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Numerical Analysis of Additive, Lévy and Feller Processes with Applications to Option Pricing

Part of the Lecture Notes in Mathematics book series (LEVY,volume 2001)

Abstract

We review the design and analysis of multiresolution (wavelet) methods for the numerical solution of the Kolmogorov equations arising, among others, in financial engineering when Lévy and Feller or additive processes are used to model the dynamics of the risky assets. In particular, the Dirichlet and free boundary problems connected to barrier and American style contracts are specified and solution algorithms based on wavelet representations of the Feller processes’ Dirichlet forms are presented. Feller processes with generators that give rise to Sobolev spaces of variable differentiation order (corresponding to a state-dependent jump intensity) are considered. A copula construction for the systematic construction of parametric multivariate Feller-Lévy processes from univariate ones is presented and the domains of the generators of the resulting multivariate Feller-Lévy processes is identified. New multiresolution norm equivalences in such Sobolev spaces allow for wavelet compression of the matrix representations of the Dirichlet forms. Implementational aspects, in particular the regularization of the process’ Dirichlet form and the singularity-free, fast numerical evaluation of moments of the Dirichlet form with respect to piecewise linear, continuous biorthogonal wavelet bases are addressed. Monte Carlo path simulation techniques for such processes by FFT and symbol localization are outlined. Numerical experiments illustrate multilevel preconditioning of the moment matrices for several exotic contracts as well as for Feller-Lévy processes with variable order jump intensities. Model sensitivity of Lévy models embedded into Feller classes is studied numerically for several types of plain vanilla, barrier and exotic contracts.

Keywords

  • Dirichlet forms
  • Feller processes
  • Kolmogorov equations
  • Lévy processes
  • Matrix compression
  • Option pricing
  • Sato processes
  • Wavelet discretization

AMS Subject Classification 2000

Primary: 45K05, 60J75, 65N30

Secondary: 45M60

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Reichmann, O., Schwab, C. (2010). Numerical Analysis of Additive, Lévy and Feller Processes with Applications to Option Pricing. In: Barndorff-Nielsen, O., Bertoin, J., Jacod, J., Klüppelberg, C. (eds) Lévy Matters I. Lecture Notes in Mathematics(), vol 2001. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14007-5_3

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