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 $O(\tau ^{2} + h^{4})$ , where τ 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.