High order finite difference schemes on non-uniform meshes for the time-fractional Black-Scholes equation

TL;DR

This paper develops a high-order finite difference scheme on non-uniform meshes for the time-fractional Black-Scholes equation, achieving fourth-order accuracy in space for specific mesh types used in finance.

Contribution

It introduces a three-point compact finite difference scheme on non-uniform meshes with proven fourth-order accuracy for certain graded meshes.

Findings

Fourth-order spatial accuracy on Tavella-Randall and quadratic meshes

Effective numerical experiments demonstrating scheme performance

Applicable to financial models with time-fractional derivatives

Abstract

We construct a three-point compact finite difference scheme on a non-uniform mesh for the time-fractional Black-Scholes equation. We show that for special graded meshes used in finance, the Tavella-Randall and the quadratic meshes the numerical solution has a fourth-order accuracy in space. Numerical experiments are discussed.