On the Solution of Fractional Option Pricing Model by Convolution Theorem

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.


Introduction
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 many mathematical models, the claims often display long-range memories, possibly due to extreme weather, natural disasters, in some cases, many stochastic dynamical systems depend not only on present and past states but also 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  Fractional Brownian motion can be applied in pricing financial derivatives. A financial derivative is an instrument whose price depends on, or is derived from the value of another asset. Often, this underlying asset is a stock. The concept of financial derivative is not new. While there remains some historical debate as to the exact date of the creation of financial derivatives, it is well accepted that the first attempt at modern derivative pricing began with the work of Charles Castelly, published in 1877. In 1969, Fisher Black and Myron Scholes got an idea that would change the world of finance forever. The central idea of their paper revolved around the discovery that one did not need to estimate the expected return of a stock in order to price an option written on that stock.
The Black-Scholes option pricing equation modeled by fractional Brownian motion is derived by replacing the standard Brownian motion involved in the classical Black-Scholes model with fractional Brownian motion which contains the Hurst exponent H. The Hurst exponent denoted by H, is a statistical measure used to classify time series. The value of H varies between 0 and 1.
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 then obtain the solution by applying the convolution theorem.

Fractional Option Pricing Model
where V is the call option price, t is the time to maturity, H is the Hurst exponent, σ is the volatility, S is the stock price and r is the discount rate.
Proof. The stock price t S follows the fractional Brownian motion process The wealth of an investor t X follows a diffusion process given by Suppose that the value of this claim at time t is given by Applying the Ito's formula for fractional Brownian motion on equation (2.5), we have ( ) .
Collecting like terms, we have ( ).
We have under the assumption of complete market that (2.14) Substituting equation (2.14) into equation (2.12), we have is the European call option price, S is the stock price at time t, t is the time to the expiration of the option, r is the discount rate, σ represents the volatility function of the underlying asset and H is the Hurst exponent.
The only τ -dependence in the integral on the right is in the integrand ( ) . As a result, we may write ( ) In other words, the partial time-derivative of ( ) is simply the total time-derivative of ( ). , Using integration by parts: Set iwx e f = and dx x u dg ∂ ∂ = to get; Recall that the boundary conditions for the heat equation on the infinite interval were: As a result, the contributions to the first term vanish and we are left with the integral. Notice that the integral is simply a multiple of the Fourier transform of u, that is We may iterate this result to obtain the Fourier transform of Thus, equation (3.30) may be re-written as: We will now take the inverse Fourier transform of both sides to retrieve ( ) To get back the solution ( ) The inversion is done via convolution theorem for Fourier transform.

Convolution Theorem for Fourier Transforms
Let ( ) F f = F and ( ) We will apply the above convolution theorem to equation (3.34 Equating the coefficients of the successive powers in w gives the following set of equations for A, B, C: The solution of this set of equations is: Substituting these values into (4.38a), we have ( ) Thus, (4.2) becomes:  We can now apply the convolution theorem to (3.34) to retrieve ( ).
, τ x u From the convolution theorem for Fourier transforms, we have