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

The linear asymptotic boundary condition, i.e. assuming 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 these methods, despite the fact that the discrete matrix equations obtained using this boundary condition loses diagonal dominance for large timesteps. 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 also show that this boundary condition satisfies sufficient conditions for stability as well.