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Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem

Published online by Cambridge University Press:  03 June 2015

Germán I. Ramírez-Espinoza  and
Matthias Ehrhardt
Germán I. Ramírez-Espinoza*
Affiliation:
EON Global Commodities, Holzstraβe 6, 40221 Düsseldorf, Germany
Matthias Ehrhardt*
Affiliation:
Lehrstuhl für Angewandte Mathematik und Numerische Analysis, Fachbereich C – Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gauβstr. 20, 42119 Wuppertal, Germany
*
Corresponding author. Email: german.rmz@gmail.com
Email: ehrhardt@math.uni-wuppertal.de
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Abstract

This work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Péclet number-is high. For Asian options, additional similar problems arise when the “spatial” variable, the stock price, is close to zero.

Here we focus on three methods: the exponentially fitted scheme, a modification of Wang’s finite volume method specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme for a general convection-diffusion equation, that is applied for the first time to option pricing problems. Special emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence. For the reduction technique proposed by Wilmott, a put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options. Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.

Keywords

65M1091B25Black-Scholes equationconvection-dominated caseexponential fitting methodsfitted finite volume methodKurganov-Tadmor schememinmod limiter
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 6 , December 2013 , pp. 759 - 790
Copyright
Copyright © Global-Science Press 2013

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