Nonstandard Finite Difference Method For European Call Option Pricing Differential Equation

In the1970s, Black and Scholes on the basis of the research on financialmarkets, pioneering for option pricing, presents the relevant theories and models.Option pricing research of Black and Scholes had also promoted the development ofmathematical finance. Innovation is the use of non-standard finite difference methodfor solving the Black-Scholes European call option pricing differential equation.The advantage of non-standard finite difference scheme is to keep positive andbounded of the solutions, which is the option price requirements. The convergenceand stability of difference scheme can be hold in the case of positive conditions.And time step can be determined by the spatial step length.In fact Black-Scholesoption pricing model with the analytical solution under certain initial and boundaryconditions, but the form of the analytical solution is too complex. And the analyticalsolution does not necessarily exist under certain initial and boundary conditions,Research of using of numerical method to solve Black-Scholes option pricing modelis clearly necessary. First descript development and perfecting process optionpricing. We suppose more press close to the price of options and stock of theoperation mechanism of the financial markets. We deduce differential equation ofBlack-Scholes option pricing using the random process and tectonic portfolio skills,combined with the basic theory such as differential equation. And we explain how toaffect the price of the option, according to the analysis of factors influencing theoption price in the differential equation.We raise principles of establishing the nonstandard finite difference scheme, wehow to select the denominator function in the process of constructing. Accordingprinciples of establishing the nonstandard finite difference scheme, first we applysub-equation method to establish nonstandard finite difference scheme of theBlack-Scholes European call option pricing differential equation. The convergenceand stability of difference scheme can be hold in the case of positive conditions. It isconcluded that the corresponding theorem. The method of correction equationanalysis is used to analyze the difference scheme. The difference scheme is goodreaction rate influence on numerical solution. Then transformation technique will beused to Black-Scholes European call option pricing differential equation. Theequation of transformation set up the nonstandard finite difference scheme. Theconvergence and stability of difference scheme can be hold under the condition ofpositive and convergence and stability theorem are obtained. Nonstandard finitedifference scheme of using the two methods to construct to be numerical simulationand the numerical solution of stability are obtained. In order to keep the Black-Scholes differential equation of European call optionpricing in financial markets requirements. We use the method of nonstandard finitedifference method to sovle the Black-Scholes differential equation of European calloption pricing. It provide more accurate option price for financial markets and atheoretical basis. It promote the development of cross field of mathematics andfinance.