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Anisotropic stable Lévy copula processes - Analytical and numerical aspects

by W. Farkas and N. Reich and Ch. Schwab

(Report number 2006-08)

Abstract
We consider the valuation of derivative contracts on baskets of risky assets whose prices are Lévy like Feller processes of tempered stable type. The dependence among the marginals' jump structure is parametrized by a Lévy copula. For marginals of regular, exponential L\'{e}vy type in the sense of [6] we show that the infinitesimal generator ${\cal A}$ of the resulting Lévy copula process is a pseudo-differential operator whose principal symbol is a distribution of anisotropic homogeneity. We analyze the jump measure of the corresponding Lévy copula processes. We prove the domains of their infinitesimal generators ${\cal A}$ are certain anisotropic Sobolev spaces. In these spaces and for a large class of L\'{e}vy copula processes, we prove a G\aa rding inequality for ${\cal A}$. We design a wavelet-based dimension-independent tensor product discretization for the efficient numerical solution of the parabolic Kolmogoroff equation $u_t+{\cal A}u = 0$ arising in valuation of derivative contracts under possibly stopped Lévy copula processes. In the wavelet basis diagonal preconditioning yields a bounded condition number of the resulting matrices.

Keywords: Lévy-copula, Lévy processes, Pseudo-differential Operators, Dirichlet Forms, Wavelet Finite Element Methods, Option Pricing

@Techreport{FRS06_357,
  author = {W. Farkas and N. Reich and Ch. Schwab},
  title = {Anisotropic stable Lévy copula processes - Analytical and numerical aspects},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2006-08},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2006/2006-08.pdf },
  year = {2006}
}

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