The Difference Scheme For The Option Pricing Model With Transaction Cost

In 1970s,Breton Forest system broke down fatefully.Then, global Finance came into free stage by fiercely competing in the Finance market.Each country confronted the serious financial riks.For adversing such risks,finance creations developed fast for the main tools on improving the ability of competition for the finance institutes.As science technologies,particular the information technology,is booming quickly.Financial engineering plays an quite important role in forging and managing financial risks hegding,arbitrage,financing and so on.So the core of the whole financial engeering is the theory and method of derivatives pricing which is a considerable positive field in finance theories.The derivatives pricing stems from 1900 when Louis Bachelier ,a France mathematician,construted the model of option pricing in his doctor thesis—The Theory of Speculation.But it is on the base of some unrealistic hypothesizes,such as the security price conformed to norm distribution,and was allowed tobe negative as well as the expectating of its return is zeros.However,he built the foundation of option pricing.Then,the theory of option pricing was going ahead continuously.Boness(1964),R.A.Samuelson(1965) modified the Bachelier′s model.To secutity price,they consider the case of having a free risk rate,conforming to log-norm distribution.But their models depended on an unknown return which was a complementary for the risks related with secutity price.So it is difficult to apply the models.Nevertheless,the miracle appeared in 1973:fisher.Black and Myron.Scholes devised a total new model which is an option pricing reformation.It overcame all the drawbacks before.It is a milestone in modern pricing theory.Black-Scholes formula depended on seven strict hypothesis among which some did not consist with the real situation so that they confined the wild application.Also,that how to price the derivatives under more general conditions is the hot point on recent pricing studies,and got a big step.Merton(1973) proposed a model basing paying diveden;Hull & White(1993) analysed the monetary option pricing moedel and stock index option model;Merton(1976)developed the jump-diffuse model;Cox & Ross(1976) deduced constant deviation flexibility model;Hull & White(1987) rencomanded the stochastic volatility model which was refined by Heston(1993).All the above models did not taken the transaction cost into consideration,but it exists in every transaction.And Leland(1985) filled the gap,then Hoggard,Whalley & Wilmott(1994) perfected it.Although the pricing theory tends to perfect,there are a lot of probems once we applies them.According to pricing theory and methods,several derivative pricing models has simple analytic formulas among the enormous financial derivatives in international finance market.For almost all the option pricing models,we can't attain the analytic solutions.But we can resort to numerical computation which is a special important tool,including three types: Lattice Method, Finite Difference Method, Monte Carto Simulation Method.Lattice method has spreaded widely in price estimation,and the results is great.But it can't solve the'dimension effect',that is,as the dimension of the problem is inceasing,the counting volume and the space of data saving are inceasing exponentially,so it is hard to deal with large dimensional computing.The finit difference method(FDM) can solve not only European option pricing models but also American option pricing models.Similarly,dimension effect exists.Even it is not easy to determine the boundary conditions. Another problem is that it is computed on the infinite domain in the converse direction.There are few researches about the application of FDM for option pricing models,especially for the non linear models.The characters of Monte Carto Simulation method are that it is flexiblethe,and it can make error of standard estimator and converge speed have nothing with dimension.So it can settle the large dimension problems with multiple underlying variables,but it needs some more times to simulate the model when the model becomes complex and we require high precision.This work introduced an Europe option pricing model with transaction cost.It is a converse direction non-linear parabolic degenerate PDE with a terminal condition.I used FDM which is called explicit form to sovle the model. And I dicussed the stability of the defference equation's solution.Then I did some experiments which showed that the results consisted with the financial theory.