Generalised class of Time Fractional Black Scholes equation and numerical analysis

Rodrigue Gnitchogna Batogna,Abdon Atangana

doi:10.3934/dcdss.2019028

Abstract

It is well known now, that a Time Fractional Black Scholes Equation (TFBSE) with a time derivative of real order \begin{document}$ \alpha $\end{document} can be obtained to describe the price of an option, when for example the change in the underlying asset is assumed to follow a fractal transmission system. Fractional derivatives as they are called were introduced in option pricing in a bid to take advantage of their memory properties to capture both major jumps over small time periods and long range dependencies in markets. Recently new derivatives of Fractional Calculus with non local and/or non singular Kernel, have been introduced and have had substantial changes in modelling of some diffusion processes. Based on consistency and heuristic arguments, this work generalises previously obtained Time Fractional Black Scholes Equations to a new class of Time Fractional Black Scholes Equations. A numerical scheme solution is also derived. The stability of the numerical scheme is discussed, graphical simulations are produced to price a double barriers knock out call option.

Highlights

If the importance to accurately price financial derivatives is at all time reminded by volumes of trade of these financial products, the constant innovation in pricing techniques on the other hand concedes that we still don’t have to date a method we could consider absolute
As new significant results in fractional calculus has recently emerged, and fractional models continuing to be vastly applied in mathematical finance, we propose here a generalised class of time fractional Black Scholes models, together with a numerical solution of such a model with the time fractional derivative defined in the sense of the recent Caputo-Fabrizio derivative with non singular Kernel [6]
In this work we considered a new class Time Fractional Black Scholes Equation (TFBSE) with the time fractional derivative defined in the sense of Caputo-Fabrizio

Summary

Introduction

If the importance to accurately price financial derivatives is at all time reminded by volumes of trade of these financial products, the constant innovation in pricing techniques on the other hand concedes that we still don’t have to date a method we could consider absolute. More recently Wenting Chen et al [9] analytically priced a double barrier options based on time fractional black scholes equation in the sense of Jumarie and Caputo time fractional derivatives. H. Zhang et al [14] proposed an implicit discrete scheme for a numerical solution of a time fractional Black Scholes model, with the time fractional derivative being defined in the sense of Guy Jumarie. As new significant results in fractional calculus has recently emerged, and fractional models continuing to be vastly applied in mathematical finance, we propose here a generalised class of time fractional Black Scholes models, together with a numerical solution of such a model with the time fractional derivative defined in the sense of the recent Caputo-Fabrizio derivative with non singular Kernel [6]. By differentiating equation (1) and using the classic Black Scholes equation to express the flux rate Y (S, τ ) as a function of other terms, we achieve the following equation

The time fractional derivative is defined as

In fact

Lα r

This is achieved when we have

Conclusion