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Zbl 1072.60052
Matache, Ana-Maria; von Petersdorff, Tobias; Schwab, Christoph
Fast deterministic pricing of options on Lévy driven assets.
(English)
[J] M2AN, Math. Model. Numer. Anal. 38, No. 1, 37-71 (2004). ISSN 0764-583X; ISSN 1290-3841/e

Summary: Arbitrage-free prices $u$ of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) $\partial_tu+{\cal A}[u]=0$. 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 $\cal A$ 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 $O(MN(\log(N))^2)$ 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.
MSC 2000:
*60H30 Appl. of stochastic analysis
60G51 Processes with independent increments
60H15 Stochastic partial differential equations
60J75 Jump processes
65M60 Finite numerical methods (IVP of PDE)
65T60 Wavelets
91B28 Finance etc.

Keywords: arbitrage-free prices; European contracts on risky assets; parabolic partial integro-differential equation; Galerkin method; Lévy price processes

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