A Compact Difference Scheme for Mixed-Type Time-Fractional Black-Scholes Equation in European Option Pricing

Abstract

The time-fractional Black-Scholes equation (TFBSE) is an important model in financial markets, widely used for estimating the prices of European options under conditions of memory effects and anomalous diffusion. Traditional models often fail to capture such dynamics, making TFBSE particularly important for accurately reflecting market behaviors over time. In this paper, we propose a novel compact difference scheme to solve the mixed-type TFBSE. Discretization in the time direction is accomplished using the L1 scheme. To achieve fourth-order discretization in the spatial direction, a compact difference method based on the reduced-order method is employed. The stability and convergence of the proposed scheme under the $L^{\infty }$ norm are established using the discrete energy method. Finally, a series of numerical examples are provided to verify the theoretical results, demonstrating both the accuracy and efficiency of the method in practical applications.