Fast deterministic pricing of options on Lévy driven assets

Matache, Ana-Maria; Petersdorff, Tobias Von; Schwab, Christoph

doi:10.1051/m2an:2004003

Matache, Ana-Maria ; Petersdorff, Tobias Von ; Schwab, Christoph

ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 37-71.

Résumé

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) $\partial_{t} u + 𝒜 [u] = 0$ . This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the $θ$ -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.

MR Zbl | 3 citations dans Numdam

DOI : 10.1051/m2an:2004003

Classification : 65N30, 60J75
Mots-clés : parabolic partial integro-differential equations, Lévy processes, Markov processes, Galerkin finite element method, wavelet, matrix compression, GMRES

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     author = {Matache, Ana-Maria and Petersdorff, Tobias Von and Schwab, Christoph},
     title = {Fast deterministic pricing of options on {L\'evy} driven assets},
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     pages = {37--71},
     publisher = {EDP-Sciences},
     volume = {38},
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     mrnumber = {2073930},
     zbl = {1072.60052},
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Matache, Ana-Maria; Petersdorff, Tobias Von; Schwab, Christoph. Fast deterministic pricing of options on Lévy driven assets. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 37-71. doi : 10.1051/m2an:2004003. http://www.numdam.org/articles/10.1051/m2an:2004003/

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