T_{2}}$ -measure:	Now we find expectation under <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> -measure: Now we find expectation under <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> -measure:

62	$\begin{array}{l} E_{Q_{T_{2}}} [\frac{P (T_{p a y}, T_{1}) P (T_{p a y}, T_{p a y})}{P (T_{p a y}, T_{2}) P (T_{p a y}, T_{2})} \| F_{t}] = \frac{P (t, T_{1}) P (t, T_{p a y})}{P (t, T_{2}) P (t, T_{2})} e^{I}, \end{array}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>,</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>,</mo></mtd></mtr></mtable></math>

63	where	where where

64	$\begin{matrix} I = \int_{t}^{T_{p a y}} ‍ (ζ^{T_{p a y}} (t) - ζ^{T_{2}} (t)) (ζ^{T_{1}} (t) - ζ^{T_{2}} (t)) d t = \frac{σ^{2}}{2 a^{3}} (e^{- a (T_{1} - T_{p a y})} - e^{- a (T_{1} + T_{p a y} - 2 t)} - \\ - e^{- a (T_{2} - T_{p a y})} + e^{- a (T_{2} + T_{p a y} - 2 t)} - e^{- a (T_{1} + T_{2} - 2 T_{p a y})} + e^{- a (T_{1} + T_{2} - 2 t)} + e^{- 2 a (T_{2} - T_{p a y})} - e^{- 2 a (T_{2} - t)}) . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>I</mi><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>t</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mfenced close="" separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mfenced open="" separators="\|"><mrow><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>2</mn><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>I</mi><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>t</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mfenced close="" separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mfenced open="" separators="\|"><mrow><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>2</mn><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>

65	Putting it all together we can write	Putting it all together we can write Putting it all together we can write

66	$\begin{matrix} i L (t, T_{p a y}, T_{1}, T_{2}) = L (t, T_{1}, T_{2}) + L (t, T_{1}, T_{2}) P (t, T_{2}) - P (t, T_{p a y}) / P (t, T_{p a y}) + \\ + \frac{P (t, T_{2})}{P (t, T_{p a y}) (T_{2} - T_{1})} (\frac{P (t, T_{1}) P (t, T_{p a y})}{P (t, T_{2}) P (t, T_{2})} e^{I} - \frac{P (t, T_{1})}{P (t, T_{2})} - \frac{P (t, T_{p a y})}{P (t, T_{2})} + 1) . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>/</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>/</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>

67	4.2. $T_{1} < T_{p a y} < T_{2}$	4.2. <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">p</mi><mi mathvariant="bold-italic">a</mi><mi mathvariant="bold-italic">y</mi></mrow></msub><mo><</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> 4.2. <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">p</mi><mi mathvariant="bold-italic">a</mi><mi mathvariant="bold-italic">y</mi></mrow></msub><mo><</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn>2</mn></mrow></msub></math>

68	Under $Q_{T_{2}}$ -measure:	Under <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> -measure: Under <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math> -measure:

69	$\begin{matrix} i L (t, T_{p a y}, T_{1}, T_{2}) = \frac{P (t, T_{2})}{P (t, T_{p a y})} E_{Q_{T_{2}}} [L (T_{1}, T_{1}, T_{2}) \frac{1}{P (T_{p a y}, T_{2})} \| F_{t}] = \\ = \frac{P (t, T_{2})}{P (t, T_{p a y})} E_{Q_{T_{2}}} [L (T_{1}, T_{1}, T_{2}) (1 + (T_{2} - T_{p a y}) L (T_{p a y}, T_{p a y}, T_{2})) \| F_{t}] = \\ = \frac{P (t, T_{2})}{P (t, T_{p a y})} (L (t, T_{1}, T_{2}) + (T_{2} - T_{p a y}) E_{Q_{T_{2}}} [L (T_{1}, T_{1}, T_{2}) L (T_{p a y}, T_{p a y}, T_{2}) \| F_{t}]) = \\ = \frac{P (t, T_{2})}{P (t, T_{p a y})} (L (t, T_{1}, T_{2}) + (T_{2} - T_{p a y}) E_{Q_{T_{2}}} [L (T_{1}, T_{1}, T_{2}) L (T_{1}, T_{p a y}, T_{2}) \| F_{t}]) . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfenced separators="\|"><mrow><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfenced separators="\|"><mrow><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>

70	Using the same technique as in Section 4.1, we can find the solution for this contract:	Using the same technique as in Section 4.1, we can find the solution for this contract: Using the same technique as in Section 4.1, we can find the solution for this contract:

71	$\begin{matrix} i L (t, T_{p a y}, T_{1}, T_{2}) = L (t, T_{1}, T_{2}) + L (t, T_{1}, T_{2}) (P (t, T_{2}) - P (t, T_{p a y})) / P (t, T_{p a y}) + \\ + \frac{P (t, T_{2})}{P (t, T_{p a y}) (T_{2} - T_{1})} (\frac{P (t, T_{1}) P (t, T_{p a y})}{P (t, T_{2}) P (t, T_{2})} e^{I} - \frac{P (t, T_{1})}{P (t, T_{2})} - \frac{P (t, T_{p a y})}{P (t, T_{2})} + 1), \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfenced separators="\|"><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfenced><mo>/</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfenced separators="\|"><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfenced><mo>/</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math>

72	where	where where

73	$\begin{matrix} I = \int_{t}^{T_{1}} ‍ ((ζ^{T_{p a y}} (t) - ζ^{T_{2}} (t)) (ζ^{T_{1}} (t) - ζ^{T_{2}} (t))) d t = \frac{σ^{2}}{2 a^{3}} (e^{- a (T_{p a y} - T_{1})} - e^{- a (T_{1} + T_{p a y} - 2 t)} - \\ - e^{- a (T_{2} - T_{1})} + e^{- a (T_{1} + T_{2} - 2 t)} - e^{- a (T_{2} + T_{p a y} - 2 T_{1})} + e^{- a (T_{2} + T_{p a y} - 2 t)} + e^{- 2 a (T_{2} - T_{1})} - e^{- 2 a (T_{2} - t)}) . \end{matrix}$ (3)	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>I</mi><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mfenced separators="\|"><mrow><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow></mfenced><mi>d</mi><mi>t</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mfenced close="" separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> (3) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>I</mi><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mfenced separators="\|"><mrow><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mo>(</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow></mfenced><mi>d</mi><mi>t</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mfenced close="" separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> (3)

74	4.3. $T_{2} < T_{p a y}$	4.3. <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">p</mi><mi mathvariant="bold-italic">a</mi><mi mathvariant="bold-italic">y</mi></mrow></msub></math> 4.3. <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">p</mi><mi mathvariant="bold-italic">a</mi><mi mathvariant="bold-italic">y</mi></mrow></msub></math>

75	Forward LIBOR rate has the following formula in this time payment case:	Forward LIBOR rate has the following formula in this time payment case: Forward LIBOR rate has the following formula in this time payment case:

76	$\begin{array}{l} i L (t, T_{p a y}, T_{1}, T_{2}) & = \frac{P (t, T_{2})}{P (t, T_{p a y}) (T_{2} - T_{1})} (\frac{P (t, T_{p a y}) P (t, T_{1})}{P (t, T_{2}) P (t, T_{2})} e^{I} - \frac{P (t, T_{p a y})}{P (t, T_{2})}), \end{array}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mtd><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mtd><mtd><mo>=</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math>

77	where $I$ is taken from (3), as both cases take place after $T_{1}$ .	where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>I</mi></math> is taken from (3), as both cases take place after <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> . where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>I</mi></math> is taken from (3), as both cases take place after <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> .

78	In the case when payment occurs after accrual period, we can prove that adjustment should be always nonpositive similarly to what we did in (Malykh, Postevoy, 2019) for payments before the end of accrual period.	In the case when payment occurs after accrual period, we can prove that adjustment should be always nonpositive similarly to what we did in (Malykh, Postevoy, 2019) for payments before the end of accrual period. In the case when payment occurs after accrual period, we can prove that adjustment should be always nonpositive similarly to what we did in (Malykh, Postevoy, 2019) for payments before the end of accrual period.

79	Theorem 2. Suppose that $P (L (T_{1}, T_{1}, T_{2}) \neq L (t, T_{1}, T_{2})) > 0$ under real-word measure. Then the forward rate $i L (t, T_{p a y}, T_{1}, T_{2})$ $<$ forward rate $L (t, T_{1}, T_{2})$ , $t < T_{1} \leq T_{2} < T_{p a y}$ .	<strong>Theorem 2.</strong> <em>Suppose that </em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="double-struck">P</mi><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>≠</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo><mo>></mo><mn>0</mn></math> <em>under real-word measure</em>. <em>Then the forward rate</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo><</mo></math> <em>forward</em> <em>rate</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> . <strong>Theorem 2.</strong> <em>Suppose that </em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="double-struck">P</mi><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>≠</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo><mo>></mo><mn>0</mn></math> <em>under real-word measure</em>. <em>Then the forward rate</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo><</mo></math> <em>forward</em> <em>rate</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> .

80	Proof. We can prove it by contradiction assuming opposite and constructing an arbitrage portfolio.	Proof<em>.</em> We can prove it by contradiction assuming opposite and constructing an arbitrage portfolio. Proof<em>.</em> We can prove it by contradiction assuming opposite and constructing an arbitrage portfolio.

