Pricing Options Under Jump-diffusion Models

Nowadays, pricing options under jump-diffusion models is a very hot topic in option pricing research. By the use of risk neutrality measure, martingale law, Girsanov theorem and multidimensional normal distribution, this article discusses the pricing of options with constant interest rate or stochastic and the pricing of foreign exchange reset options under jump-diffusion models. Finally, this article demonstrates the simulation under jump-diffusion models using the finite difference method and obtains some valuable results.Pricing options with constant interest rate under jump-diffusion models is always a very important problem in option pricing research.Generally, we use backward stochastic differential equation to solve this kind of problem. But this article embarks from another side, solving this problem by use of martingale law. Firstly, we establish the money market model under jump-diffusion models, risk neutrality measure, equivalent martingale, then get the pricing formula using Girsanov theorem.Pricing options with stochastic interest rate under jump-diffusion models is always a difficulty in option pricing research. There are little research about this, mostly using backward stochastic differential equation. This article applies the method in diffusion models to jump-diffusion models successfully and gets the pricing formula by martingale law when the interest rate obeys the Vasicek model.Pricing foreign exchange reset options under jump-diffusion models refers to multidimensional conditions and reset time. Firstly, we establish a multidimensional money market model under jump-diffusion models and define a multidimensional risk neutrality measure;finally, by use of martingale law and multidimensional normal distribution, we get the pricing formula of this option under special conditions.The finite difference method is the most essential one in option pricing simulation, widely applied in simulation of diffusion models. This article realizes the simulation in jump-diffusion using the finite difference method.