Valuation in Incomplete Markets

III Introduction Black-Scholes model is the most celebrated example of option pricing and hedging in complete market using no arbitrage theory and martingale methods. Harrison and Kreps (1979)[HK79] studied the problem in the discrete time case, and Harrison and Pliska (1981) [HP81] and (1983) extended the results to the continuous time case. Since from these papers, the modern theory of contingent claim valuation has been developed with the mathematical foundation of Martingale and Stochastic Integrals. In these papers, the concept of market completeness is discussed at the beginning and it is shown that the market is complete if and only if its vector price process has a certain martingale representation property. In the case of complete market all contingent claims have a definite price due to the existence of a unique martingale measure and a self-financing trading strategy which replicates the pay-off of the claim. The “arbitrage-based” approach while is very useful for complete market, is usefulness for incomplete market. In such markets contingent claim typically cannot be replicated exactly using self-financing portfolios. The case in which the problem appears, are for example when the volatility of the underlying stock is stochastic, or the hypothesis of the B-S model are not satisfied. One possible approach then, is to relax the requirement that portfolios be self-financed, by requiring that the gain process is equal not to zero, but is a martingale. A portfolio with this property was called mean-self-financed by Föllmer and Sondermann (1986)[FS86]. These authors introduced this notion and then established that for any square- integrable contingent claim and discounted price processes that are martingales, there exists a portfolio whose value is almost surely equal to that of contingent claim at time T and that possesses a certain “risk-minimizing” optimality property; such a portfolio is unique and mean-self-financing. The Föllmer and Sondermann [FS86] approach was extended by Schweizer and Föllmer and Schweizer[FS89] to semimartingale processes; in these papers, mean-self-financing portfolios were characterized in terms of a stochastic functional equation, which has a solution a under the assumption that a certain minimal martingale measure exists. This measure has the property that although it turns prices into martingales, it does not otherwise change the structure of the model. Using this equivalent martingale measures, Schweizer (1992b)[S92] solved a mean-variance hedging problem in a particular incomplete