Jump - Diffusion Model Under Uncertain Equity Pricing Issues

Contingent claims valuation is one of the kernel problems on financial mathematics. A mass of finance practice has indicated that there is a serious warp between the hypothesis of Black-Scholes model about the underlying asset price and the realistic markets. The solutions to contingent claims are determined by the alteration law of the underlying asset prices. So, the improvement on Black-Scholes model attracts wide attention, and many new models are proposed. This dissertation contributes to option pricing problem in jump-diffusion models and proposes a new asset prices model. This thesis has the following contents:1. Complete security markets with discontinuous security price are studied. Using martingale method, general pricing formula of European contingent claims is derived and European option and put-call parity is analyzed. These results are gained under underlying asset price being driven by a jump-diffusion process.2. A new method of option pricing (i.e. insurance actuary pricing) is introduced. It is proved that martingale pricing is consistent with insurance actuary pricing in a Poisson jump-diffusion model. Pricing warps on Black-Scholes model are analyzed.3. A new model is proposed, where asset prices are given by the combination of finite state Q process stochastic volatility and a compound Poisson process. The general formula of European call option pricing has been derived, and the results of Hull and White are generalized. At last, an analysis and numerical simulations for jump-diffusion model with q process volatility are given.4. The problem of contingent claims valuation is discussed when the underlying asset price is a jump-diffusion process under stochastic interest rates. Using martingale method, pricing formula of European contingent claims is derived and put-call parity is analyzed. Pricing formula of European option is also given when risk asset pays continuous dividends.