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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: [email protected]
∗∗ Postal address: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email address: [email protected]
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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.

References

Asmussen, S. (2000). Ruin Probabilities. World Scientific, River Edge, NJ.Google Scholar
Avram, F., Palmowski, Z. and Pistorius, M. R. (2014). On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk-process in the presence of a penalty function. Preprint. Available at http://uk.arxiv.org/abs/1110.4965.Google Scholar
Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68.Google Scholar
Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
Broadie, M., Glasserman, P. and Kou, S. G. (1999). Connecting discrete and continuous path-dependent options. Finance Stoch. 3, 55–82.Google Scholar
Carr, P. (1998). Randomization and the American put. Rev. Financial Studies 11, 597–626.Google Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation. J. Business 75, 305–333.Google Scholar
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Chen, Z., Feng, L. and Lin, X. (2012). Simulating Lévy processes from their characteristic functions and financial applications. ACM Trans. Model. Comput. Simul. 22, 26 pp.Google Scholar
Doney, R. A. (2004). Stochastic bounds for Lévy processes. Ann. Prob. 32, 1545–1552.Google Scholar
Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16, 91–106.Google Scholar
Ferreiro-Castilla, A. and Schoutens, W. (2012). The β-Meixner model. J. Comput. Appl. Math. 236, 2466–2476.Google Scholar
Ferreiro-Castilla, A., Kyprianou, A. E., Scheichl, R. and Suryanarayana, G. (2014). Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorization. Stoch. Process. Appl. 124, 985–1010.Google Scholar
Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Operat. Res. 56, 607–617.CrossRefGoogle Scholar
Glasserman, P. and Liu, Z. (2010). Sensitivity estimates from characteristic functions. Operat. Res. 58, 1611–1623.Google Scholar
Hubalek, F. and Kyprianou, A. E. (2011). Old and new examples of scale functions for spectrally negative Lévy processes. In Seminar on Stochastic Analysis, Random Fields and Applications, Birkhäuser, Basel, pp. 119–145.Google Scholar
Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801.Google Scholar
Kuznetsov, A. (2010). Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Prob. 20, 1801–1830.Google Scholar
Kuznetsov, A. (2010). Wiener–Hopf factorization for a family of Lévy processes related to theta functions. J. Appl. Prob. 47, 1023–1033.Google Scholar
Kuznetsov, A., Kyprianou, A. E. and Pardo, J. C. (2012). Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Prob. 22, 1101–1135.Google Scholar
Kuznetsov, A., Kyprianou, A. E., Pardo, J. C. and van Schaik, K. (2011). A Wiener–Hopf Monte Carlo simulation technique for Lévy processes. Ann. Appl. Prob. 21, 2171–2190.Google Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Kyprianou, A. E., Pardo, J. C. and Rivero, V. (2010). Exact and asymptotic n-tuple laws at first and last passage. Ann. Appl. Prob. 20, 522–564.Google Scholar
Lundberg, F. (1903). Approximerad Framställning av Sannolikhetsfunktionen. Återförsäkring av Kollektivrisker. Almqvist. och Wiksell, Uppsala.Google Scholar
Merton, R. C. (1976). Option pricing when the underlying stock returns are discontinuous. J. Financial Econom. 3, 125–144.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Schoutens, W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives. John Wiley, Chichester.Google Scholar
Schoutens, W. and Cariboni, J. (2009). Lévy Processes in Credit Risk. John Wiley, Chichester.Google Scholar
Schoutens, W. and van Damme, G. (2011). The β-variance gamma model. Rev. Deriv. Res. 14, 263–282.Google Scholar
Song, R. and Vondraček, Z. (2008). On suprema of Lévy processes and application in risk theory. Ann. Inst. H. Poincaré Prob. Statist. 44, 977–986.Google Scholar
Vigon, V. (2002). Simplifiez vos Lévy en titillant la factorisation de Wiener–Hopf. , Laboratoire de Mathématiques de L'INSA de Rouen.Google Scholar