Hilbert transform, spectral filters and option pricing

Abstract

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 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.

Fluctuation Identities with Continuous Monitoring and Their Application to Price Barrier Options

“…(18). The latter can be calculated directly, while so far only an iterative solution has been found (Fusai et al, 2016;Phelan et al, 2017) to the coupled Eqs. (20) and (21).…”

“…In Section 4 we show numerical results comparing the error convergence obtained using the Spitzer identities for continuous monitoring with the performance of the closely related method using the Spitzer identities for discrete monitoring (Green et al, 2010;Fusai et al, 2016;Phelan et al, 2017).…”

“…We describe the pricing procedure for single-barrier down-and-out options as an example, but the use of the Spitzer identities is equally applicable to other types of barrier options and also to lookback options; the pricing formulae described by Green et al (2010) include methods for single-barrier up-and-out and knock-in options. The pricing method is adapted from the scheme by Fusai et al (2016) and Phelan et al (2017) using the relationship between Φ(ξ, q) and Φ c (ξ, s) described in Section 2.1.1.…”

“…To improve the error of the decomposition in the continuously monitored case we can improve the slope of the function on the input to the Hilbert transform by using a spectral filter. We use an exponential filter which has previously achieved good results in option pricing applications (Ruijter et al, 2015;Phelan et al, 2017). The filter is described by Eq.…”

“…This method follows the approach suggested by Green et al (2010) and is based on the Fusai, Germano and Marazzina (FGM) method (Fusai et al, 2016) with spectral filtering (Phelan et al, 2017). While the latter is for discrete monitoring and thus in the Fourier-z domain, here we operate in the Fourier-Laplace domain.…”