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.
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Notes
With reference to the notation in Fusai et al. (2016), \([1/\varPhi _-(\xi ,q)]_{l+}=[e^{-il\xi }/\varPhi _-(\xi ,q)]_{+}\), due to the shift theorem.
The online supplementary material which accompanies this paper demonstrates the robustness of the pricing algorithm with regards to the variation of parameters.
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The support of the Economic and Social Research Council (ESRC) in funding the Systemic Risk Centre (Grant Number ES/K002309/1) and of the Engineering and Physical Sciences Research Council (EPSRC) in funding the UK Centre for Doctoral Training in Financial Computing and Analytics (Grant Number 1482817) are gratefully acknowledged.
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Appendix A
Appendix A
Table 4 contains all the parameters used for the numerical experiments which produced the results presented in Sect. 4.
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Phelan, C.E., Marazzina, D., Fusai, G. et al. Hilbert transform, spectral filters and option pricing. Ann Oper Res 282, 273–298 (2019). https://doi.org/10.1007/s10479-018-2881-4
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DOI: https://doi.org/10.1007/s10479-018-2881-4