81	Assume that there is a forward rate on the market and $i L (t, T_{p a y}, T_{1}, T_{2}) \geq L (t, T_{1}, T_{2})$ . Without loss of generality let $(T_{2} - T_{1}) = 1$ year. Without loss of generality let $(T_{2} - T_{1}) = 1$ year. Consider the following strategy:	Assume that there is a forward rate on the market and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>≥</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> . Without loss of generality let <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>1</mn></math> year. Without loss of generality let <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>1</mn></math> year. Consider the following strategy: Assume that there is a forward rate on the market and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>≥</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> . Without loss of generality let <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>1</mn></math> year. Without loss of generality let <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>1</mn></math> year. Consider the following strategy:

82	1) time t: buy FRA with $K = L (t, T_{1}, T_{2})$ , $N = 1$ and sell iFRA with payment date $T_{p a y}$ , $K = i L (t, T_{p a y}, T_{1}, T_{2})$ and $N = P (t, T_{1}) / P (t, T_{2})$ . Portfolio value $V_{t} = 0$ ;	1) time <em>t</em>: buy FRA with <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi><mo>=</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi><mo>=</mo><mn>1</mn></math> and sell iFRA with payment date <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi><mo>=</mo><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi><mo>=</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>/</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> . Portfolio value <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>V</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn>0</mn></math> ; 1) time <em>t</em>: buy FRA with <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi><mo>=</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi><mo>=</mo><mn>1</mn></math> and sell iFRA with payment date <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi><mo>=</mo><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi><mo>=</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>/</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> . Portfolio value <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>V</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn>0</mn></math> ;

83	2) $T_{1}$ : LIBOR rate is fixed and we enter into forward contract to buy $(L (T_{1}, T_{1}, T_{2}) - L (t, T_{1}, T_{2})) P (T_{1}, T_{2}) / P (T_{1}, T_{p a y})$ number of zero-coupon bonds (ZCB) with maturity $T_{p a y}$ at time $T_{2}$ . It costs us $F = L (T_{1}, T_{1}, T_{2}) - L (t, T_{1}, T_{2})$ ;	2) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> : LIBOR rate is fixed and we enter into forward contract to buy <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>/</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></math> number of zero-coupon bonds (ZCB) with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> at time <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> . It costs us <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>F</mi><mo>=</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> ; 2) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> : LIBOR rate is fixed and we enter into forward contract to buy <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>/</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></math> number of zero-coupon bonds (ZCB) with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> at time <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> . It costs us <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>F</mi><mo>=</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> ;

84	3) $T_{2}$ : FRA settlement occurs. Portfolio value is	3) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> : FRA settlement occurs. Portfolio value is 3) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> : FRA settlement occurs. Portfolio value is

85	$V_{T_{2}} = (L (T_{1}, T_{1}, T_{2}) - L (t, T_{1}, T_{2})) - F + (L (T_{1}, T_{1}, T_{2}) - L (t, T_{1}, T_{2}) P (T_{2}, T_{p a y}) P (T_{1}, T_{2}) / P (T_{1}, T_{p a y}));$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>=</mo><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mo>-</mo><mi>F</mi><mo>+</mo><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>/</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfenced><mo>;</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>=</mo><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mo>-</mo><mi>F</mi><mo>+</mo><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>/</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfenced><mo>;</mo></math>

86	4) $T_{p a y}$ : iFRA settlement occurs	4) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> : iFRA settlement occurs 4) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> : iFRA settlement occurs

87	$V_{T_{p a y}} = (L (T_{1}, T_{1}, T_{2}) - L (t, T_{1}, T_{2})) \frac{P (T_{1}, T_{2})}{P (T_{1}, T_{p a y})} + \frac{P (t, T_{1})}{P (t, T_{2})} (i L (t, T_{1}, T_{2}) - L (T_{1}, T_{1}, T_{2})) .$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub><mo>=</mo><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mo>.</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub><mo>=</mo><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mo>.</mo></math>

88	We use the fact that $(T_{2} - T_{1}) L (t, T_{1}, T_{2}) = P (t, T_{1}) / P (t, T_{2}) - 1$ and that $P (t, T_{1}) \geq P (t, T_{2})$ $\forall t \leq T_{1} \leq T_{2}$ . Now we can rewrite out portfolio value:	We use the fact that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>/</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mn>1</mn></math> and that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>≥</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>∀</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> . Now we can rewrite out portfolio value: We use the fact that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>/</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mn>1</mn></math> and that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>≥</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>∀</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> . Now we can rewrite out portfolio value:

89	$\begin{matrix} V_{T_{p a y}} \geq (L (T_{1}, T_{1}, T_{2}) - L (t, T_{1}, T_{2})) (\frac{P (T_{1}, T_{2})}{P (T_{1}, T_{p a y})} - \frac{P (t, T_{1})}{P (t, T_{2})}) = = (\frac{P (T_{1}, T_{1})}{P (T_{1}, T_{2})} - \frac{P (t, T_{1})}{P (t, T_{2})}) \times \\ \times (\frac{P (T_{1}, T_{2})}{P (T_{1}, T_{p a y})} - \frac{P (t, T_{1})}{P (t, T_{2})}) = \frac{(P (T_{1}, T_{1}) P (t, T_{2}) - P (T_{1}, T_{2}) P (t, T_{1})) (P (T_{1}, T_{2}) P (t, T_{2}) - P (T_{1}, T_{p a y}) P (t, T_{1}))}{P (T_{1}, T_{2}) P (t, T_{2}) P (T_{1}, T_{p a y}) P (t, T_{2})} \geq \\ \geq \frac{P (T_{1}, T_{2}) P (T_{1}, T_{p a y}) (P (t, T_{2}) - P (t, T_{1}))^{2}}{P (T_{1}, T_{2}) P (t, T_{2}) P (T_{1}, T_{p a y}) P (t, T_{2})} \geq 0 . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub><mo>≥</mo><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mo>=</mo><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>×</mo></mtd></mtr><mtr><mtd><mo>×</mo><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mfrac><mrow><mo>(</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>)</mo><mo>(</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>≥</mo></mtd></mtr><mtr><mtd><mo>≥</mo><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>(</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>≥</mo><mn>0</mn><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub><mo>≥</mo><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mo>=</mo><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>×</mo></mtd></mtr><mtr><mtd><mo>×</mo><mfenced separators="\|"><mrow><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mfrac><mrow><mo>(</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>)</mo><mo>(</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>)</mo></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>≥</mo></mtd></mtr><mtr><mtd><mo>≥</mo><mfrac><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>(</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>)</mo><mi>P</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>≥</mo><mn>0</mn><mo>.</mo></mtd></mtr></mtable></math>

90	It’s worth noting that $P (V_{T_{p a y}} > 0) > 0$ , because of our assumption, that $P (L (T_{1}, T_{1}, T_{2}) \neq L (t, T_{1}, T_{2})) > 0$ . We managed to construct an arbitrage portfolio which contradicts to our assumption of no-arbitrage. Hence, $i L (t, T_{p a y}, T_{1}, T_{2}) < L (t, T_{1}, T_{2})$ .	It’s worth noting that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="double-struck">P</mi><mo>(</mo><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub><mo>></mo><mn>0</mn><mo>)</mo><mo>></mo><mn>0</mn></math> , because of our assumption, that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="double-struck">P</mi><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>≠</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo><mo>></mo><mn>0</mn></math> . We managed to construct an arbitrage portfolio which contradicts to our assumption of no-arbitrage. Hence, <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo><</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> . It’s worth noting that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="double-struck">P</mi><mo>(</mo><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></mrow></msub><mo>></mo><mn>0</mn><mo>)</mo><mo>></mo><mn>0</mn></math> , because of our assumption, that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="double-struck">P</mi><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>≠</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo><mo>></mo><mn>0</mn></math> . We managed to construct an arbitrage portfolio which contradicts to our assumption of no-arbitrage. Hence, <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo><</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> .

91	5. In-arrears FRA behavior	5. In-arrears FRA behavior 5. In-arrears FRA behavior

92	Using results from Section 4.1–4.3 we proved the common formula (2). We can also find the limit of adjustments when $T_{1} \to \infty$ . We denote $τ_{1} = T_{2} - T_{1}$ , $τ_{2} = T_{1} - T_{p a y}$ , $τ_{3} = T_{2} - T_{p a y}$ . Then	Using results from Section 4.1–4.3 we proved the common formula (2). We can also find the limit of adjustments when <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>→</mo><mi mathvariant="normal">∞</mi></math> . We denote <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> . Then Using results from Section 4.1–4.3 we proved the common formula (2). We can also find the limit of adjustments when <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>→</mo><mi mathvariant="normal">∞</mi></math> . We denote <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>a</mi><mi>y</mi></mrow></msub></math> . Then

93	$\begin{array}{l} \underset{T_{1} \to \infty}{l i m} C A & = \frac{1}{τ_{1}} e x p \{(\frac{θ}{a} - \frac{σ^{2}}{2 a}) τ_{1}\} (e^{J} - 1), \end{array}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>C</mi><mi>A</mi></mtd><mtd><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mfenced separators="\|"><mrow><mfrac><mrow><mi>θ</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></mfenced><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mfenced separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>J</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>C</mi><mi>A</mi></mtd><mtd><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mfenced separators="\|"><mrow><mfrac><mrow><mi>θ</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></mfenced><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mfenced separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>J</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math>

94	Where	Where Where

95	$J = (σ^{2} / 2 a^{3}) (e^{- a \| τ_{2} \|} - e^{- a ((τ_{3}) 1_{τ_{2} > 0} + τ_{1} 1_{τ_{2} \leq 0})} + e^{- 2 a ((τ_{3}) 1_{τ_{2} > 0} + τ_{1} 1_{τ_{2} \leq 0})} - e^{- a (\| τ_{2} \| + τ_{3} + (- τ_{2})^{+})}) .$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>J</mi><mo>=</mo><mfenced separators="\|"><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfenced><mfenced close="" separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>\|</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>\|</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><mo>(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≤</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo></mrow></mfenced><mfenced open="" separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><mo>(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≤</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><mo>\|</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>\|</mo><mo>+</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mo>(</mo><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></msup></mrow></mfenced><mo>.</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>J</mi><mo>=</mo><mfenced separators="\|"><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfenced><mfenced close="" separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>\|</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>\|</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><mo>(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≤</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></msup><mo>+</mo></mrow></mfenced><mfenced open="" separators="\|"><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mo>(</mo><mo>(</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≤</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></msup><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mo>(</mo><mo>\|</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>\|</mo><mo>+</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mo>(</mo><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></msup></mrow></mfenced><mo>.</mo></math>

96	Using these properties convexity adjustment with different payment date properties is given in fig. 1.	Using these properties convexity adjustment with different payment date properties is given in fig. 1. Using these properties convexity adjustment with different payment date properties is given in fig. 1.

97	Fig. 1. Comparison of adjustments: CA (convexity adjustment) for forward LIBOR rate with t=0 ; θ=0.035 ; τ=0.5 ; r(t)=5% (bps — 1 basis point is equivalent to 0.01% (1/100th of a percent) or 0.0001 in decimal form)	<p><img src="http://ras.jes.su/images/publication_images/27217/image1.png" class="image-formula"/> <strong>Fig</strong><strong>.</strong><strong> 1</strong>. Comparison of adjustments: CA (convexity adjustment) for forward LIBOR rate with</p> <p>t=0 ;</p> <p>θ=0.035 ;</p> <p>τ=0.5 ;</p> <p>r(t)=5% (bps — 1 basis point is equivalent to 0.01% (1/100th of a percent) or 0.0001 in decimal form)</p> <p><img src="http://ras.jes.su/images/publication_images/27217/image1.png" class="image-formula"/> <strong>Fig</strong><strong>.</strong><strong> 1</strong>. Comparison of adjustments: CA (convexity adjustment) for forward LIBOR rate with</p> <p>t=0 ;</p> <p>θ=0.035 ;</p> <p>τ=0.5 ;</p> <p>r(t)=5% (bps — 1 basis point is equivalent to 0.01% (1/100th of a percent) or 0.0001 in decimal form)</p>

98	6. Quanto in-arrears FRA	<p>6. Quanto in-arrears FRA</p> <p>6. Quanto in-arrears FRA</p>

99	We consider another exotic modification of FRA — quanto FRA.	We consider another exotic modification of FRA — quanto FRA. We consider another exotic modification of FRA — quanto FRA.

