Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Heath Windcliff, Peter A. Forsyth, Ken R.Vetzal

Abstract

ABSTRACT

The linear asymptotic boundary condition, which assumes that the second derivative of the value of the derivative security vanishes as the asset price becomes large, is commonly used in practice. To our knowledge there have been no rigorous studies of the stability of this method despite the fact that the discrete matrix equations obtained using this boundary condition lose diagonal dominance for large time steps. In this paper, we demonstrate that the discrete equations obtained using this boundary condition satisfy necessary conditions for stability for a finite-difference discretization. Computational experiments show that this boundary condition satisfies sufficient conditions for stability as well.