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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) \(\partial_tu+{\mathcal A}[u]=0\). 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 \(\mathcal A\) 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:
60H30 Applications of stochastic analysis (to PDEs, etc.)
60G51 Processes with independent increments; Lévy processes
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J75 Jump processes (MSC2010)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65T60 Numerical methods for wavelets
91G20 Derivative securities (option pricing, hedging, etc.)
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