100	Definition 7. Quanto FRA is a forward contract, where buyer of this contract at $t \leq T_{1}$ with maturity $T_{2}$ , fixed rate $K$ in d-currency (domestic) units and principal $N$ in f-currency (foreign) units, agrees on following obligations with counterparties at $T_{2}$ :	<strong>Definition </strong><strong>7</strong><strong>.</strong> Quanto FRA is a forward contract, where buyer of this contract at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> , fixed rate <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi></math> in d-currency (domestic) units and principal <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi></math> in f-currency (foreign) units, agrees on following obligations with counterparties at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> : <strong>Definition </strong><strong>7</strong><strong>.</strong> Quanto FRA is a forward contract, where buyer of this contract at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> , fixed rate <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi></math> in d-currency (domestic) units and principal <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi></math> in f-currency (foreign) units, agrees on following obligations with counterparties at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> :

101	byuer pay $(T_{2} - T_{1}) K N$ f-currency units; receive $(T_{2} - T_{1}) L (T_{1}, T_{1}, T_{2}) N$ f-currency units, where $L$ — LIBOR rate in d-currency units.	<ol> <li>byuer <em>pay</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi><mi>N</mi></math> f-currency units; </li> <li><em>receive</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>N</mi></math> f-currency units, where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi></math> — LIBOR rate in d-currency units. </li></ol> <ol> <li>byuer <em>pay</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi><mi>N</mi></math> f-currency units; </li> <li><em>receive</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>N</mi></math> f-currency units, where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi></math> — LIBOR rate in d-currency units. </li></ol>

102	Definition 8. Quanto in-arrears FRA (iqFRA) is a forward contract, where buyer of this contract at $t \leq T_{1}$ with maturity $T_{2}$ , fixed rate $K$ in d-currency units and principal $N$ in f-currency units, agrees on following obligations with counterparties at $T_{1}$ :	<strong>Definition 8.</strong> Quanto in-arrears FRA (iqFRA) is a forward contract, where buyer of this contract at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> , fixed rate <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi></math> in d-currency units and principal <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi></math> in f-currency units, agrees on following obligations with counterparties at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> : <strong>Definition 8.</strong> Quanto in-arrears FRA (iqFRA) is a forward contract, where buyer of this contract at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> , fixed rate <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi></math> in d-currency units and principal <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi></math> in f-currency units, agrees on following obligations with counterparties at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> :

103	byuer pay $(T_{2} - T_{1}) K N$ f-currency units, receive $(T_{2} - T_{1}) L (T_{1}, T_{1}, T_{2}) N$ f-currency units, where $L$ — LIBOR rate in d-currency units.	<ol> <li>byuer <em>pay</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi><mi>N</mi></math> f-currency units, </li> <li><em>receive</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>N</mi></math> f-currency units, where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi></math> — LIBOR rate in d-currency units. </li></ol> <ol> <li>byuer <em>pay</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi><mi>N</mi></math> f-currency units, </li> <li><em>receive</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>N</mi></math> f-currency units, where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi></math> — LIBOR rate in d-currency units. </li></ol>

104	Let $N$ = 1.	Let <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi></math> = 1. Let <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi></math> = 1.

105	By $i q L$ we denote forward rate of iqFRA contract. Notation $E_{Q_{T_{1}}^{f}}$ means mathematical expectation by forward measure $T_{1}$ of payments in f-currency. Then $i q L (t, T_{1}, T_{2}) = E_{Q_{T_{1}}^{f}} [L (T_{1}, T_{1}, T_{2}) \| F_{t}] .$	By <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>q</mi><mi>L</mi></math> we denote forward rate of iqFRA contract. Notation <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></msub></math> means mathematical expectation by forward measure <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> of payments in f-currency. Then <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></msub><mo>[</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>]</mo><mo>.</mo></math> By <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>q</mi><mi>L</mi></math> we denote forward rate of iqFRA contract. Notation <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></msub></math> means mathematical expectation by forward measure <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> of payments in f-currency. Then <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></msub><mo>[</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>]</mo><mo>.</mo></math>

106	We need to change measure to $Q_{T_{1}}^{d}$ for payments in d-currency. Radon–Nikodym derivative is	We need to change measure to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></math> for payments in d-currency. Radon–Nikodym derivative is We need to change measure to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></math> for payments in d-currency. Radon–Nikodym derivative is

107	$\frac{d Q_{T_{1}}^{d}}{d Q_{T_{1}}^{f}} = \frac{P_{d} (T_{1}, T_{1})}{P_{d} (t, T_{1})} \frac{P_{f} (t, T_{1}) X (t)}{P_{f} (T_{1}, T_{1}) X (T_{1})},$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><mi>d</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow><mrow><mi>d</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>X</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><mi>d</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow><mrow><mi>d</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>X</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>,</mo></math>

108	where $X (t)$ — spot exchange rate at time $t$ . Then	where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> — spot exchange rate at time <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi></math> . Then where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> — spot exchange rate at time <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi></math> . Then

109	$i q L (t, T_{1}, T_{2}) = E_{Q_{T_{1}}^{d}} [L (T_{1}, T_{1}, T_{2}) \frac{d Q_{T_{1}}^{f}}{d Q_{T_{1}}^{d}} \| F_{t}] = \frac{P_{d} (t, T_{1})}{P_{f} (t, T_{1}) X (t)} E_{Q_{T_{1}}^{d}} [L (T_{1}, T_{1}, T_{2}) \frac{P_{f} (T_{1}, T_{1})}{P_{d} (T_{1}, T_{1})} X (T_{1}) \| F_{t}] .$ (4)	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfrac><mrow><mi>d</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow><mrow><mi>d</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></mfrac><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mi>X</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>.</mo></math> (4) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfrac><mrow><mi>d</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow><mrow><mi>d</mi><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></mfrac><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mi>X</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>.</mo></math> (4)

110	We use the fact that the forward exchange rate with maturity $T$ is $X_{T} (t) = P_{f} (t, T) X (t) / P_{d} (t, T) .$ Then $\begin{array}{l} i q L (t, T_{1}, T_{2}) = {(X_{T_{1}} (t))}^{- 1} E_{Q_{T_{1}}^{d}} [X_{T_{1}} (T_{1}) L (T_{1}, T_{1}, T_{2}) \| F_{t}] . \end{array}$	We use the fact that the forward exchange rate with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>T</mi></math> is <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>X</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>.</mo></math> Then <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> We use the fact that the forward exchange rate with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>T</mi></math> is <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>X</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>.</mo></math> Then <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>

111	To calculate this expectation we need to:	To calculate this expectation we need to: To calculate this expectation we need to:

112	find SDE for process $X_{T_{1}} (t)$ in forward measure $Q_{T_{1}}^{d}$ ; find joint distribution of $X_{T_{1}} (T_{1}) L (T_{1}, T_{1}, T_{2})$ in forward measure $Q_{T_{1}}^{d}$ .	<ol> <li>find SDE for process <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> in forward measure <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></math> ; </li> <li>find joint distribution of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> in forward measure <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></math> . </li></ol> <ol> <li>find SDE for process <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> in forward measure <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></math> ; </li> <li>find joint distribution of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> in forward measure <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></math> . </li></ol>

113	First, write SDE of major processes:	First, write SDE of major processes: First, write SDE of major processes:

114	$\frac{d P_{d} (t, T)}{P_{d} (t, T)} = r_{d} (t) d t + σ_{P_{d}} B_{P_{d}} (t, T) d W_{d, P_{d}}^{Q} (t),$ $\frac{d P_{f} (t, T)}{P_{f} (t, T)} = r_{f} (t) d t + σ_{P_{f}} B_{P_{f}} (t, T) d W_{f, P_{f}}^{Q} (t),$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mfrac><mo>=</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mi>Q</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mfrac><mo>=</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>f</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>Q</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mfrac><mo>=</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mi>Q</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mfrac><mo>=</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>f</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>Q</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math>

115	$\frac{d X (t)}{X (t)} = (r_{d} (t) - r_{f} (t)) d t + σ_{X} d W_{X}^{Q} (t) .$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><mi>d</mi><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>X</mi></mrow><mrow><mi>Q</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>.</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><mi>d</mi><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>X</mi></mrow><mrow><mi>Q</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>.</mo></math>

116	$W_{d, P_{d}}^{Q}$ means wiener process for process $P_{d}$ in measure $Q$ in currency $d$ . To find SDE of $X_{T_{1}} (t)$ in risk-neutral measure $Q$ we need to write $P_{f} (t, T)$ in currency $d$ . Changing the measure we get $\frac{d P_{f} (t, T)}{P_{f} (t, T)} = (r_{f} (t) - ζ_{P_{f}}^{T} (t) σ_{X} ρ_{P_{f}, X}) + ζ_{P_{f}}^{T} (t) d W_{d, P_{f}}^{Q} (t),$ where $ζ_{P_{f}}^{T} (t) = σ_{P_{f}} B_{P_{f}} (t, T)$ and $ρ_{P_{f}, X}$ — correlation between $P_{f}$ and $X$ . Now write SDE of $X_{T_{1}} (t)$ :	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mi>Q</mi></mrow></msubsup></math> means wiener process for process <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></math> in measure <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>Q</mi></math> in currency <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi></math> . To find SDE of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> in risk-neutral measure <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>Q</mi></math> we need to write <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></math> in currency <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi></math> . Changing the measure we get <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>T</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>T</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>Q</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math> where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>T</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub></math> — correlation between <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>X</mi></math> . Now write SDE of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> : <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mi>Q</mi></mrow></msubsup></math> means wiener process for process <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></math> in measure <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>Q</mi></math> in currency <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi></math> . To find SDE of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> in risk-neutral measure <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>Q</mi></math> we need to write <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></math> in currency <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi></math> . Changing the measure we get <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>T</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>T</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>Q</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math> where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><mi>T</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>T</mi><mo>)</mo></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub></math> — correlation between <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>X</mi></math> . Now write SDE of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> :

