Instabilities of explicit finite difference schemes with ghost points on the diffusion equation

TL;DR

This paper investigates how ghost points affect the stability of explicit finite difference schemes for the diffusion equation, with implications for financial models like Black-Scholes.

Contribution

It provides a detailed analysis of ghost point impacts on stability in explicit schemes, extending understanding to financial applications.

Findings

Ghost points influence stability conditions of explicit schemes.

Results applicable to a wide range of financial derivatives.

Guidelines for implementing ghost points in practice.

Abstract

Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models.