An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

TL;DR

This paper introduces a stable and accurate numerical method combining exponential B-splines and finite differences to solve the time-fractional Black-Scholes equation for European options, with applications to various option types.

Contribution

It develops a novel numerical scheme that is unconditionally stable and convergent for solving the time-fractional Black-Scholes model, enhancing option pricing accuracy.

Findings

The method is unconditionally stable as shown by von-Neumann analysis.

It achieves second-order convergence in space and (2−μ) order in time.

Numerical examples validate the method's accuracy and applicability to different European options.

Abstract

In this paper a time-fractional Black-Scholes model (TFBSM) is considered to study the price change of the underlying fractal transmission system. We develop and analyze a numerical method to solve the TFBSM governing European options. The numerical method combines the exponential B-spline collocation to discretize in space and a finite difference method to discretize in time. The method is shown to be unconditionally stable using von-Neumann analysis. Also, the method is proved to be convergent of order two in space and $2-\mu$ is time, where $\mu$ is order of the fractional derivative. We implement the method on various numerical examples in order to illustrate the accuracy of the method, and validation of the theoretical findings. In addition, as an application, the method is used to price several different European options such as the European call option, European put option, and European double barrier knock-out call option.