In this paper, we present a finite volume method for a two-dimensional Black-Scholes equation with stochastic volatility governing European option pricing. In this work, we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conservative form. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presented.
Huang, C-S., Hung, C-H., & Wang, S. (2006). A Fitted Finite Volume Method for the Valuation of Options on Assets with Stochastic Volatilities. Computing, 77(3), 297-320. https://doi.org/10.1007/s00607-006-0164-4