A Novel Numerical Method and Its Analysis for a Two‐Dimensional Time Fractional Black‐Scholes PDEs for European Options

ABSTRACT

In this paper, we develop and analyze a high‐order numerical scheme for a two‐dimensional Caputo time‐fractional Black–Scholes (TFBS) equation whose solution exhibits a weak singularity at the initial time . To effectively handle this singular behavior, a fast L2‐ discretization on a graded mesh is employed in the temporal direction, while the Galerkin finite element method (FEM) is used for the spatial discretization. A rigorous stability and convergence analysis of the proposed scheme is performed. The theoretical analysis establishes that the scheme achieves second‐order accuracy in the spatial direction and a convergence rate of order in the temporal direction. The proposed method is validated through a test problem with a known analytical solution. Moreover, its practical applicability is further demonstrated by solving two real‐world financial problems involving the pricing of European call and put options. The numerical results confirm that the graded mesh yields significantly more accurate solutions compared to the uniform mesh due to its ability to capture the weak singularity present in the solution at the initial time.