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Hilbert transform, spectral filters and option pricing

Phelan, C. E., Marazzina, D., Fusai, G. ORCID: 0000-0001-9215-2586 and Germano, G. (2018). Hilbert transform, spectral filters and option pricing. Annals of Operations Research, doi: 10.1007/s10479-018-2881-4

Abstract

We show how spectral filters 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 (Math Finance 18(3):337–384, 2008) and Fusai et al. (Eur J Oper Res 251(4):124–134, 2016) to price discretely monitored barrier options where the underlying asset price is modelled by an exponential Lévy 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.

Publication Type: Article
Additional Information: This is a post-peer-review, pre-copyedit version of an article published in Annals of Operations Research. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10479-018-2881-4
Publisher Keywords: double-barrier options, discrete monitoring, L´evy processes, Spitzer identity, Wiener-Hopf factorisation, Hilbert transform, Fourier transform, FFT, z-transform, sinc function, Gibbs phenomenon, spectral filtering
Subjects: Q Science > QA Mathematics
Departments: Cass Business School > Finance
URI: http://openaccess.city.ac.uk/id/eprint/20052
[img] Text - Accepted Version
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