The Study Of Option Price Model Under Jump-diffusion Process With Time-dependent Parameters

Financial mathematics is a new developing branch of science, it is now being paid close attention to in the domain of international finance and applied mathematics. Uncertain pricing is one core of financial mathematics study, It involves the theories of modern finance such as asset pricing theory, investment combination theory and it involves theories of modern mathematics, such as stochastic analyzing and optimizing theory too. Effective management of risk occupies the right evaluation of derivative securities. The critical thing is that the financial derivative securities exist reasonably and develop properly is how to value its fair price. As financial market develop, more and more new option appear and the option pricing also become complicate, This dissertation is intended to study option pricing problems, so as to establish the mathematic module of option pricing with jump-diffusion process by means of mathematical tools such as martingale theory and stochastic analysis, to deduce reload option,option with power payoffs and exchange option pricing when stock price driven by jump-diffuse process.Reload option is an exotic European call option, the holder can operate the option on a given day before expiration date, and when operate it, the option is in the money, the strike price and the option number all change. The paper assumes that the stock price process driver by non-homogeneous Poisson jump-diffusion process, and the expect rateμ(t), volatilityσ(t) and the riskless rate r(t) are function of time, under risk neutral pricing model ,we obtain the price of reload stock pricing formula . Option with power payoffs is also exotic option, it's payoff function is [h(S(T))-K]~+. Under the assumption that stocks price process driven by Poisson jump-diffusion process, and parameters are function of time, based on the theory of equivalent martingale measure transformation, we get the pricing formulas of European options with power payoffs. We also discussed the option pricing with multiple sources of jumps. At last we also discuss the pricing of exchange option that the assets have different weight. Assumes that the two stock price processes are jump-diffusion process, under the condition that the expected rate and volatility are function of time, using physical probabilistic measure of price process and the principle of fair premium, we obtain the pricing formula of exchange options.