Generalised class of Time Fractional Black Scholes equation and numerical analysis

Rodrigue Gnitchogna Batogna; Abdon Atangana

doi:10.3934/dcdss.2019028

Article Contents

2019, Volume 12, Issue 3: 435-445. Doi: 10.3934/dcdss.2019028

This issue Previous Article Preface: New trends on numerical analysis and analytical methods with their applications to real world problems Next Article New exact solutions for some fractional order differential equations via improved sub-equation method

Generalised class of Time Fractional Black Scholes equation and numerical analysis

Rodrigue Gnitchogna Batogna^1,2, , and
Abdon Atangana^3,

1.
Department of Mathematics and Applied Mathematics, Faculty of Natural and Agricultural Sciences, University of the Free State, P.O. Box 339 Bloemfontein 9300, South Africa
2.
Department of Mathematics University of Namibia, Private bag 13301 Windhoek, Namibia
3.
Institute of Groundwater Studies, University of the Free State, IB 56 UFS P.O. Box 339 Bloemfontein 9300, South Africa

* Corresponding author: Rodrigue Gnitchogna Batogna
* Corresponding author: Rodrigue Gnitchogna Batogna

Received: April 30, 2017

Revised: September 30, 2017

Published: October 2018

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Abstract

It is well known now, that a Time Fractional Black Scholes Equation (TFBSE) with a time derivative of real order $ \alpha $ 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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Double barrier option price solutions. Model parameters are $\sigma = 0.45, r = 0.03, T = 1, K = 10, DO = 3, UO = 15$

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Figure 2. Approximate solutions from equation (15) Double barrier option prices approximate solutions. Model parameters are $\sigma = 0.45, r = 0.03, T = 1, K = 10, DO = 3 ,UO = 15$

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Figure 3. Approximate solutions from equation (15) Double barrier option prices approximate solutions. Model parameters are $\sigma = 0.45, r = 0.03, T = 1, K = 10, DO = 3 ,UO = 15$

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