Fast deterministic pricing of options on Lévy driven assets
Ana-Maria Matache1, Tobias von Petersdorff2 and Christoph Schwab3
Seminar for Applied Mathematics,
ETH-Zentrum, 8092 Zürich, Switzerland.
2 Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
3 Seminar for Applied Mathematics, ETH-Zentrum, 8092 Zürich, Switzerland. firstname.lastname@example.org.
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) . 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 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(Nlog(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.
Mathematics Subject Classification: 65N30 / 60J75
Key words: Parabolic partial integro-differential equations / Lévy processes / Markov processes / Galerkin finite element method / wavelet / matrix compression / GMRES.
© EDP Sciences, SMAI, 2004