Study On Several Option Pricing Problems With Jump-diffusion Market

Option pricing has been a research hot-spot and also been a cutting-edge issue ofmathematical finance region so that the research on it has a profound theoretical and practicalsignificance. This paper uses the martingale method and differential equation method withfree boundary problem in a main line with optimal stopping. Respectively, we have a study onthe American option,the Russian option,and the Game option pricing in a diffusion marketand a jump-diffusion market.The main research includes:For American option pricing, we discuss the infinite flat American Option PricingProblems (or perpetual American option) and limited level of American Option Pricing. Forthe perpetual American Option Pricing Problems, we give the martingale representationmodel about the pricing problem, and then we gain the free boundary problem with which thevalue function satisfies. We gain the value function and optimal stopping boundary byapplying the method of the undetermined coefficients. For a limited level of the AmericanOption Pricing problem, we give the martingale representation model of the pricingproblem,then we obtain the free boundary parabolic problem which is a moving boundarywaiting for solving.For the free boundary parabolic problem,we study the regular nature ofthe optimal stopping boundary by applying the optimal stopping theory,and then,we gain abetter results by applying the theoretical analysis into analyzing the American Option Pricingexercise boundary. From a mathematical angle, the American Option Pricing problem underthe proliferation of the market boils down to the free boundary problem for parabolicequations,the American Option Pricing problem under the jump-diffusion market boils downto the free boundary problem of the parabolic differential-integral equation.The essentialdifference above is that the infinitesimal generator integral operator which the jump-diffusiongenerates brings essential difficulties for modeling and solving the problem. The main difference between the Russian and the American option is the revenuefunction,which results to the basic similar approach.In the part of the Russian option pricingproblem, we have a research on the martingale representation model and the solving of thevalue function concerning to the perpetual Russian option, also we apply the transformsimplified method to discuss the limited level of the Russian Option Pricing problem.For thesimplified one-dimensional problem,we give the corresponding model of the freeboundary,and study the related nature of the value function.For the Game Options, we give the general martingale representation model, and obtainthe value function and the stopping boundary by solving the perpetual Game Option Pricingproblem.For the barrier Game Options, we give the corresponding differential equation modeland the solving. At last, we have a research on the perpetual Game Options withjump-diffusion model and solve that.