A spectral analysis method for stochastic systems as applies to financial mathematics exemplified with the Black—Scholes model

Аuthors

Kozhevnikov A. S.^*, Rybackov K. A.^**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: AlexKozhevnikov@yandex.ru
**e-mail: rkoffice@mail.ru

Abstract

We consider the Black-Scholes model, which is used to estimate stock prices and European options. The problem of finding stock price probability density function, expected stock price and its dispersion as well as the option premium is studied in the paper. To solve the analysis problem which consists in finding of probability density function for stock price we develop the algorithm based on spectral form of mathematical description. We also use the Monte-Carlo method to control an accuracy of approximate solution. The Black-Scholes formula is used to calculate the option premium. Simulated results are analyzed for different variants of the stock price and values of the option behaviors.

Keywords:

volatility, Black-Scholes model, option, spectral method, spectral characteristic, stochastic system, probability density function