Convergence of a fitted finite volume method for pricing two dimensional assets with stochastic volatilities

Abstract

In this article, we provide the rigorous mathematical convergence proof both in space and time of the two dimensional Black Scholes equation with stochastic volatility. The spatial approximation of this three dimensional problem is performed using the finite volume method coupled with a fitted technique to tackle the degeneracy in the Black Scholes operator, while the temporal discretization is performed using implicit Euler method. We provide a mathematical rigorous convergence proof in space and time of the full discretized scheme. Numerical results are presented to validate our theoretical results.