Option Pricing In Incomplete Markets Based On The Methodological Perspective

This thesis is devoted to several methods of option pricing in incomplete markets, where risks are more than tradable risky assets. There are several methods to deal with those pricing and portfolio problems. Among two main methods are studied in this thesis.Using the first method, that is the Minimal Martingale Measure Methods, we deal with option pricing with risk-minimization criterion in an incomplete market when the dynamics of the risky underlying asset is governed by a jump diffusion equation with stochastic volatility. We obtain the Radon--Nikodym derivative for the minimal martingale measure and a partial integro-differential equation (PIDE) of European option. The finite difference method is employed to compute the European option valuation of PIDE.Using the second method, that is General Equilibrium framework Methods, we study the equity premium and option pricing under jump diffusion model with stochastic volatility. Then, we obtain the pricing kernel which likes the physical and risk-neutral densities and moments in the economy. Moreover, the exact expression of option valuation is given by Fourier transformation methods. Finally, we discuss the relationship of central moments between the physical measure and risk-neutral measure.