117	$\begin{matrix} d (X_{T_{1}}) = \frac{\partial X_{T_{1}}}{\partial P_{d}} d P_{d} + \frac{\partial X_{T_{1}}}{\partial P_{f}} d P_{f} + \frac{\partial X_{T_{1}}}{\partial X} d X + 0.5 \frac{\partial^{2} X_{T_{1}}}{\partial {P_{d}}^{2}} (d P_{d})^{2} + \frac{\partial^{2} X_{T_{1}}}{\partial P_{d} \partial P_{f}} (d P_{d}) (d P_{f}) + \\ + \frac{\partial^{2} X_{T_{1}}}{\partial P_{d} \partial X} (d P_{d}) (d X) + \frac{\partial^{2} X_{T_{1}}}{\partial P_{f}^{2}} (d P_{f})^{2} . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>d</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfrac><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>+</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></mfrac><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>+</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><mi>X</mi></mrow></mfrac><mi>d</mi><mi>X</mi><mo>+</mo><mn>0.5</mn><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msup><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>(</mo><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>∂</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></mfrac><mo>(</mo><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>(</mo><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>∂</mo><mi>X</mi></mrow></mfrac><mo>(</mo><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>(</mo><mi>d</mi><mi>X</mi><mo>)</mo><mo>+</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo>(</mo><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>d</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfrac><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>+</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></mfrac><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>+</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><mi>X</mi></mrow></mfrac><mi>d</mi><mi>X</mi><mo>+</mo><mn>0.5</mn><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msup><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>(</mo><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>∂</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></mfrac><mo>(</mo><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>(</mo><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>∂</mo><mi>X</mi></mrow></mfrac><mo>(</mo><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>(</mo><mi>d</mi><mi>X</mi><mo>)</mo><mo>+</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow><mrow><mo>∂</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo>(</mo><mi>d</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>.</mo></mtd></mtr></mtable></math>

118	Switching to $Q^{T_{1}}$ -measure:	Switching to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math> -measure: Switching to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math> -measure:

119	$d (X_{T_{1}}) / X_{T_{1}} = - ζ_{P_{d}}^{T_{1}} (t) d W_{d, P_{d}}^{T_{1}} (t) + ζ_{P_{f}}^{T_{1}} (t) d W_{d, P_{f}}^{T_{1}} (t) + σ_{X} d W_{d, X}^{T_{1}} (t),$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>)</mo><mo>/</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>=</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>)</mo><mo>/</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>=</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math>

120	$\begin{matrix} d l n (X_{T_{1}}) = - ζ_{P_{d}}^{T_{1}} (t) d W_{d, P_{d}}^{T_{1}} (t) + ζ_{P_{f}}^{T_{1}} (t) d W_{d, P_{f}}^{T_{1}} (t) + σ_{X} d W_{d, X}^{T_{1}} (t) - \\ - 0.5 {(- ζ_{P_{d}}^{T_{1}} (t) d W_{d, P_{d}}^{T_{1}} (t) + ζ_{P_{f}}^{T_{1}} (t) d W_{d, P_{f}}^{T_{1}} (t) + σ_{X} d W_{d, X}^{T_{1}} (t))}^{2}, \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>d</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>)</mo><mo>=</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mn>0.5</mn><msup><mrow><mfenced separators="\|"><mrow><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>d</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>)</mo><mo>=</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mn>0.5</mn><msup><mrow><mfenced separators="\|"><mrow><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mtd></mtr></mtable></math>

121	$\begin{matrix} X_{T_{1}} (T_{1}) = X_{T_{1}} (t) e x p (\int_{t}^{T_{1}} ‍ (- ζ_{P_{d}}^{T_{1}} (t) d W_{d, P_{d}}^{T_{1}} (t) + ζ_{P_{f}}^{T_{1}} (t) d W_{d, P_{f}}^{T_{1}} (t) + σ_{X} d W_{d, X}^{T_{1}} (t)) - \\ - 0.5 \int_{t}^{T_{1}} ‍ {(- ζ_{P_{d}}^{T_{1}} (t) d W_{d, P_{d}}^{T_{1}} (t) + ζ_{P_{f}}^{T_{1}} (t) d W_{d, P_{f}}^{T_{1}} (t) + σ_{X} d W_{d, X}^{T_{1}} (t))}^{2}) . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced close="" separators="\|"><mrow><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mfenced separators="\|"><mrow><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mfenced open="" separators="\|"><mrow><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><msup><mrow><mfenced separators="\|"><mrow><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced close="" separators="\|"><mrow><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mfenced separators="\|"><mrow><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mfenced open="" separators="\|"><mrow><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><msup><mrow><mfenced separators="\|"><mrow><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>

122	Now we need to get $L (t, T_{1}, T_{2})$ in $Q^{T_{1}} - m e a s u r e$ Remember that $L (t, T_{1}, T_{2}) = (P_{d} (t, T_{1}) / P_{d} (t, T_{2}) - 1) / (T_{2} - T_{1})$ . Then, recall that:	Now we need to get <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> in <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>-</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi></math> Remember that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfenced separators="\|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>/</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mn>1</mn></mrow></mfenced><mo>/</mo><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced></math> . Then, recall that: Now we need to get <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> in <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>-</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi></math> Remember that <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfenced separators="\|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>/</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mn>1</mn></mrow></mfenced><mo>/</mo><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced></math> . Then, recall that:

123	$d (\frac{P_{d} (t, T_{1})}{P_{d} (t, T_{2})}) = \frac{P_{d} (t, T_{1})}{P_{d} (t, T_{2})} (ζ_{P_{d}}^{T_{1}} (t) - ζ_{P_{d}}^{T_{2}} (t)) (d W_{d, P_{d}}^{Q} (t) - ζ_{P_{d}}^{T_{2}} (t) d t) .$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mfenced separators="\|"><mrow><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mi>Q</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi></mrow></mfenced><mo>.</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mfenced separators="\|"><mrow><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mi>Q</mi></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi></mrow></mfenced><mo>.</mo></math>

124	Changing measure to $Q^{T_{1}}$ :	Changing measure to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math> : Changing measure to <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math> :

125	$d (\frac{P_{d} (t, T_{1})}{P_{d} (t, T_{2})}) = \frac{P_{d} (t, T_{1})}{P_{d} (t, T_{2})} ((ζ_{P_{d}}^{T_{1}} (t) - ζ_{P_{d}}^{T_{2}} (t)) d W_{d, P_{d}}^{T_{1}} (t) + {(ζ_{P_{d}}^{T_{1}} (t) - ζ_{P_{d}}^{T_{2}} (t))}^{2} d t),$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi></mrow></mfenced><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mfenced separators="\|"><mrow><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi></mrow></mfenced><mo>,</mo></math>

126	$d l n (\frac{P_{d} (t, T_{1})}{P_{d} (t, T_{2})}) = (ζ_{P_{d}}^{T_{1}} (t) - ζ_{P_{d}}^{T_{2}} (t)) d W_{d, P_{d}}^{T_{1}} (t) + 0.5 {(ζ_{P_{d}}^{T_{1}} (t) - ζ_{P_{d}}^{T_{2}} (t))}^{2} d t,$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mn>0.5</mn><msup><mrow><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>=</mo><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mn>0.5</mn><msup><mrow><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math>

127	$\frac{P_{d} (T_{1}, T_{1})}{P_{d} (T_{1}, T_{2})} = \frac{P_{d} (t, T_{1})}{P_{d} (t, T_{2})} e x p \{\frac{1}{2} \int_{t}^{T_{1}} ‍ {(ζ_{P_{d}}^{T_{1}} (t) - ζ_{P_{d}}^{T_{2}} (t))}^{2} d t + \int_{t}^{T_{1}} ‍ (ζ_{P_{d}}^{T_{1}} (t) - ζ_{P_{d}}^{T_{2}} (t)) d W_{d, P_{d}}^{T_{1}} (t)\} .$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><msup><mrow><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>+</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mo>.</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><msup><mrow><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>+</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mfenced separators="\|"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mo>.</mo></math>

128	So,	So, So,

129	$\begin{matrix} X_{T_{1}} (T_{1}) \frac{P_{d} (T_{1}, T_{1})}{P_{d} (T_{1}, T_{2})} = X_{T_{1}} (t) \frac{P_{d} (t, T_{1})}{P_{d} (t, T_{2})} e x p (0.5 \int_{t}^{T_{1}} ‍ (- 2 ζ_{P_{d}}^{T_{1}} (t) ζ_{P_{d}}^{T_{2}} (t) + ζ_{P_{d}}^{T_{2}} (t)^{2} + 2 ζ_{P_{d}}^{T_{1}} (t) ζ_{P_{f}}^{T_{1}} (t) ρ_{P_{d}, P_{f}} - \\ - ζ_{P_{f}}^{T_{1}} (t)^{2} + 2 ζ_{P_{d}}^{T_{1}} (t) σ_{X} ρ_{P_{d}, X} - 2 ζ_{P_{f}}^{T_{1}} (t) σ_{X} ρ_{P_{f}, X} - σ_{X}^{2}) d t + \int_{t}^{T_{1}} ‍ - ζ_{P_{d}}^{T_{2}} (t) d W_{d, P_{d}}^{T_{1}} (t) + \\ \int_{t}^{T_{1}} ‍ ζ_{P_{f}}^{T_{1}} (t) d W_{d, P_{f}}^{T_{1}} (t) + \int_{t}^{T_{1}} ‍ σ_{X} d W_{d, X}^{T_{1}} (t)) . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced close="" separators="\|"><mrow><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mfenced close="" separators="\|"><mrow><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfenced><mi>d</mi><mi>t</mi><mo>+</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced close="" separators="\|"><mrow><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mfenced close="" separators="\|"><mrow><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfenced><mi>d</mi><mi>t</mi><mo>+</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>

130	Using Cholesky decomposition, we decompose correlated wiener processes on independent ones:	Using Cholesky decomposition, we decompose correlated wiener processes on independent ones: Using Cholesky decomposition, we decompose correlated wiener processes on independent ones:

131	$d W_{d, P_{d}}^{T_{1}} (t) = a_{11} d B_{1} (t) + a_{12} d B_{2} (t) + a_{13} d B_{3} (t),$ $d W_{d, P_{f}}^{T_{1}} (t) = a_{21} d B_{1} (t) + a_{22} d B_{2} (t) + a_{23} d B_{3} (t),$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>11</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>12</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>13</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>21</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>22</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>23</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>11</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>12</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>13</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>21</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>22</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>23</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math>

132	$d W_{d, X}^{T_{1}} (t) = a_{31} d B_{1} (t) + a_{32} d B_{2} (t) + a_{33} d B_{3} (t),$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>31</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>32</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>33</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msubsup><mrow><mi>W</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>31</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>32</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>33</mn></mrow></msub><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo></math>

133	where $d B_{1} (t)$ , $d B_{2} (t)$ , and $d B_{3} (t)$ – uncorrelated Wiener processes in $Q^{T_{1}}$ -measure, $a_{i j}$ are the elements of the covariance matrix square root. So, expectation of lognormal random variable:	where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> , and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> – uncorrelated Wiener processes in <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math> -measure, <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> are the elements of the covariance matrix square root. So, expectation of lognormal random variable: where <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> , <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> , and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>d</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> – uncorrelated Wiener processes in <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math> -measure, <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> are the elements of the covariance matrix square root. So, expectation of lognormal random variable:

