Fast deterministic pricing of options on Lévy driven assets
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 1, p. 37-71
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) ${\partial }_{t}u+𝒜\left[u\right]=0$. 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 $O\left(MN{\left(log\left(N\right)\right)}^{2}\right)$ operations and $O\left(Nlog\left(N\right)\right)$ 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.
DOI : https://doi.org/10.1051/m2an:2004003
Classification:  65N30,  60J75
@article{M2AN_2004__38_1_37_0,
author = {Matache, Ana-Maria and Petersdorff, Tobias Von and Schwab, Christoph},
title = {Fast deterministic pricing of options on L\'evy driven assets},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {38},
number = {1},
year = {2004},
pages = {37-71},
doi = {10.1051/m2an:2004003},
zbl = {1072.60052},
mrnumber = {2073930},
language = {en},
url = {http://www.numdam.org/item/M2AN_2004__38_1_37_0}
}

Matache, Ana-Maria; Petersdorff, Tobias Von; Schwab, Christoph. Fast deterministic pricing of options on Lévy driven assets. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 1, pp. 37-71. doi : 10.1051/m2an:2004003. http://www.numdam.org/item/M2AN_2004__38_1_37_0/

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