Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation

Abiodun Ezekiel Owoyemi,Ira Sumiati,Endang Rusyaman,Sukono Sukono

doi:10.46336/ijqrm.v1i4.83

Abiodun Ezekiel Owoyemi, Ira Sumiati + Show 2 more

Open Access

https://doi.org/10.46336/ijqrm.v1i4.83

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Abstract

Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems.

Highlights

Fractional calculus is a field of calculus relating to derivatives and integral with the fractional order
Based on the background of the problem and previous studies that have been presented, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes partial differential equation and the boundary conditions for the European option pricing problem
The solution of the fractional Black-Scholes equation (15) using the Laplace decomposition method is as follows v0 = max{ex − 1, 0}, vn+1

Summary

Introduction

Fractional calculus is a field of calculus relating to derivatives and integral with the fractional order. This decomposition method can be an effective and useful procedure for solving differential or integral equations without linearization, perturbation, or discretization (Adomian, 1988) This method is able to solve natural or fractional-order differential equations, ordinary or partial, with initial or boundary value problems, with constant or variable coefficients, linear or nonlinear, homogeneous or non-homogenous (Ray and Bera, 2005; Tatari et al, 2007; Saeed and Rahman, 2010; Duan et al, 2012; Bougoffa and Rach, 2013; Abushammala, 2014; Al awawdah, 2016; Sumiati et al, 2019a). Based on the background of the problem and previous studies that have been presented, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes partial differential equation and the boundary conditions for the European option pricing problem

Fractional Calculus

Laplace Decomposition Method

Numerical Illustration

Conclusion