A posteriori grid method for a time-fractional Black-Scholes equation

In this paper, a posteriori grid method for solving a time-fractional Black-Scholes equation governing European options is studied. The possible singularity of the exact solution complicates the construction of the discretization scheme for the time-fractional Black-Scholes equation. The $L1$ method on an arbitrary grid is used to discretize the time-fractional derivative and the central difference method on a piecewise uniform grid is used to discretize the spatial derivatives. Stability properties and a posteriori error analysis for the discrete scheme are studied. Then, an adapted a posteriori grid is constructed by using a grid generation algorithm based on a posteriori error analysis. Numerical experiments show that the $L1$ method on an adapted a posteriori grid is more accurate than the method on the uniform grid.