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Fast deterministic pricing of optionson Lévy driven assets

Published online by Cambridge University Press:  15 February 2004

Ana-Maria Matache ,
Tobias von Petersdorff  and
Christoph Schwab
Ana-Maria Matache
Affiliation:
RiskLab and Seminar for Applied Mathematics, ETH-Zentrum, 8092 Zürich, Switzerland.
Tobias von Petersdorff
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
Christoph Schwab
Affiliation:
Seminar for Applied Mathematics, ETH-Zentrum, 8092 Zürich, Switzerland. [email protected].
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Abstract

Arbitrage-free prices u of European contracts on risky assets whoselog-returns are modelled by Lévy processes satisfya parabolic partial integro-differential equation (PIDE) $\partial_t u + {\mathcal{A}}[u] = 0$ .This PIDE is localized tobounded domains and the error due to this localization isestimated. The localized PIDE is discretized by theθ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for ${\mathcal{A}}$ can be replaced by a sparse matrix in the wavelet basis, and the linear systemsin each implicit time step are solved approximativelywith GMRES in linear complexity. The total work of the algorithm for M time steps is bounded byO(MN(log(N))2) operations and O(Nlog(N)) memory.The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solutionin the same complexity as finite difference approximationsof the standard Black–Scholes equation.Computational examples for various Lévy price processes are presented.

Keywords

Parabolic partial integro-differential equationsLévy processesMarkov processesGalerkin finite element methodwaveletmatrix compressionGMRES.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 1 , January 2004 , pp. 37 - 71
Copyright
© EDP Sciences, SMAI, 2004

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