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Applying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy Processes to Path Functionals

Part of: Stochastic processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  30 January 2018

Albert Ferreiro-Castilla  and
Kees van Schaik
Albert Ferreiro-Castilla*
Affiliation:
University of Bath
Kees van Schaik*
Affiliation:
University of Manchester
*
∗ Postal address: Direcció D'Inversions en Accions, Banc Sabadall, Carrer del Sena, 12, Sant Cugat del Vallès 08174, Spain. Email address: aferreiro.c@gmail.com
∗∗ Postal address: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email address: kees.vanschaik@manchester.ac.uk
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Abstract

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In this paper we apply the recently established Wiener-Hopf Monte Carlo simulation technique for Lévy processes from Kuznetsov et al. (2011) to path functionals; in particular, first passage times, overshoots, undershoots, and the last maximum before the passage time. Such functionals have many applications, for instance, in finance (the pricing of exotic options in a Lévy model) and insurance (ruin time, debt at ruin, and related quantities for a Lévy insurance risk process). The technique works for any Lévy process whose running infimum and supremum evaluated at an independent exponential time can be sampled from. This includes classic examples such as stable processes, subclasses of spectrally one-sided Lévy processes, and large new families such as meromorphic Lévy processes. Finally, we present some examples. A particular aspect that is illustrated is that the Wiener-Hopf Monte Carlo simulation technique (provided that it applies) performs much better at approximating first passage times than a ‘plain’ Monte Carlo simulation technique based on sampling increments of the Lévy process.

Keywords

Wiener-Hopf decompositionMonte Carlo simulationmultilevel Monte CarloLévy processexotic option pricingfirst passage timeovershootinsurance risk process

MSC classification

Primary: 65C05: Monte Carlo methods 68U20: Simulation 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 129 - 148
Copyright
© Applied Probability Trust 

Footnotes

Partially supported by a Royal Society Newton International Fellowship.

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