A Fitted Finite Volume Method for the Valuation of Options on Assets with Stochastic Volatilities

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.

[1]  D. N. De G. Allen,et al.  RELAXATION METHODS APPLIED TO DETERMINE THE MOTION, IN TWO DIMENSIONS, OF A VISCOUS FLUID PAST A FIXED CYLINDER , 1955 .

[2]  Song Wang,et al.  A new non-conforming Petrov-Galerkin finite-element method with triangular elements for a singularly perturbed advection-diffusion problem , 1994 .

[3]  Song Wang,et al.  A novel fitted finite volume method for the Black-Scholes equation governing option pricing , 2004 .

[4]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[5]  Eduardo S. Schwartz.  The valuation of warrants: Implementing a new approach , 1977 .

[6]  P. Wilmott,et al.  Option pricing: Mathematical models and computation , 1994 .

[7]  Georges Courtadon,et al.  A More Accurate Finite Difference Approximation for the Valuation of Options , 1982, Journal of Financial and Quantitative Analysis.

[8]  Y. Kwok.  Mathematical models of financial derivatives , 2008 .

[9]  Peter A. Forsyth,et al.  Penalty methods for American options with stochastic volatility , 1998 .

[10]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[11]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[12]  Song Wang,et al.  A Novel Exponentially Fitted Triangular Finite Element Method for an Advection-Diffusion Problem with Boundary Layers , 1997 .

[13]  Kok Lay Teo,et al.  Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation , 2006 .

[14]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[15]  Song Wang,et al.  An exponentially fitted finite volume method for the numerical solution of 2D unsteady incompressible flow problems , 1994 .