Fast deterministic pricing of options on Lévy driven assets

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

ESAIM: Mathematical Modelling and Numerical Analysis , 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) ${\partial }_{t}u+𝒜\left[u\right]=0$. 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 $O\left(MN{(log\left(N\right))}^2\right)$ operations and $O(Nlog(N\left)\right)$ 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.

Détail

DOI : 10.1051/m2an:2004003

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