Abstract
The classical Black-Scholes equation driven by Brownian motion has no memory, therefore it is proper to replace the Brownian motion with fractional Brownian motion (FBM) which has long-memory due to the presence of the Hurst exponent. In this paper, the option pricing equation modeled by fractional Brownian motion is obtained. It is further reduced to a one-dimensional heat equation using Fourier transform and then a solution is obtained by applying the convolution theorem.
Highlights
In many areas of science, there has been an increasing interest in the investigation of the systems incorporating memory or after effect, that is, there is the effect of delay on state equations
In this paper we intend to reduce the Black-Scholes option pricing equation modeled by fractional Brownian motion to a one-dimensional heat equation using Fourier transform and obtain the solution by applying the convolution theorem
The inversion is done via convolution theorem for Fourier transform
Summary
In many areas of science, there has been an increasing interest in the investigation of the systems incorporating memory or after effect, that is, there is the effect of delay on state equations. The claims often display long-range memories, possibly due to extreme weather, natural disasters, in some cases, many stochastic dynamical systems depend on present and past states and contain the derivatives with delays. In such cases, class of stochastic differential equations driven by fractional Brownian motion provides an important tool for describing and analyzing. In this paper we intend to reduce the Black-Scholes option pricing equation modeled by fractional Brownian motion to a one-dimensional heat equation using Fourier transform and obtain the solution by applying the convolution theorem
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