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Fast deterministic pricing of options on Lévy driven assets. (English) Zbl 1072.60052
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) ${\partial }_{t}u+𝒜\left[u\right]=0$. 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.
##### MSC:
 60H30 Applications of stochastic analysis 60G51 Processes with independent increments; Lévy processes 60H15 Stochastic partial differential equations 60J75 Jump processes 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 65T60 Wavelets (numerical methods) 91B28 Finance etc. (MSC2000)