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Publication Detail
Hilbert transform, spectral filters and option pricing
  • Publication Type:
    Journal article
  • Publication Sub Type:
  • Authors:
    Phelan CE, Marazzina D, Fusai G, Germano G
  • Publisher:
    Springer Verlag
  • Publication date:
  • Journal:
    Annals of Operations Research
  • Article number:
    arXiv:1706.09755 [q-fin.CP]
  • Status:
    Published online
  • Print ISSN:
  • Language:
We show how spectral filtering techniques can improve the convergence of numerical schemes which use discrete Hilbert transforms based on a sinc function expansion, and thus ultimately on the fast Fourier transform. This is relevant, for example, for the computation of fluctuation identities, which give the distribution of the maximum or the minimum of a random path, or the joint distribution at maturity with the extrema staying below or above barriers. We use as examples the methods by Feng and Linetsky (2008) and Fusai, Germano and Marazzina (2016) to price discretely monitored barrier options where the underlying asset price is modelled by a L\'evy process. Both methods show exponential convergence with respect to the number of grid points in most cases, but are limited to polynomial convergence under certain conditions. We relate these rates of convergence to the Gibbs phenomenon for Fourier transforms and achieve improved results with spectral filtering.
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