Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options

European Journal of Operational Research, Volume 251, Issue 1, 16 May 2016, Pages 124-134

30 Pages Posted: 20 May 2016

See all articles by Gianluca Fusai

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

Daniele Marazzina

Polytechnic University of Milan - Department of Mathematics

Date Written: May 18, 2016

Abstract

The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Lévy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Lévy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points.

Keywords: Path-dependent options, Hilbert transform, Lévy process, Spitzer identity, Wiener-Hopf factorization

Suggested Citation

Fusai, Gianluca and Germano, Guido and Marazzina, Daniele, Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options (May 18, 2016). European Journal of Operational Research, Volume 251, Issue 1, 16 May 2016, Pages 124-134. Available at SSRN: https://ssrn.com/abstract=2781434 or http://dx.doi.org/10.2139/ssrn.2781434

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

Daniele Marazzina (Contact Author)

Polytechnic University of Milan - Department of Mathematics ( email )

Via Bonardi, 9
Milano, MI 20133
Italy

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