Novel numerical techniques based on mimetic finite difference method for pricing two dimensional options

Abstract

The Black–Scholes differential operator which underlies the option pricing of European and American options is known to be degenerate close to the boundary at zero. At this singularity, important properties of the differential operator are lost and the classical finite difference scheme applied to this problem fails to give accurate approximations as it is no longer monotone. In this paper novel numerical techniques based on mimetic finite difference method are proposed for accurately pricing European and American options. More precisely, we propose the mimetic and fitted mimetic finite difference methods, which are techniques that preserve and conserve general properties of the continuum operator in the discrete case. The fitted method further handles the degeneracy of the underlying partial differential equations (PDE). Those spatial discretization methods are coupled with the Euler implicit method for time discretization. Several numerical simulations are performed to demonstrate the robustness of our methods comparing to standard fitted finite volume method for both European and American put options.