Pricing Options In Incomplete Market

Option pricing is extremely important in financial market. In complete market, there are mature theories of option pricing, which can be viewed from several different angles. However, in incomplete market, option pricing just begins to prosper, when the local-equilibrium principle is proposed. It is a coincident concept for option pricing in both complete and incomplete market. The aim of this thesis is to develop the local-equilibrium principle.In this thesis, we discuss how to price options with stochastic volatility in incom-plete market. Under the power utility preference, we show that the option fair price depends on the current position of the portfolio, which is associate with the current wealth of the portfolio. Furthermore, we verify the put-call-parity and give an explicit expression for the market-price-of-risk and the optimal hedging scheme which is dif-ferent from delta hedging. We also propose an asymptotic expansion, which not only explains the position dependence theory of option pricing, but also gives us a good ap-proximation. In addition, we show many numerical examples which illustrated various points.Under the exponential utility preference, we develop the analytical asymptotic for-mulas for the fair price and trading price of options, and we also obtain the expansions of the corresponding implied volatilities, which are deduced through regular perturba-tions for small volatility of volatility. With these expansions, the surface of the implied volatility, such as the skew and smile, can be explained sufficiently. Particularly, we show how these prices of options depend on the position. The position effect plays a significant role on options pricing, because it can dominate the surface of the implied volatility under some circumstance. Furthermore, compared with the numerical results, such asymptotic formulas can be regard as good approximations.