Carolyn E. Phelan, Daniele Marazzina, Gianluca Fusai, Guido Germano
We show how the convergence of numerical schemes which use discrete Hilbert transforms based on a sinc function expansion and thus ultimately on the fast Fourier transform can be improved with spectral filtering techniques. This is relevant e.g. in 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 a barrier. We use as examples the schemes by Feng and Linetsky (2008) and Fusai, Germano and Marazzina (2016) to price discretely monitored barrier options modelled with Lévy processes. Both methods show exponential convergence on the grid size in most cases but are limited to polynomial convergence under certain conditions. We relate these rates of convergence to the widely studied issue of the Gibbs phenomenon for Fourier transforms and achieve improved results with spectral filtering.