Pricing Barrier Options Underlying The Market Of Jump-diffusion In A Stochastic Interest Rate Environment

The maximum of stochastic process is often encountered in the pricing of exotic options,especially when the stochastic process is a Jump-Diffusion process, the problem get more complicated because the distribution of the maximum is difficult to derive . There is no doubt that we should use Jump-Diffusion process to describe changing behaviors of financial markets instead of Diffusion process, meanwhile the pricing is also more complex. Information dissemination,market quotations and trade practices are three important factors which effect trade speed and transaction volume, and a reasonable pricing is the premise which active exchange market.The paper mainly considered two critical issues: one is how to describe the behavior of market; the other is the joint distribution of the maximum and terminal value of Jump-Diffusion process, and the two issues are also the key of pricing of other exotic options.This paper made use of Fortet method in direct computing the joint distribution of the maximum and terminal value of Jump-Diffusion process and barrier options is priced.First chose proper models of interest rate and stock price, then obtained the expressions of the solutions to the above two stochastic differential equations by It? formula. Via equivalent martingale measure, the expressions were transformed into easier expressions in risk-neutral measure. Based on the property of the martingale, we got the mathematical expectation of discounted future return of barrier options. Owing to the Fortet method, the joint distribution of the maximum and terminal value of Jump-Diffusion process was computed directly, and the approximation of the price of barrier options. Finally, parameters of the models were estimated and analyzed.The program of the paper's numerical calculation was written in C#, the statistical functions quoted; the graphics were plotted by Matlab.