Finite Difference Method For European Call Option Pricing Differential Equation

This paper mainly discusses the Black-Scholes European option pricingdifferential equation when stock price volatility is constant and non-constant.Under two conditions talked above, this paper take intensive research intocalculation process of finite difference method and its stability.Research field of this paper is a new crossed subject between the theory ofnumerical methods for solving differential equations and option pricing theory offinancial math. In1970, Black and Scholes got innovative achievements on theclassic formula of option pricing in financial market. This discovery has greatlypromoted the combination of subjects of finance and math, and also advanced therapid development of finance market.The article makes a systematic summarize of solving parabolic partialdifferential equation by finite difference method, discusses compatibility,stability and convergence of various formats, and establishes the foundation forsolving option pricing differential equation by finite difference method.This article makes an in-depth analysis of the option which is kind offinancial derivatives, and its related theories. In this process, some assumption inaccordance with the real financial market. are set in order to take use of theoriesof stochastic processes, portfolio theory and differential equations to deduceBlack-Scholes European option pricing differential equation.Based on the mesh refine in area of differential equation set solution, eachformulae with initial and boundary conditions in the differential equations aretaken the difference, and the related finite difference schemes and its solvingmethods are given out. The European call option pricing equation withouttransaction cost is firstly solved by the finite difference method, and the analysisof difference scheme truncating error is taken, then the Fourier analysis method isused to discuss the compatibility, stability and convergence of the method, finallythe corresponding theorem is shown. Furthermore, this paper discusses Europeancall option pricing equation without transaction cost but with non-constant stockprice volatility which is more similar with the reality, by taking the same waybefore. Under the two conditions talked above, took to its solution, and thedifference scheme stability analysis. Finally, through numerical simulation test, the conclusion is verified.Research of this paper has great value on improving option pricing theory,and how to have more stable and accurate access to get the price of option, andpromoting that mathematical theory is more widely used in the financial market.