134	$E_{Q_{T_{1}}^{d}} [X_{T_{1}} (T_{1}) \frac{P_{d} (T_{1}, T_{1})}{P_{d} (T_{1}, T_{2})} \| F_{t}] = X_{T_{1}} (t) \frac{P_{d} (t, T_{1})}{P_{d} (t, T_{2})} e^{I},$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>,</mo></math>

135	where	where where

136	$\begin{matrix} I = 0.5 \int_{t}^{T_{1}} ‍ (- 2 ζ_{P_{d}}^{T_{1}} (t) ζ_{P_{d}}^{T_{2}} (t) + ζ_{P_{d}}^{T_{2}} (t)^{2} + 2 ζ_{P_{d}}^{T_{1}} (t) ζ_{P_{f}}^{T_{1}} (t) ρ_{P_{d}, P_{f}} - ζ_{P_{f}}^{T_{1}} (t)^{2} + 2 ζ_{P_{d}}^{T_{1}} (t) σ_{X} ρ_{P_{d}, X} - \\ - 2 ζ_{P_{f}}^{T_{1}} (t) σ_{X} ρ_{P_{f}, X} - σ_{X}^{2}) d t + 0.5 \int_{t}^{T_{1}} ‍ (- ζ_{P_{d}}^{T_{2}} (t) a_{11} + ζ_{P_{f}}^{T_{1}} (t) a_{21} + σ_{X} a_{31})^{2} d t + \\ + 0.5 \int_{t}^{T_{1}} ‍ (- ζ_{P_{d}}^{T_{2}} (t) a_{12} + ζ_{P_{f}}^{T_{1}} (t) a_{22} + σ_{X} a_{32})^{2} d t + 0.5 \int_{t}^{T_{1}} ‍ (- ζ_{P_{d}}^{T_{2}} (t) a_{13} + ζ_{P_{f}}^{T_{1}} (t) a_{23} + σ_{X} a_{33})^{2} d t . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>I</mi><mo>=</mo><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>21</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>31</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>32</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>+</mo><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>13</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>23</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>33</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi>I</mi><mo>=</mo><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo><mi>d</mi><mi>t</mi><mo>+</mo><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>21</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>31</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>32</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>+</mo><mn>0.5</mn><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>13</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>23</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>33</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>.</mo></mtd></mtr></mtable></math>

137	Calculating this expression, we get the equation for the in-arrears quanto FRA:	Calculating this expression, we get the equation for the in-arrears quanto FRA: Calculating this expression, we get the equation for the in-arrears quanto FRA:

138	$\begin{array}{l} i q L (t, T_{1}, T_{2}) = \frac{1}{T_{2} - T_{1}} (\frac{P_{d} (t, T_{1})}{P_{d} (t, T_{2})} e^{I} - \frac{1}{X_{T_{1}} (t)}), \end{array}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>I</mi></mrow></msup><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math>

139	where calculation of $I$ is given in Appendix	where calculation of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>I</mi></math> is given in Appendix where calculation of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>I</mi></math> is given in Appendix

140	Convexity adjustment for this exotic forward contract is $\begin{array}{l} i q C A (t, T_{1}, T_{2}) = i q L (t, T_{1}, T_{2}) - L (t, T_{1}, T_{2}) . \end{array}$ Fig. 2–3 show $i q C A (t, T_{1}, T_{2})$ with different parameters.	Convexity adjustment for this exotic forward contract is <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>q</mi><mi>C</mi><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>.</mo></mtd></mtr></mtable></math> Fig. 2–3 show <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>q</mi><mi>C</mi><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> with different parameters. Convexity adjustment for this exotic forward contract is <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>i</mi><mi>q</mi><mi>C</mi><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>.</mo></mtd></mtr></mtable></math> Fig. 2–3 show <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mi>q</mi><mi>C</mi><mi>A</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> with different parameters.

141	Fig. 2. Convexity adjustment for quanto in-arrears FRA with te following parameters: t=0 ; σd=σf=10% ; T2-T1=0.5 ; θf=θd=0.035 ; rd(t)=5% ; rf(t)=10% ; ρPd,Pf=ρPd,X=ρPf,X=0.3 ; X(t)=1	<p><img src="data:image/png;base64,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" alt="" /></p> <p><strong>Fig</strong><strong>.</strong><strong> </strong><strong>2</strong>. Convexity adjustment for quanto in-arrears FRA with te following parameters:</p> <p>t=0 ;</p> <p>σd=σf=10% ;</p> <p>T2-T1=0.5 ;</p> <p>θf=θd=0.035 ;</p> <p>rd(t)=5% ;</p> <p>rf(t)=10% ;</p> <p>ρPd,Pf=ρPd,X=ρPf,X=0.3 ;</p> <p>X(t)=1</p> <p><img src="data:image/png;base64,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" alt="" /></p> <p><strong>Fig</strong><strong>.</strong><strong> </strong><strong>2</strong>. Convexity adjustment for quanto in-arrears FRA with te following parameters:</p> <p>t=0 ;</p> <p>σd=σf=10% ;</p> <p>T2-T1=0.5 ;</p> <p>θf=θd=0.035 ;</p> <p>rd(t)=5% ;</p> <p>rf(t)=10% ;</p> <p>ρPd,Pf=ρPd,X=ρPf,X=0.3 ;</p> <p>X(t)=1</p>

142	Fig. 3. Convexity adjustment for quanto in-arrears FRA with the following parameters: t=0 ; σd=σf=10% ; θ=0.035; T2-T1=0.5 ; a=0.7 ; r(t)=5% ; ρPd,X=ρPf,X=0 ; ρPd,Pf=1 , both rates are identical	<p><img 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" alt="" /></p> <p><strong>Fig</strong><strong>.</strong><strong> 3</strong><strong>. </strong>Convexity adjustment for quanto in-arrears FRA with the following parameters:</p> <p>t=0 ;</p> <p>σd=σf=10% ;</p> <p>θ=0.035;</p> <p>T2-T1=0.5 ;</p> <p>a=0.7 ;</p> <p>r(t)=5% ;</p> <p>ρPd,X=ρPf,X=0 ;</p> <p>ρPd,Pf=1 , both rates are identical</p> <p><img src="data:image/png;base64,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" alt="" /></p> <p><strong>Fig</strong><strong>.</strong><strong> 3</strong><strong>. </strong>Convexity adjustment for quanto in-arrears FRA with the following parameters:</p> <p>t=0 ;</p> <p>σd=σf=10% ;</p> <p>θ=0.035;</p> <p>T2-T1=0.5 ;</p> <p>a=0.7 ;</p> <p>r(t)=5% ;</p> <p>ρPd,X=ρPf,X=0 ;</p> <p>ρPd,Pf=1 , both rates are identical</p>

143	In case when both rates are from the same currency market, adjustment term is similar to the in-arrears one, which is shown in the fig. 3.	In case when both rates are from the same currency market, adjustment term is similar to the in-arrears one, which is shown in the fig. 3. In case when both rates are from the same currency market, adjustment term is similar to the in-arrears one, which is shown in the fig. 3.

144	7. Quanto in-arrears options	<p>7. Quanto in-arrears options</p> <p>7. Quanto in-arrears options</p>

145	As a part of our study of quanto in-arrears contracts we also consider quanto in-arrears options on interest rate – caplet and floorlet.	As a part of our study of quanto in-arrears contracts we also consider quanto in-arrears options on interest rate – caplet and floorlet. As a part of our study of quanto in-arrears contracts we also consider quanto in-arrears options on interest rate – caplet and floorlet.

146	Definition 9. An in-arrears quanto caplet (floorlet) is a European-style call (put) option on interest rate which is fixed at $T_{1}$ . Buyer of this option at $t \leq T_{1}$ with maturity $T_{1}$ , strike $K$ and principal amount $N$ is offered with the following rights at time $T_{1}$ :	<strong>Definition 9.</strong> An in-arrears quanto caplet (floorlet) is a European-style call (put) option on interest rate which is fixed at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> . Buyer of this option at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> , strike <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi></math> and principal amount <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi></math> is offered with the following rights at time <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> : <strong>Definition 9.</strong> An in-arrears quanto caplet (floorlet) is a European-style call (put) option on interest rate which is fixed at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> . Buyer of this option at <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>t</mi><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> with maturity <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> , strike <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi></math> and principal amount <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>N</mi></math> is offered with the following rights at time <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math> :

147	pay (receive) $(T_{2} - T_{1}) K N$ f-currency units; receive (pay) $(T_{2} - T_{1}) L (T_{1}, T_{1}, T_{2}) N$ f-currency units, while $L$ and $K$ are set in d-currency units.	<ol> <li><em>pay (receive)</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi><mi>N</mi></math> f-currency units; </li> <li><em>receive (pay)</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>N</mi></math> f-currency units, while <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi></math> are set in d-currency units. </li></ol> <ol> <li><em>pay (receive)</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi><mi>N</mi></math> f-currency units; </li> <li><em>receive (pay)</em> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>N</mi></math> f-currency units, while <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>L</mi></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>K</mi></math> are set in d-currency units. </li></ol>

148	Formulas for option prices are given below:	Formulas for option prices are given below: Formulas for option prices are given below:

149	$q C p l (t, T_{1}, T_{2}, K) = (T_{2} - T_{1}) P_{f} (t, T_{1}) E_{Q_{T_{1}}^{f}} [{(L (T_{1}, T_{1}, T_{2}) - K)}^{+} \| F_{t}],$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal"> </mi><mi>q</mi><mi>C</mi><mi>p</mi><mi>l</mi><mi mathvariant="normal"> </mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>K</mi><mo>)</mo><mo>=</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal"> </mi><mi>q</mi><mi>C</mi><mi>p</mi><mi>l</mi><mi mathvariant="normal"> </mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>K</mi><mo>)</mo><mo>=</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>,</mo></math>

150	$q F l (t, T_{1}, T_{2}, K) = (T_{2} - T_{1}) P_{f} (t, T_{1}) E_{Q_{T_{1}}^{f}} [{(K - L (T_{1}, T_{1}, T_{2}))}^{+} \| F_{t}] .$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal"> </mi><mi>q</mi><mi>F</mi><mi>l</mi><mi mathvariant="normal"> </mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>K</mi><mo>)</mo><mo>=</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><mi>K</mi><mo>-</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>.</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal"> </mi><mi>q</mi><mi>F</mi><mi>l</mi><mi mathvariant="normal"> </mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>K</mi><mo>)</mo><mo>=</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><mi>K</mi><mo>-</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>.</mo></math>

151	First, we find price of qCpl. We switch to d-currency — as in Section 6:	First, we find price of qCpl. We switch to d-currency — as in Section 6: First, we find price of qCpl. We switch to d-currency — as in Section 6:

