Stability and convergence analysis of discretizations of the Black-Scholes PDE with the linear boundary condition

TL;DR

This paper analyzes the stability and convergence of finite difference discretizations of the Black-Scholes PDE with a linear boundary condition, providing new sufficient conditions and supporting numerical experiments.

Contribution

It establishes sufficient stability and convergence conditions for discretizations with the linear boundary condition, extending previous eigenvalue-based results.

Findings

Stability can be achieved under new sufficient conditions.

Convergence is not impaired by growth in the matrix exponential norm.

Numerical experiments confirm theoretical results.

Abstract

In this paper we consider the stability and convergence of numerical discretizations of the Black-Scholes partial differential equation (PDE) when complemented with the popular linear boundary condition. This condition states that the second derivative of the option value vanishes when the underlying asset price gets large and is often applied in the actual numerical solution of PDEs in finance. To our knowledge, the only theoretical stability result in the literature up to now pertinent to the linear boundary condition has been obtained by Windcliff, Forsyth and Vetzal (2004) who showed that for a common discretization a necessary eigenvalue condition for stability holds. In this paper, we shall present sufficient conditions for stability and convergence when the linear boundary condition is employed. We deal with finite difference discretizations in the spatial (asset) variable and a subsequent implicit discretization in time. As a main result we prove that even though the maximum norm of exp(tM) ($t\ge 0$) can grow with the dimension of the semidiscrete matrix M, this generally does not impair the convergence behavior of the numerical discretizations. Our theoretical results are illustrated by ample numerical experiments.