Computer simulation of derivative markets using Black-Scholes model

Abstract

In recent years diverse mathematical physics models became widely used in practice as a great implement for describing economical and physical phenomena. Modeling of the market of derivatives, in particular case, computation of European option rational price, is carried out with Black-Scholes equation, which in terms of constant volatility and non-dividend situation has the well-known analytical solve. Several generalizations of Black-Scholes model were suggested to reach better conformity between the results of mathematical modeling and the real market figures. In these generalized models volatility depends on the desirable function in a rather complicated way. Surveyed are Leland model, Frey-Patie model and also the model of indefinite volatility, which vary from minimum to maximum value. In order to calculate the numerical solution of this mixed task the scheme with scales, which has the second approximation order on the coordinate and time, is used for the equation of parabolic type. Series of computing experiments which allow to compare the results of analytic and numerical analysis were conducted for the different range of parameters. According to the results of computer simulation, generalized non-linear Black-Scholes equation describes the real market simulation describes the real market situation with a very close approximation.