A computational scheme for uncertain volatility model in option pricing

In this paper we develop a novel numerical scheme for a nonlinear partial differential equation arising from the uncertain volatility model in option pricing. The fitted finite volume method is developed for the space discretization with implicit scheme in time discretization, which results in a nonlinear discrete system. We prove that this method is consistent, stable and monotone, hence it ensures the convergence to the viscosity solution. We also propose an iteration scheme for the nonlinear discrete scheme and show its convergence property. Numerical experiments are implemented to verify the efficiency and usefulness of this method.

[1]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

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

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

[4]  D. Talay,et al.  Numerical Methods in Finance: Contents , 1997 .

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

[6]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[7]  Song Wang,et al.  Convergence of a fitted finite volume method for the penalized Black–Scholes equation governing European and American Option pricing , 2007, Numerische Mathematik.

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

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

[10]  J. Hull.  Options, Futures, and Other Derivatives , 1989 .

[11]  Jesper Andreasen,et al.  Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Pricing , 1999 .

[12]  D. Duffy.  Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach , 2006 .

[13]  G. Thompson,et al.  Optimal Control Theory: Applications to Management Science and Economics , 2000 .

[14]  Terry Lyons,et al.  Uncertain volatility and the risk-free synthesis of derivatives , 1995 .

[15]  M. Avellaneda,et al.  Pricing and hedging derivative securities in markets with uncertain volatilities , 1995 .

[16]  G. Barles,et al.  Numerical Methods in Finance: Convergence of Numerical Schemes for Degenerate Parabolic Equations Arising in Finance Theory , 1997 .

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

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

[19]  Kai Zhang,et al.  Pricing options under jump diffusion processes with fitted finite volume method , 2008, Appl. Math. Comput..

[20]  Song Wang,et al.  A Fitted Finite Volume Method for the Valuation of Options on Assets with Stochastic Volatilities , 2006, Computing.