Hilbert Transform, Spectral Filtering and Option Pricing

30 Pages Posted: 4 Jul 2017

See all articles by Carolyn Phelan

Carolyn Phelan

University College London - Financial Computing and Analytics Group, Department of Computer Science

Daniele Marazzina

Polytechnic University of Milan - Department of Mathematics

Gianluca Fusai

Università del Piemonte Orientale Dipartimento di Studi per l'Economia e l'Impresa; Sir John Cass Business School - City, University of London

Guido Germano

University College London

Date Written: June 29, 2017

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.

Keywords: Double-Barrier Options, Discrete Monitoring, Lévy Processes, Spitzer Identity, Wiener-Hopf Factorisation, Hilbert Transform, Fourier Transform, FFT, Z-Transform, Sinc Function, Gibbs Phenomenon, Spectral Filtering

JEL Classification: G12, C02

Suggested Citation

Phelan, Carolyn and Marazzina, Daniele and Fusai, Gianluca and Germano, Guido, Hilbert Transform, Spectral Filtering and Option Pricing (June 29, 2017). Available at SSRN: https://ssrn.com/abstract=2995391 or http://dx.doi.org/10.2139/ssrn.2995391

Carolyn Phelan (Contact Author)

University College London - Financial Computing and Analytics Group, Department of Computer Science ( email )

Gower Street
London, WC1E 6BT
United Kingdom

Daniele Marazzina

Polytechnic University of Milan - Department of Mathematics ( email )

Via Bonardi, 9
Milano, MI 20133
Italy

Gianluca Fusai

Università del Piemonte Orientale Dipartimento di Studi per l'Economia e l'Impresa ( email )

Via Perrone, 18
Novara, 28100
Italy

HOME PAGE: http://https://upobook.uniupo.it/gianluca.fusai

Sir John Cass Business School - City, University of London ( email )

106 Bunhill Row
London, EC2Y 8HB
Great Britain

HOME PAGE: http:// www.cass.city.ac.uk/experts/G.Fusai

Guido Germano

University College London ( email )

Department of Computer Science
Gower Street
London, WC1E 6BT
United Kingdom
+44 20 3108 7105 (Phone)
+44 20 7387 1397 (Fax)

HOME PAGE: http://www.cs.ucl.ac.uk/people/G.Germano

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