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Fast deterministic pricing of options on Lévy driven assets

by A. M. Matache and T. von Petersdorff and Ch. Schwab

(Report number 2002-11)

Abstract
A partial integro-differential equation (PIDE) $\partial_t u + {\mathcal{A}}[u] = 0$ for European contracts on assets with general jump-diffusion price process of Lévy type is derived. The 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 space. The full Galerkin matrix for ${\mathcal{A}}$ can be replaced with a sparse matrix in the wavelet basis, and the linear systems for each time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by $O(MN (ln N)^2)$ operations and $O(N ln(N))$ 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 (VG, CGMY) are presented.

Keywords: Option pricing, Lévy processes, partial integro-differential equation (PIDE), wavelet discretization

@Techreport{MvS02_297,
  author = {A. M. Matache and T. von Petersdorff and Ch. Schwab},
  title = {Fast deterministic pricing of options on Lévy driven assets},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2002-11},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2002/2002-11.pdf },
  year = {2002}
}

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