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

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 (LBC). 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 LBC was obtained by Windcliff et al. (2004, Analysis of the stability of the linear boundary condition for the Black-Scholes equation, J. Comput. Finance, 8, 65-92) 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 LBC 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 e(tM) (t >= 0) can grow with the dimension of the semidiscrete matrix M, this generally does not impair the convergence behaviour of the numerical discretizations. Our theoretical results are illustrated by ample numerical experiments.