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Convergence of a high-order compact finite difference scheme for a nonlinear Black–Scholes equation

Published online by Cambridge University Press:  15 March 2004

Bertram Düring ,
Michel Fournié  and
Ansgar Jüngel
Bertram Düring
Affiliation:
Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universität Mainz, Germany, duering@uni-mainz.de.
Michel Fournié
Affiliation:
UMR-CNRS 5640, Laboratoire MIP, Université Paul Sabatier, Toulouse, France.
Ansgar Jüngel
Affiliation:
Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universität Mainz, Germany, duering@uni-mainz.de.
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Abstract

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.

Keywords

High-order compact finite differencesnumerical convergenceviscosity solutionfinancial derivatives.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 2 , March 2004 , pp. 359 - 369
Copyright
© EDP Sciences, SMAI, 2004

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