152	$\begin{matrix} E_{Q_{T_{1}}^{f}} [(L (T_{1}, T_{1}, T_{2}) - K)^{+} \| F_{t}] = E_{Q_{T_{1}}^{d}} [{(L (T_{1}, T_{1}, T_{2}) - K)}^{+} \frac{P_{f} (T_{1}, T_{1})}{P_{d} (T_{1}, T_{1})} X (T_{1}) \| F_{t}] \frac{P_{d} (t, T_{1})}{P_{f} (t, T_{1}) X (t)} = \\ = \frac{E_{Q_{T_{1}}^{d}}}{X_{T_{1}} (t)} [{(L (T_{1}, T_{1}, T_{2}) - K)}^{+} X_{T_{1}} (T_{1}) \| F_{t}] = \\ = \frac{E_{Q_{T_{1}}^{d}}}{X_{T_{1}} (t)} [\frac{P_{d} (T_{1}, T_{1})}{P_{d} (T_{1}, T_{2})} X_{T_{1}} (T_{1}) 1_{\frac{P_{d} (T_{1}, T_{1})}{P_{d} (T_{1}, T_{2})} > 1 + (T_{2} - T_{1}) K} \| F_{t}] - \frac{E_{Q_{T_{1}}^{d}}}{X_{T_{1}} (t)} [(1 + (T_{2} - T_{1}) K) X_{T_{1}} (T_{1}) 1_{\frac{P_{d} (T_{1}, T_{1})}{P_{d} (T_{1}, T_{2})} > 1 + (T_{2} - T_{1}) K} \| F_{t}] . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi><msup><mrow><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mi>X</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mfenced open="[" close="]" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mfenced open="[" close="" separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced><mfenced open="" close="]" separators="\|"><mrow><msub><mrow><mn>1</mn></mrow><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>></mo><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi></mrow></msub><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>-</mo><mfrac><mrow><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mfenced open="[" close="]" separators="\|"><mrow><mfenced separators="\|"><mrow><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi></mrow></mfenced><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mn>1</mn></mrow><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>></mo><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi></mrow></msub><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>f</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><mo>(</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi><msup><mrow><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub><mfenced open="[" close="]" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfrac><mi>X</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>X</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mfenced open="[" close="]" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><mi>L</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi></mrow></mfenced></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mfenced open="[" close="" separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced><mfenced open="" close="]" separators="\|"><mrow><msub><mrow><mn>1</mn></mrow><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>></mo><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi></mrow></msub><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>-</mo><mfrac><mrow><msub><mrow><mi>E</mi></mrow><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>d</mi></mrow></msubsup></mrow></msub></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mfenced open="[" close="]" separators="\|"><mrow><mfenced separators="\|"><mrow><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi></mrow></mfenced><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mn>1</mn></mrow><mrow><mfrac><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>></mo><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi></mrow></msub><mo>\|</mo><msub><mrow><mi mathvariant="script">F</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>

153	Calculating both mathematical expectations, we come to the analytical formula of the quanto in-arrears option price:	Calculating both mathematical expectations, we come to the analytical formula of the quanto in-arrears option price: Calculating both mathematical expectations, we come to the analytical formula of the quanto in-arrears option price:

154	$\begin{matrix} q C p l (t, T_{1}, T_{2}, K) = P_{f} (t, T_{1}) (\{P_{d} (t, T_{1}) / P_{d} (t, T_{2})\} e x p \{0.5 J_{0}\} e x p \{0.5 (J_{1} + J_{2} + J_{3})\} \times \\ \times N (\sqrt{J_{1}} - l) N (\sqrt{J_{2}} - l) N (\sqrt{J_{3}} - l) - (1 + (T_{2} - T_{1}) K) e x p \{- 0.5 Q_{0}\} \times \\ \times e x p \{0.5 (Q_{1} + Q_{2} + Q_{3})\} N (\sqrt{Q_{1}} - l) N (\sqrt{Q_{2}} - l) N (\sqrt{Q_{3}} - l)), \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi mathvariant="normal"> </mi><mi>q</mi><mi>C</mi><mi>p</mi><mi>l</mi><mi mathvariant="normal"> </mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>K</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mfenced close="" separators="\|"><mrow><mfenced open="{" close="}" separators="\|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>/</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mn>0.5</mn><msub><mrow><mi>J</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mn>0.5</mn><mo>(</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></mrow></mfenced><mo>×</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mo>×</mo><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>J</mi></mrow><mrow><mn>3</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo><mo>-</mo><mfenced separators="\|"><mrow><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi></mrow></mfenced><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>0.5</mn><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo>×</mo></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>×</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mn>0.5</mn><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></mrow></mfenced><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>Q</mi></mrow><mrow><mn>3</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mi mathvariant="normal"> </mi><mi>q</mi><mi>C</mi><mi>p</mi><mi>l</mi><mi mathvariant="normal"> </mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>K</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mfenced close="" separators="\|"><mrow><mfenced open="{" close="}" separators="\|"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>/</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mfenced><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mn>0.5</mn><msub><mrow><mi>J</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mn>0.5</mn><mo>(</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></mrow></mfenced><mo>×</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mo>×</mo><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>J</mi></mrow><mrow><mn>3</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo><mo>-</mo><mfenced separators="\|"><mrow><mn>1</mn><mo>+</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>K</mi></mrow></mfenced><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>0.5</mn><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mo>×</mo></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>×</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mn>0.5</mn><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></mrow></mfenced><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo><mi>N</mi><mo>(</mo><msqrt><msub><mrow><mi>Q</mi></mrow><mrow><mn>3</mn></mrow></msub></msqrt><mo>-</mo><mi>l</mi><mo>)</mo></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math>

155	Where	Where Where

156	$J_{0} = \int_{t}^{T_{1}} ‍ (- 2 ζ_{P_{d}}^{T_{1}} (t) ζ_{P_{d}}^{T_{2}} (t) + ζ_{P_{d}}^{T_{2}} (t)^{2} + 2 ζ_{P_{d}}^{T_{1}} (t) ζ_{P_{f}}^{T_{1}} (t) ρ_{P_{d}, P_{f}} - ζ_{P_{f}}^{T_{1}} (t)^{2} + 2 ζ_{P_{d}}^{T_{1}} (t) σ_{X} ρ_{P_{d}, X} - 2 ζ_{P_{f}}^{T_{1}} (t) σ_{x} ρ_{P_{f}, X} - σ_{X}^{2}) d t,$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>J</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>J</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo><mi>d</mi><mi>t</mi><mo>,</mo></math>

157	$J_{1} = \int_{t}^{T_{1}} ‍ (- ζ_{P_{d}}^{T_{2}} (t) a_{11} + ζ_{P_{f}}^{T_{1}} (t) a_{21} + σ_{X} a_{31})^{2} d t,$ $J_{2} = \int_{t}^{T_{1}} ‍ (- ζ_{P_{d}}^{T_{2}} (t) a_{12} + ζ_{P_{f}}^{T_{1}} (t) a_{22} + σ_{X} a_{32})^{2} d t,$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>21</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>31</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>32</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>21</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>31</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>32</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math>

158	$J_{3} = \int_{t}^{T_{1}} ‍ (- ζ_{P_{d}}^{T_{2}} (t) a_{13} + ζ_{P_{f}}^{T_{1}} (t) a_{23} + σ_{X} a_{33})^{2} d t,$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>J</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>13</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>23</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>33</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>J</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>13</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>23</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>33</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math>

159	$Q_{0} = \int_{t}^{T_{1}} ‍ (ζ_{P_{d}}^{T_{1}} (t)^{2} - 2 ζ_{P_{d}}^{T_{1}} (t) ζ_{P_{f}}^{T_{1}} (t) ρ_{P_{d}, P_{f}} - 2 ζ_{P_{d}}^{T_{1}} (t) σ_{X} ρ_{P_{d}, X} + ζ_{P_{f}}^{T_{1}} (t)^{2} + 2 ζ_{P_{f}}^{T_{1}} (t) σ_{X} ρ_{P_{f}, X} + σ_{X}^{2}) d t,$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>-</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo><mi>d</mi><mi>t</mi><mo>,</mo></math>

160	$Q_{1} = \int_{t}^{T_{1}} ‍ (- ζ_{P_{d}}^{T_{1}} (t) a_{11} + ζ_{P_{f}}^{T_{1}} (t) a_{21} + σ_{X} a_{31})^{2} d t,$ $Q_{2} = \int_{t}^{T_{1}} ‍ (- ζ_{P_{d}}^{T_{1}} (t) a_{12} + ζ_{P_{f}}^{T_{1}} (t) a_{22} + σ_{X} a_{32})^{2} d t,$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>21</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>31</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>32</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>21</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>31</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>32</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math>

161	$Q_{3} = \int_{t}^{T_{1}} ‍ (- ζ_{P_{d}}^{T_{1}} (t) a_{13} + ζ_{P_{f}}^{T_{1}} (t) a_{23} + σ_{X} a_{33})^{2} d t,$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>13</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>23</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>33</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>Q</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mrow><msubsup><mo stretchy="false">∫</mo><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi mathvariant="normal">‍</mi></mrow></mrow><mo>(</mo><mo>-</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>13</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>23</mn></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>33</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo></math>

162	calculations of which are given in Appendix.	calculations of which are given in Appendix. calculations of which are given in Appendix.

163	We will find the floorlet price using put-call parity of European options:	We will find the floorlet price using put-call parity of European options: We will find the floorlet price using put-call parity of European options:

164	$q F l (t, T_{1}, T_{2}, K) = q C p l (t, T_{1}, T_{2}, K) - (T_{2} - T_{1}) P_{f} (t, T_{1}) (i q L (t, T_{1}, T_{2}) - K) .$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal"> </mi><mi>q</mi><mi>F</mi><mi>l</mi><mi mathvariant="normal"> </mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>K</mi><mo>)</mo><mo>=</mo><mi mathvariant="normal"> </mi><mi>q</mi><mi>C</mi><mi>p</mi><mi>l</mi><mi mathvariant="normal"> </mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>K</mi><mo>)</mo><mo>-</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>(</mo><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi><mo>)</mo><mo>.</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal"> </mi><mi>q</mi><mi>F</mi><mi>l</mi><mi mathvariant="normal"> </mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>K</mi><mo>)</mo><mo>=</mo><mi mathvariant="normal"> </mi><mi>q</mi><mi>C</mi><mi>p</mi><mi>l</mi><mi mathvariant="normal"> </mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>K</mi><mo>)</mo><mo>-</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>(</mo><mi>i</mi><mi>q</mi><mi>L</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>-</mo><mi>K</mi><mo>)</mo><mo>.</mo></math>

165	Fig. 4–5 show differences in quanto in arrears and standard caplet and floorlet prices with different parameters.	Fig. 4–5 show differences in quanto in arrears and standard caplet and floorlet prices with different parameters. Fig. 4–5 show differences in quanto in arrears and standard caplet and floorlet prices with different parameters.

