  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.
