A high-order and fast scheme with variable time steps for the time-fractional Black-Scholes equation

TL;DR

This paper develops a high-order, efficient numerical scheme with variable time steps for the time-fractional Black-Scholes equation, addressing initial singularities and achieving high accuracy in both time and space.

Contribution

It introduces a novel finite difference scheme using nonuniform Alikhanov formula and SOE technique for improved efficiency and accuracy in solving the time-fractional Black-Scholes equation.

Findings

The scheme is stable and convergent with second-order in time and fourth-order in space.

Numerical experiments confirm the theoretical accuracy and efficiency.

The method effectively handles initial singularities in solutions.

Abstract

In this paper, a high-order and fast numerical method is investigated for the time-fractional Black-Scholes equation. In order to deal with the typical weak initial singularities of the solution, we construct a finite difference scheme with variable time steps, where the fractional derivative is approximated by the nonuniform Alikhanov formula and the sum-of-exponentials (SOE) technique. In the spatial direction, an average approximation with fourth-order accuracy is employed. The stability and the convergence with second-order in time and fourth-order in space of the proposed scheme are religiously derived by the energy method. Numerical examples are given to demonstrate the theoretical statement.