A compact finite difference scheme for solving fractional Black-Scholes option pricing model

In this work, we introduce an efficient compact finite difference (CFD) method for solving the time-fractional Black-Scholes (TFBS) option pricing model. The time-fractional derivative is described using Caputo-Fabrizio (C-F) fractional derivative, and a compact finite difference method is employed to discretize the spatial derivative. The main contribution of this work is to develop a high-order discrete scheme for the TFBS model. In the numerical scheme, we have developed a convergence rate of O(tau 2+h4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\tau <^>{2} + h<^>{4})$\end{document}, where tau denotes the temporal step and h represents the spatial step. To verify the effectiveness of the proposed method, we have conducted stability analysis and error estimation using the Fourier method. Furthermore, a series of numerical experiments were conducted, and the numerical results demonstrated the theoretical order of accuracy and illustrated the effectiveness of the proposed method.