Title: Matrix-based numerical modelling of financial differential equations

Authors: Robert Piche, Juho Kanniainen

Addresses: Tampere University of Technology, P.O. Box 553, Tampere, FI-33101, Finland. ' Tampere University of Technology, P.O. Box 553, Tampere, FI 33101, Finland

Abstract: Differentiation matrices provide a compact and unified formulation for a variety of differential equation discretisation and time stepping algorithms. This paper illustrates their use for solving three differential equations of finance: the classic Black-Scholes equation (linear initial-boundary value problem), American option pricing problem (linear complementarity problem), and an optimal maintenance and shutdown model (non-linear boundary value problem with free boundary). We present numerical results that show the advantage of an L-stable time-stepping method over the Crank-Nicolson method, and results that show how spectral collocation methods are superior for boundary value problems with smooth solutions, while finite difference methods are superior for option-pricing problems.

Keywords: differentiation matrix; finite difference method; Chebyshev spectral collocation; method of lines; Runge-Kutta; option pricing; American options; optimal shutdown; optimal maintenance; financial differential equations.

DOI: 10.1504/IJMMNO.2009.030089

International Journal of Mathematical Modelling and Numerical Optimisation, 2009 Vol.1 No.1/2, pp.88 - 100

Published online: 09 Dec 2009 *

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