A fast implicit difference scheme with nonuniform discretized grids for the time-fractional Black–Scholes model

Abstract

The solution of the time-fractional Black–Scholes (TFBS) equation often exhibits a weak singularity at initial time and possible non-physical oscillations in the computed solution due to the degeneracy of the BS differential operator. To address this issue, we combine a modified graded mesh and a piecewise uniform mesh for temporal and spatial discretizations, respectively. Then we use the fast approximation (rather than the direct approximation) of the L1 scheme for the Caputo derivative to establish an implicit difference method for the TFBS model. Our analysis shows the stability and convergence of the proposed scheme, as well as the α-nonrobust error bounds. Finally, numerical results are presented to show the effectiveness of the proposed method.