Novel fitted schemes based on mimetic finite difference method for options pricing - Attipoe, David Sena (Mathematics and Applied Mathematics, 2022)
Numerical methods have been increasingly important for finding approximate solutions of partial differential equations (PDEs) describing financial models since only a few of them have analytical solutions. Indeed, in the pricing of derivative securities such as European options, the underlying PDE, the so called Black-Scholes equation, is known to have a closed-form solution when the coefficients are constant. In the case of an American put option, however, there is no analytical solution, even for constant coefficients. In this thesis, we propose alternative schemes based on mimetic finite difference to overcome the known limitations of the finite difference method while pricing options. The standard mimetic finite difference method is known in fluid dynamics to preserve important properties of the continuous problem in the discrete case thereby resulting in more accurate approximations. The underlying Black-Scholes differential operator is known to be degenerate at the boundary when the stock price equals zero. At this singularity, important properties of the PDE are lost. A negative consequence here is that the classical finite difference scheme applied to such problems is no longer monotone and hence fails to give an accurate approximation when the stock price is small. Therefore, more sophisticated techniques that are adapted to handle the degeneracy must be sought. Our proposed scheme, a fitted local approximation method, is able to handle the degeneracy of the Black-Scholes differential operator near the boundary at zero. The novel combined schemes are called fitted mimetic finite difference methods and are used for spatial discretization of the Black-Scholes PDE in one and two dimensional domains. Furthermore, rigorous mathematical convergence proofs of the methods for the one dimensional case are provided where the standard Euler method is used for temporal discretization. Numerical simulations show that the proposed numerical methods (in one and two dimensional domains) applied to both European and American options are more accurate compared to the standard finite difference method and the standard fitted finite volume methods.