CALCULATION OF EXOTIC OPTIONS IN INCOMPLETE MARKETS

Elena Shelemeh

We consider incomplete market with discreet time and without transaction costs, consisting of a risky and a risk-free assets. Return on risky asset is supposed to take on each step one of three values, so that increase, decrease and constancy of risky asset’s price are possible. There is exotic option in the market. In the article exotic option is a contract, for which payoff and moment of execution depend on some event, such as risky asset’s price reaches some given value. To calculate such an option in the incomplete market minimax approach was adopted. The approach reflekts seller’s point of view and implies the following. Seller’s risk function is supposed to be exponential and to depend on the option’s initial price and on deficit of seller’s portfolio at the execution moment. Participants of the market do not know the distribution of risky asset’s price. The seller is prudent, i.e. the seller assumes that the distribution is the worst one (it maximazes seller’s expected risk) and seeks to minimize expected risk by portfolio management. For this problem we have derived a new recurrent equation describing evolution of minimax value for seller’s expected risk. A self-financing portfolio has been found, which minimizes maximum value of seller’s expected risk (optimal portfolio). It has been proved that a corresponding portfolio with consumption is a superhaging one with respect to any discrete measure and it’s capital does not exceed capital of any other superhaging portfolio. Thus it is impossible to improve obtained solution choosing another risk functions. By use of this resuts formulas for superhedging portfolios of binary and barrier options have been derived.