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Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs

Published online by Cambridge University Press:  12 June 2009

Rafael Company ,
Lucas Jódar  and
José-Ramón Pintos
Rafael Company
Affiliation:
Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Edificio 8G, piso 2, P.O. Box 46022, Valencia, Spain. rcompany@imm.upv.es; ljodar@imm.upv.es; jrpt60@gmail.com
Lucas Jódar
Affiliation:
Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Edificio 8G, piso 2, P.O. Box 46022, Valencia, Spain. rcompany@imm.upv.es; ljodar@imm.upv.es; jrpt60@gmail.com
José-Ramón Pintos
Affiliation:
Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Edificio 8G, piso 2, P.O. Box 46022, Valencia, Spain. rcompany@imm.upv.es; ljodar@imm.upv.es; jrpt60@gmail.com
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Abstract

This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical approximation of the Gamma of the option, previously obtained. Results are illustrated with numerical examples.

Keywords

Nonlinear Black-Scholes equationoption pricingnumerical analysistransaction costs.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 6 , November 2009 , pp. 1045 - 1061
Copyright
© EDP Sciences, SMAI, 2009

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