166	Fig. 4. Quanto in-arrears caplet price vs. Standard caplet price with the following parameters: t=0 ; σd=σf=10% ; Ti+1-Ti=0.5 ; θf=θd=0.035 ; rd(t)=5% ; rf(t)=10% ; ρPd,Pf=ρPd,X=ρPf,X=0.3 ; X(t)=1	<p><img 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" alt="" /></p> <p><strong>Fig</strong><strong>.</strong><strong> 4</strong><strong>. </strong>Quanto in-arrears caplet price vs. Standard caplet price with the following parameters:</p> <p>t=0 ;</p> <p>σd=σf=10% ;</p> <p>Ti+1-Ti=0.5 ;</p> <p>θf=θd=0.035 ;</p> <p>rd(t)=5% ;</p> <p>rf(t)=10% ;</p> <p>ρPd,Pf=ρPd,X=ρPf,X=0.3 ;</p> <p>X(t)=1</p> <p><img src="data:image/png;base64,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" alt="" /></p> <p><strong>Fig</strong><strong>.</strong><strong> 4</strong><strong>. </strong>Quanto in-arrears caplet price vs. Standard caplet price with the following parameters:</p> <p>t=0 ;</p> <p>σd=σf=10% ;</p> <p>Ti+1-Ti=0.5 ;</p> <p>θf=θd=0.035 ;</p> <p>rd(t)=5% ;</p> <p>rf(t)=10% ;</p> <p>ρPd,Pf=ρPd,X=ρPf,X=0.3 ;</p> <p>X(t)=1</p>

167	Fig. 5. Quanto in-arrears floorlet price vs. Standard floorlet price with the following parameters: t=0 ; σd=σf=10% ; Ti+1-Ti=0.5 ; θf=θd=0.035 ; rd(t)=5% ; rf(t)=10% ; ρPd,Pf=ρPd,X=ρPf,X=0.3 ; X(t)=1	<p><img 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" alt="" /></p> <p><strong>Fig</strong><strong>.</strong><strong> 5</strong><strong>. </strong>Quanto in-arrears floorlet price vs. Standard floorlet price with the following parameters:</p> <p>t=0 ;</p> <p>σd=σf=10% ;</p> <p>Ti+1-Ti=0.5 ;</p> <p>θf=θd=0.035 ;</p> <p>rd(t)=5% ;</p> <p>rf(t)=10% ;</p> <p>ρPd,Pf=ρPd,X=ρPf,X=0.3 ;</p> <p>X(t)=1</p> <p><img src="data:image/png;base64,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" alt="" /></p> <p><strong>Fig</strong><strong>.</strong><strong> 5</strong><strong>. </strong>Quanto in-arrears floorlet price vs. Standard floorlet price with the following parameters:</p> <p>t=0 ;</p> <p>σd=σf=10% ;</p> <p>Ti+1-Ti=0.5 ;</p> <p>θf=θd=0.035 ;</p> <p>rd(t)=5% ;</p> <p>rf(t)=10% ;</p> <p>ρPd,Pf=ρPd,X=ρPf,X=0.3 ;</p> <p>X(t)=1</p>

168	8. Conclusion	8. Conclusion 8. Conclusion

169	We derived the formula for calculating the forward LIBOR rate in FRA when payment is settled at different dates. It was proved that the convexity adjustment to the vanilla forward rate should be negative when payment takes place after forward period. Next, we studied quanto in-arrears FRA and checked, that it equals in-arrears FRA in case when rates and principal are from the same currency market, which is shown in the fig. 3. Finally, we briefly studied quanto in-arrears option contracts and found that their prices are greater than those of vanilla options.	We derived the formula for calculating the forward LIBOR rate in FRA when payment is settled at different dates. It was proved that the convexity adjustment to the vanilla forward rate should be negative when payment takes place after forward period. Next, we studied quanto in-arrears FRA and checked, that it equals in-arrears FRA in case when rates and principal are from the same currency market, which is shown in the fig. 3. Finally, we briefly studied quanto in-arrears option contracts and found that their prices are greater than those of vanilla options. We derived the formula for calculating the forward LIBOR rate in FRA when payment is settled at different dates. It was proved that the convexity adjustment to the vanilla forward rate should be negative when payment takes place after forward period. Next, we studied quanto in-arrears FRA and checked, that it equals in-arrears FRA in case when rates and principal are from the same currency market, which is shown in the fig. 3. Finally, we briefly studied quanto in-arrears option contracts and found that their prices are greater than those of vanilla options.

170	Appendix	<em><strong>Appendix</strong></em> <em><strong>Appendix</strong></em>

171	Here is the calculation of the integral from the Section 6: $\begin{array}{l} I = I_{0} + . . . + I_{3}, \end{array}$ where calculations of $I_{i},$ $i = 0, . . ., 3$ , are given below:	Here is the calculation of the integral from the Section 6: <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>I</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo></mtd></mtr></mtable></math> where calculations of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mn>3</mn></math> , are given below: Here is the calculation of the integral from the Section 6: <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable columnalign="left"><mtr><mtd><mi>I</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo></mtd></mtr></mtable></math> where calculations of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mn>3</mn></math> , are given below:

172	$\begin{matrix} I_{0} = 0.5 ({(σ_{P_{d}} / a_{P_{d}})}^{2} (T_{1} - t - \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{2} - t)\}}{a_{P_{d}}} - \frac{1}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{1} - t)\}}{a_{P_{d}}} + \\ + \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{2 a_{P_{d}}} - \frac{e x p \{- a_{P_{d}} (T_{2} + T_{1} - 2 t)\}}{2 a_{P_{d}}}) + {(σ_{P_{d}} / a_{P_{d}})}^{2} (T_{1} - t - \frac{2 e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}}} + \\ + \frac{2 e x p \{- a_{P_{d}} (T_{2} - t)\}}{a_{P_{d}}} + \frac{e x p \{- 2 a_{P_{d}} (T_{2} - T_{1})\}}{2 a_{P_{d}}} - \frac{e x p \{- 2 a_{P_{d}} (T_{2} - t)\}}{2 a_{P_{d}}}) + \end{matrix} \begin{matrix} + 2 ρ_{P_{d}, P_{f}} \frac{σ_{P_{d}} σ_{P_{f}}}{a_{P_{d}} a_{P_{f}}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}} + \frac{1}{a_{P_{d}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{d}}} + \frac{1}{a_{P_{d}} + a_{P_{f}}} - \\ - \frac{e x p \{- (a_{P_{d}} + a_{P_{f}}) (T_{1} - t)\}}{a_{P_{d}} + a_{P_{f}}}) - {(σ_{P_{f}} / a_{P_{f}})}^{2} (T_{1} - t - \frac{2}{a_{P_{f}}} + \frac{2 e x p \{- a_{P_{f}} (T_{2} - t)\}}{a_{P_{f}}} + \frac{1}{2 a_{P_{f}}} - \\ - \frac{e x p \{- 2 a_{P_{f}} (T_{1} - t)\}}{2 a_{P_{f}}}) + 2 σ_{X} ρ_{P_{d}, X} \frac{σ_{P_{d}}}{a_{P_{d}}} (T_{1} - t - \frac{1}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{1} - t)\}}{a_{P_{d}}}) - \\ - 2 σ_{X} ρ_{P_{f}, X} \frac{σ_{P_{f}}}{a_{P_{f}}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}}) - σ_{X}^{2} (T_{1} - t), \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0.5</mn><mfenced close="" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr></mtable><mtable><mtr><mtd><mo>+</mo><mn>2</mn><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mfenced separators="\|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0.5</mn><mfenced close="" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr></mtable><mtable><mtr><mtd><mo>+</mo><mn>2</mn><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mfenced separators="\|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math>

173	$\begin{matrix} I_{i} = 0.5 ({(σ_{P_{d}} a_{1 i} / a_{P_{d}})}^{2} (T_{1} - t - \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}}} + \frac{2 e x p \{- a_{P_{d}} (T_{2} - t)\}}{a_{P_{d}}} + \frac{e x p \{- 2 a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}}} + \\ - \frac{e x p \{- 2 a_{P_{d}} (T_{2} - t)\}}{2 a_{P_{d}}}) - 2 a_{1 i} a_{2 i} \frac{σ_{P_{d}} σ_{P_{f}}}{a_{P_{d}} a_{P_{f}}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}} - \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{2 a_{P_{d}}} + \\ + \frac{e x p \{- a_{P_{d}} (T_{2} - t)\}}{2 a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}} + a_{P_{f}}} - \frac{e x p \{- 2 a_{P_{d}} (T_{2} - t) - a_{P_{f}} (T_{1} - t)\}}{a_{P_{d}} + a_{P_{f}}}) - 2 a_{1 i} a_{2 i} σ_{X} \frac{σ_{P_{d}}}{a_{P_{d}}} \times \\ \times (T_{1} - t - \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{d} - t)\}}{a_{P_{d}}}) + {(\frac{σ_{P_{d}}}{a_{P_{d}}} a_{2 i})}^{2} (T_{1} - t - \frac{2}{a_{P_{f}}} + \frac{2 e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}} + \\ + \frac{1}{2 a_{P_{f}}} - \frac{e x p \{- 2 a_{P_{f}} (T_{1} - t)\}}{2 a_{P_{f}}}) + 2 a_{2 i} a_{3 i} σ_{X} \frac{σ_{P_{f}}}{a_{P_{f}}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}}) + σ_{X}^{2} a_{3 i}^{2} (T_{1} - t), \end{matrix} i = 1,2, 3 .$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>0.5</mn><mfenced close="" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mrow></mfenced><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>×</mo></mtd></mtr><mtr><mtd><mo>×</mo><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mfenced open="" separators="\|"><mrow><mo>-</mo><mi>t</mi></mrow></mfenced><mo>,</mo></mtd></mtr></mtable><mi>i</mi><mo>=</mo><mn>1,2</mn><mo>,</mo><mn>3</mn><mo>.</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>0.5</mn><mfenced close="" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mrow></mfenced><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>×</mo></mtd></mtr><mtr><mtd><mo>×</mo><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mfenced open="" separators="\|"><mrow><mo>-</mo><mi>t</mi></mrow></mfenced><mo>,</mo></mtd></mtr></mtable><mi>i</mi><mo>=</mo><mn>1,2</mn><mo>,</mo><mn>3</mn><mo>.</mo></math>

174	Below are the calculations of the integrals from the Section 7:	Below are the calculations of the integrals from the Section 7: Below are the calculations of the integrals from the Section 7:

