DG method for numerical option pricing under the merton short rate model

One of the possible improvements of the classical Black–Scholes option pricing model is to incorporate the stochastic nature of the short rate dynamics in option valuation. In this paper, we present the numerical scheme, based on the discontinuous Galerkin method, for European option pricing when the short rate follows the Merton model. The pricing function satisfies a partial differential equation with two underlying variables — stock price and short rate value. With a localization to a bounded spatial domain, including setting the proper boundary conditions, the governing equation is discretized by the discontinuous Galerkin method over a finite element grid and Crank-Nicolson time integration is applied, consequently. As a result the numerical scheme is represented by a sequence of linear algebraic systems with sparse matrices. Moreover, the numerical simulations reflect the capability of the scheme presented.