Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

Sangkwon Kim,Junseok Kim,Chaeyoung Lee,Wonjin Lee,Darae Jeong,Soobin Kwak,Youssri Hassan Youssri

doi:10.1155/2021/9984473

Sangkwon Kim, Junseok Kim + Show 5 more

Open Access

https://doi.org/10.1155/2021/9984473

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Abstract

In this study, we present an accurate and efficient nonuniform finite difference method for the three-dimensional (3D) time-fractional Black–Scholes (BS) equation. The operator splitting scheme is used to efficiently solve the 3D time-fractional BS equation. We use a nonuniform grid for pricing 3D options. We compute the three-asset cash-or-nothing European call option and investigate the effects of the fractional-orderαin the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.

Highlights

IntroductionWe consider the following 3D version of the time-fractional Black–Scholes (BS) model [1]:
We consider the following 3D version of the time-fractional Black–Scholes (BS) model [1]:∂αu ∂tα ðx, y, z, tÞ + L BSuðx, y, z, ð1Þ= 0 for ðx, y, z, tÞ ∈ Ω × 1⁄20, TÞ, where 0 < α < 1 and L BS = σ2x x2 ∂2u ∂x2 σ2y y2
In this study, we present an accurate and efficient nonuniform finite difference method for the 3D time-fractional BS model

Summary

Introduction

We consider the following 3D version of the time-fractional Black–Scholes (BS) model [1]:. The author in [24] proposed the application of homotopy analysis method (HAM) for pricing European call option based on timefractional BS equation He demonstrated the accuracy, effectiveness, and suitability of HAM through comparative tests. In [26], option derivatives were numerically priced using the θ-method for the time-fractional BS equation These schemes are both first-order and second-order accurate in HH x1 = 0 x2 x3 h h h xi–1 xi K xi+1 xi+2. At each time step, the numerical solution scheme should be fast To satisfy these conditions, in this study, we present an accurate and efficient nonuniform finite difference method for the 3D time-fractional BS model. We provide the MATLAB code for the numerical implementation for the three-asset cash-ornothing option

Numerical Solutions

Numerical Experiments

Three-Asset Options with Nonuniform Mesh

Conclusions