A Hybrid Interpolating Galerkin Meshless Method for the Time-Fractional Black–Scholes Equations with Error Estimations

Capitalizing on the high-precision advantages inherent in meshless methods, this paper develops a high-accuracy meshless algorithm for the fractional Black–Scholes option pricing model. By constructing shape functions using the interpolating moving least squares with non-singular weights and building spatial discrete equations through the Galerkin weak form, while approximating the time fractional derivative using the L1 scheme and L1-2 scheme, respectively, a hybrid interpolating Galerkin meshless method for the fractional Black–Scholes model is proposed. Furthermore, stability analysis and error estimations are performed for the proposed meshless method. The convergence orders with respect to both temporal and spatial discretizations are rigorously proven for the fully discrete system when coupled with the L1 and L1-2 schemes, respectively. Numerical examples confirm the effectiveness and high computational accuracy of the proposed method, demonstrating that the numerical convergence orders align well with the theoretically proven ones.