The Numerical Methods Of Option Pricing

The numerical solution of partial differential equations(PDEs) is an important area of scientific computing,since there are so many processes,e.g.engineering,physics,biology,and cven economics.This method using PDE theory is greatly convenience.The option pricing and volatility estimate is financial project,financial mathematics problem of lcading cdge as well as a hot one at present.At the same time,option is one of the core tools of financial derivative security,which plays an important role in the effective management of risk and investment.American options on dividend-paying stock is a free boundary problem of parabolic partial differential equations.So, researching more effective numerical methods be able to solve this problem is also important.Numerical methods for the American option pricing on dividend-paying stock are few,such as the binomial tree methods and standard finite differential methods.However,the binomial tree methods neglect the possibility of non-fluctuating prices,and computation time is too long;the ordinary finite differential method is lack of analysis of free boundary and the accuracy is low. Therefore,this paper transforms unknown and unbounded variable regin to bounded regin theory through variable meshes method.This regin approaches to the real regin when the number of nodes add to.So we can get numerical solution of this problem using partial differential theory.The part of this introduction has done the reviewing of generality to the pricing theory,including development of early and modern option pricing.The second part introduces the economical background and the basic concept of financial derivatives and optimal exercise boundary. Then we educe the differential form of Black-Scholes.In the third part of this paper,option pricing problem is transformed to heat equation,through theory of free boundary.What's more,we get the numerical solution of this problem by compact different method in extending finite regin,and numerical experiments show that the new method is very efficient for option pricing problems.In the fourth part,on the base of theory of free boundary,we get the discontinuous Galerkin method of option pricing.The relevant conclusions are made in final chapter.