Vol. 2, No. 4, 2009

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On the consistency of finite difference approximations of the Black–Scholes equation on nonuniform grids

Myles D. Baker and Daniel D. Sheng

Vol. 2 (2009), No. 4, 479–494
Abstract

The Black–Scholes equation has been used for modeling option pricing extensively. When the volatility of financial markets creates irregularities, the model equation is difficult to solve numerically; for this reason nonuniform grids are often used for greater accuracy. This paper studies the numerical consistency of popular explicit, implicit and leapfrog finite difference schemes for solving the Black–Scholes equation when nonuniform meshes are utilized. Mathematical tools including Taylor expansions are used throughout our analysis. The consistency ensures the basic reliability of the finite difference schemes based on choices of temporal and variable spatial derivative approximations. Truncation error terms are derived and discussed, and numerical experiments using C, C++ and Matlab are given to illustrate our discussions. We show that, though orders of accuracy are lower compared with their peers on uniform grids, nonuniform algorithms are easy to implement and use for turbulent financial markets.

Keywords
finite difference approximation, difference algorithm, explicity, implicity, consistency, accuracy, matrix equations
Mathematical Subject Classification 2000
Primary: 65G50, 65L12, 65N06, 65N15
Secondary: 65L70, 65L80, 65D25
Milestones
Received: 4 June 2009
Revised: 23 July 2009
Accepted: 27 July 2009
Published: 28 October 2009

Communicated by Johnny Henderson
Authors
Myles D. Baker
Department of Mathematics
Baylor University
Waco, TX 76798-7328
United States
http://www.baylor.edu/math/
Daniel D. Sheng
Westwood High School
Austin, TX 78750
United States
http://schools.roundrockisd.org/Westwood/