Convergence of the mimetic finite difference and fitted mimetic finite difference method for options pricing

Abstract

We present in this paper two novel numerical spatial discretization techniques based on the mimetic finite difference method for a degenerated partial differential equation (PDE) in one dimension. This PDE is well known as the Black-Scholes PDE which govern option pricing. To handle the degeneracy of the PDE, a novel fitted mimetic finite difference scheme is proposed together with the standard mimetic finite difference method. The temporal discretization is performing using standard implicit scheme. Furthermore rigorous convergence proofs in appropriate normed spaces are proposed. We validate the theoretical results by presenting numerical results and simulations. Those numerical experiments show that our two novel schemes outperform the standard finite difference method and the standard fitted finite volume method in terms of accuracy.