Numerical methods for the valuation of American options under jump-diffusion processes

The purpose of this dissertation is studying the numerical valuation of American and European option prices under jump-diffusion processes. Due to the jump part, the market is incomplete and so it is impossible to construct a hedging portfolio with stocks and riskless assets. Contrary to the case of a complete market in which only one equivalent martingale measure exists, there are infinite numbers of equivalent martingale measures in an incomplete market. Our research here is focusing on risk minimizing strategy and its associated minimal martingale measure under the jump-diffusion processes.; Based on this risk minimizing hedging strategy, we characterize the dynamics of a risky asset and derive the valuation formula for an option price. Under the minimal martingale measure, we obtain an analytical formula for a European option price. The main contribution of this dissertation is to extend Kim (1990)'s early exercise premium representation based on a decomposition method in order to calculate an American option price under jump-diffusion processes as a summation of a European option price and early exercise premiums.; We derive the early exercise premium representation under jump-diffusion processes with various distributions of jump size---lognormal, jump-to-ruin, bivariate and double exponential distribution. In calculating an optimal boundary, we modify and extend numerical methods previously used in the pure diffusion processes---Kim's integral equation method, and Ju's approximation scheme by multipiece exponential functions. Also we apply Richardson extrapolation scheme and modify MacMillan-Zhang's analytical method to calculate American option prices in a faster way.; We implement two previous procedures: a binomial lattice method of Amin (1993) and a semi-implicit finite difference method of Zhang (1997) and compare them with our extended integral equation method. The numerical performance of the extended integral equation method is found to be superior to the previous methods in that the former shows a smaller relative root-mean-square error, possesses a lower degree of algorithmic complexity and converges faster than the two previous methods.