Analysis of a Finite Difference Method for a Time-Fractional Black–Scholes Equation

The goal of this paper is to give an error analysis of a finite difference method for a time-fractional Black–Scholes equation with weakly singular solutions. The time Gerasimov-Caputo derivative is discretized by the L1 scheme on a graded mesh designed to compensate for the initial singularities, and a standard finite difference method is used for spatial discretization on a uniform mesh. A discrete comparison principle is presented for the fully discrete scheme, and stability and convergence of the scheme in maximum norm are established by constructing some appropriate barrier functions. Furthermore, an α-robust pointwise error estimate of the fully discrete scheme on a uniform mesh is given. Finally, some numerical results are presented to show the sharpness of the error estimate.