Fast deterministic pricing of options on Lévy driven assets
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Date
2002-07
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Report
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yes
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Abstract
Arbitrage-free prices $u$ of European contracts on risky assets whose logreturns are modelled by Lévy processes satisfy a parabolic parabolic partial integrodifferential equation (PIDE) $\partial_t u + {\mathcal{A}}[u] = 0$. This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the $\theta$-scheme in time and a wavelet Galerkin method with $N$ degrees of freedom in log-price space. The dense matrix for $A$ can be replaced by a sparse matrix in the wavelet basis, and the linear systems in each implicit time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for $M$ time steps is bounded by $O (M N (log (N))^2) operations and $O(N log (N))$ memory. The deterministic algorithm gives optimal convergence rates ( up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black-Scholes equation. Computational examples for various Lévy price processes are presented.
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published
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Volume
2002-11
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Seminar for Applied Mathematics, ETH Zurich
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Subject
Option pricing; Lévy processes; partial integro-differential equation (PIDE); wavelet discretization
Organisational unit
02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
03435 - Schwab, Christoph / Schwab, Christoph