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.

• Legal notice:
  

  Copyright (c) Pielaszek Research, all rights reserved.
  The above materials, including auxiliary resources, are subject to Publisher's copyright and the Author(s) intellectual rights. Without limiting Author(s) rights under respective Copyright Transfer Agreement, no part of the above documents may be reproduced without the express written permission of Pielaszek Research, the Publisher. Express permission from the Author(s) is required to use the above materials for academic purposes, such as lectures or scientific presentations.
  In every case, proper references including Author(s) name(s) and URL of this webpage: https://science24.com/paper/32427 must be provided.

1. Consistency of local-stochastic volatility models in the FX market with respect to spot inversion and multiplication
2. Stability of calibration procedures: fractals in the Black-Scholes model