175	$\begin{matrix} J_{0} = - 2 ({(σ_{P_{d}} / a_{P_{d}})}^{2} (T_{1} - t - \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{2} - t)\}}{a_{P_{d}}} - \frac{1}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{1} - t)\}}{a_{P_{d}}} + \\ + \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{2 a_{P_{d}}} - \frac{e x p \{- a_{P_{d}} (T_{2} + T_{1} - 2 t)\}}{2 a_{P_{d}}}) + {(σ_{P_{d}} / a_{P_{d}})}^{2} (T_{1} - t - \frac{2 e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}}} + \\ + \frac{2 e x p \{- a_{P_{d}} (T_{2} - t)\}}{a_{P_{d}}} + \frac{e x p \{- 2 a_{P_{d}} (T_{2} - T_{1})\}}{2 a_{P_{d}}} - \frac{e x p \{- 2 a_{P_{d}} (T_{2} - t)\}}{2 a_{P_{d}}}) + \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>J</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mo>-</mo><mn>2</mn><mfenced close="" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>J</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mo>-</mo><mn>2</mn><mfenced close="" separators="\|"><mrow><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr></mtable></math>

176	$\begin{matrix} + \frac{2 σ_{P_{d}} σ_{P_{f}}}{a_{P_{d}} a_{P_{f}}} ρ_{P_{d}, P_{f}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}} + \frac{1}{a_{P_{d}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{d}}} + \frac{1}{a_{P_{d}} + a_{P_{f}}} - \\ - \frac{e x p \{- (a_{P_{d}} + a_{P_{f}}) (T_{1} - t)\}}{a_{P_{d}} + a_{P_{f}}}) - {(σ_{P_{f}} / a_{P_{f}})}^{2} (T_{1} - t - \frac{2}{a_{P_{f}}} + \frac{2 e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}} + \frac{1}{2 a_{P_{f}}} - \\ - \frac{e x p \{- 2 a_{P_{f}} (T_{1} - t)\}}{2 a_{P_{f}}}) - 2 σ_{X} ρ_{P_{d}, X} \frac{σ_{P_{f}}}{a_{P_{f}}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}}) - σ_{X}^{2} (T_{1} - t), \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mo>+</mo><mfrac><mrow><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mfenced separators="\|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><mo>+</mo><mfrac><mrow><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mfenced separators="\|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfenced><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math>

177	$\begin{matrix} J_{i} = {(\frac{σ_{P_{d}}}{a_{P_{d}}} a_{1 i})}^{2} (T_{1} - t - \frac{2 e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}}} + \frac{2 e x p \{- a_{P_{d}} (T_{2} - t)\}}{a_{P_{d}}} + \frac{e x p \{- 2 a_{P_{d}} (T_{2} - T_{1})\}}{2 a_{P_{d}}} - \\ + \frac{e x p {\{- 2 a_{P_{d}} (T_{2} - t)\}}^{}}{2 a_{P_{d}}}) - \frac{2 a_{1 i} a_{2 i}}{a_{P_{d}} a_{P_{f}}} σ_{P_{d}} σ_{P_{f}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}} - \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}}} + \\ + \frac{e x p \{- a_{P_{d}} (T_{2} - t)\}}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}} + a_{P_{f}}} - \frac{e x p {\{- a_{P_{d}} (T_{2} - t) - a_{P_{f}} (T_{1} - t)\}}^{}}{a_{P_{d}} + a_{P_{f}}}) - \\ - 2 a_{1 i} a_{3 i} σ_{X} \frac{σ_{P_{d}}}{a_{P_{d}}} (T_{1} - t - + \frac{e x p \{- a_{P_{d}} (T_{2} - T_{1})\}}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{2} - t)\}}{a_{P_{d}}}) + \\ + {(a_{2 i} \frac{σ_{P_{f}}}{a_{P_{f}}})}^{2} (T_{1} - t - \frac{2}{a_{P_{f}}} + \frac{2 e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}} + \frac{1}{2 a_{P_{f}}} - \frac{e x p \{- 2 a_{P_{f}} (T_{1} - t)\}}{2 a_{P_{f}}}) + \\ + 2 a_{2 i} a_{3 i} σ_{X} \frac{σ_{P_{f}}}{a_{P_{f}}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p {\{- a_{P_{f}} (T_{1} - t)\}}^{}}{a_{P_{f}}}) + (σ_{X} a_{3 i})^{2} (T_{1} - t), i = 1,2, 3 . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>J</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow/></msup></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mfrac><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow/></msup></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow/></msup></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mo>(</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>i</mi><mo>=</mo><mn>1,2</mn><mo>,</mo><mn>3</mn><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>J</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow/></msup></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mfrac><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow/></msup></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow/></msup></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mo>(</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>i</mi><mo>=</mo><mn>1,2</mn><mo>,</mo><mn>3</mn><mo>.</mo></mtd></mtr></mtable></math>

178	$\begin{matrix} Q_{0} = {(\frac{σ_{P_{d}}}{a_{P_{d}}})}^{2} (T_{1} - t - \frac{2}{a_{P_{d}}} + \frac{2 e x p {\{- a_{P_{d}} (T_{1} - t)\}}^{}}{a_{P_{d}}} + \frac{1}{2 a_{P_{d}}} - \frac{e x p \{- 2 a_{P_{d}} (T_{1} - t)\}}{2 a_{P_{d}}}) - \\ - 2 ρ_{P_{d}, P_{f}} \frac{σ_{P_{d}} σ_{P_{f}}}{a_{P_{d}} a_{P_{f}}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}} - \frac{1}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{1} - t)\}}{a_{P_{d}}} + \frac{1}{a_{P_{d}} + a_{P_{f}}} - \\ - \frac{e x p \{- (a_{P_{d}} + a_{P_{f}}) (T_{1} - t)\}}{a_{P_{d}} + a_{P_{f}}}) - 2 σ_{X} ρ_{P_{d}, X} \frac{σ_{P_{d}}}{a_{P_{d}}} (T_{1} - t - \frac{1}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{1} - t)\}}{a_{P_{d}}}) + \\ + {(\frac{σ_{P_{f}}}{a_{P_{f}}})}^{2} (T_{1} - t - \frac{2}{a_{P_{f}}} + \frac{2 e x p {\{- a_{P_{f}} (T_{1} - t)\}}^{}}{a_{P_{f}}} + \frac{1}{2 a_{P_{f}}} - \frac{e x p \{- 2 a_{P_{f}} (T_{1} - t)\}}{2 a_{P_{f}}}) + \\ + 2 σ_{X} ρ_{P_{f}, X} \frac{σ_{P_{f}}}{a_{P_{f}}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}}) + σ_{X}^{2} (T_{1} - t) . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow/></msup></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow/></msup></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mo>+</mo><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow/></msup></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow/></msup></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mo>+</mo><mn>2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mi>X</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo><mo>.</mo></mtd></mtr></mtable></math>

179	$\begin{matrix} Q_{i} = {(\frac{a_{1 i} σ_{P_{d}}}{a_{P_{d}}})}^{2} (T_{1} - t - \frac{2}{a_{P_{d}}} + \frac{2 e x p \{- a_{P_{d}} (T_{1} - t)\}}{a_{P_{d}}} + \frac{1}{2 a_{P_{d}}} - \frac{e x p \{- 2 a_{P_{d}} (T_{1} - t)\}}{2 a_{P_{d}}}) - \frac{2 a_{1 i} a_{2 i} σ_{P_{d}} σ_{P_{f}}}{a_{P_{d}} a_{P_{f}}} \times \\ \times (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P_{f}}} - \frac{1}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{1} - t)\}}{a_{P_{d}}} + \frac{1}{a_{P_{d}} + a_{P_{f}}} - \frac{e x p \{- (a_{P_{d}} + a_{P_{f}}) (T_{1} - t)\}}{a_{P_{d}} + a_{P_{f}}}) - \\ - \frac{2 a_{1 i} a_{3 i} σ_{X} σ_{P_{d}}}{a_{P_{d}}} (T_{1} - t - \frac{1}{a_{P_{d}}} + \frac{e x p \{- a_{P_{d}} (T_{1} - t)\}}{a_{P_{d}}}) + {(\frac{a_{2 i} σ_{P_{f}}}{a_{P_{f}}})}^{2} (T_{1} - t - \frac{2}{a_{P_{f}}} + \frac{2 e x p \{- a_{P_{f}} (T_{1} - t)\}}{a_{P - f}} + \\ + \frac{1}{2 a_{P_{f}}} - \frac{e x p \{- 2 a_{P_{f}} (T_{1} - t)\}}{2 a_{P_{f}}}) + \frac{2 a_{2 i} a_{3 i} σ_{X} σ_{P_{f}}}{a_{P_{f}}} (T_{1} - t - \frac{1}{a_{P_{f}}} + \frac{e x p {\{\}}^{- a_{P_{f}} (T_{1} - t)}}{a_{P_{f}}}) + (σ_{X} a_{3 i})^{2} (T_{1} - t), i = 1,2, 3 . \end{matrix}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>Q</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mfrac><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>×</mo></mtd></mtr><mtr><mtd><mo>×</mo><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>P</mi><mo>-</mo><mi>f</mi></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfrac><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow/></mfenced></mrow><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mo>(</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>i</mi><mo>=</mo><mn>1,2</mn><mo>,</mo><mn>3</mn><mo>.</mo></mtd></mtr></mtable></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mtable><mtr><mtd><msub><mrow><mi>Q</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mfrac><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>×</mo></mtd></mtr><mtr><mtd><mo>×</mo><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfenced open="" separators="\|"><mrow><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>)</mo><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="\|"><mrow><mfrac><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced close="" separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>P</mi><mo>-</mo><mi>f</mi></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" separators="\|"><mrow><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="{" close="}" separators="\|"><mrow><mo>-</mo><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mfrac><mrow><mn>2</mn><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mfenced separators="\|"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><msup><mrow><mfenced open="{" close="}" separators="\|"><mrow/></mfenced></mrow><mrow><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo></mrow></msup></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo><mo>(</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>X</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn><mi>i</mi></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>-</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>i</mi><mo>=</mo><mn>1,2</mn><mo>,</mo><mn>3</mn><mo>.</mo></mtd></mtr></mtable></math>

Библиография

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7. Malykh N.O., Postevoy I.S. (2019). Calculation of the convexity adjustment to the forward rate in the Vasicek model for the forward in-arrears contracts on LIBOR rate. Theory of Probability and Mathematical Statistics, 99, 189–198. Available at: https://www.ams.org/journals/tpms/2019-99-00/S0094-9000-2020-01089-9/

8. Mcinerney D., Zastawniak T. (2015). Stochastic Interest Rates. Cambridge: Cambridge University Press.

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