The Annals of Applied Probability

A Wiener–Hopf Monte Carlo simulation technique for Lévy processes

A. Kuznetsov, A. E. Kyprianou, J. C. Pardo, and K. van Schaik

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We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general Lévy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr’s so-called “Canadization” technique as well as Doney’s method of stochastic bounds for Lévy processes; see Carr [Rev. Fin. Studies 11 (1998) 597–626] and Doney [Ann. Probab. 32 (2004) 1545–1552]. We rely fundamentally on the Wiener–Hopf decomposition for Lévy processes as well as taking advantage of recent developments in factorization techniques of the latter theory due to Vigon [Simplifiez vos Lévy en titillant la factorization de Wiener–Hopf (2002) Laboratoire de Mathématiques de L’INSA de Rouen] and Kuznetsov [Ann. Appl. Probab. 20 (2010) 1801–1830]. We illustrate our Wiener–Hopf Monte Carlo method on a number of different processes, including a new family of Lévy processes called hypergeometric Lévy processes. Moreover, we illustrate the robustness of working with a Wiener–Hopf decomposition with two extensions. The first extension shows that if one can successfully simulate for a given Lévy processes then one can successfully simulate for any independent sum of the latter process and a compound Poisson process. The second extension illustrates how one may produce a straightforward approximation for simulating the two-sided exit problem.

Article information

Ann. Appl. Probab., Volume 21, Number 6 (2011), 2171-2190.

First available in Project Euclid: 23 November 2011

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Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods 68U20: Simulation [See also 65Cxx]

Lévy processes exotic option pricing Wiener–Hopf factorization


Kuznetsov, A.; Kyprianou, A. E.; Pardo, J. C.; van Schaik, K. A Wiener–Hopf Monte Carlo simulation technique for Lévy processes. Ann. Appl. Probab. 21 (2011), no. 6, 2171--2190. doi:10.1214/10-AAP746.

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  • [1] Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Probab. 15 2062–2080.
  • [2] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [3] Bertoin, J. (1997). Regularity of the half-line for Lévy processes. Bull. Sci. Math. 121 345–354.
  • [4] Boyarchenko, S. I. and Levendorskii, S. Z. (2002). Non-Gaussian Merton–Black–Scholes Theory. Advanced Series on Statistical Science & Applied Probability 9. World Scientific, River Edge, NJ.
  • [5] Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2011). Explicit identities for Lévy processes associated to symmetric stable processes. Bernoulli 17 34–59.
  • [6] Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2010). On the Lamperti stable processes. Probab. Math. Statist. 30 1–28.
  • [7] Carr, P. (1998). Randomization and the American Put. Rev. Fin. Studies 11 597–626.
  • [8] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.
  • [9] Doney, R. A. (2004). Stochastic bounds for Lévy processes. Ann. Probab. 32 1545–1552.
  • [10] Hubalek, F. and Kyprianou, A. E. (2010). Old and new examples of scale functions for spectrally negative Lévy processes. In Sixth Seminar on Stochastic Analysis, Random Fields and Applications (R. Dalang, M. Dozzi and F. Russo, eds.). Progress in Probability 63 119–145. Birkhäuser, Basel.
  • [11] Jeffrey, A., ed. (2007). Table of Integrals, Series, and Products, 7th ed. Academic Press, Amsterdam.
  • [12] Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 1766–1801.
  • [13] Kuznetsov, A. (2010). Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Probab. 20 1801–1830.
  • [14] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • [15] Kyprianou, A. E. and Loeffen, R. (2005). Lévy processes in finance distinguished by their coarse and fine path properties. In Exotic Option Pricing and Advanced Lévy Models (A. E. Kyprianou, W. Schoutens and P. Willmott, eds.) 1–28. Wiley, Chichester.
  • [16] Kyprianou, A. E., Pardo, J. C. and Rivero, V. (2010). Exact and asymptotic n-tuple laws at first and last passage. Ann. Appl. Probab. 20 522–564.
  • [17] Kyprianou, A. E. and Rivero, V. (2008). Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 13 1672–1701.
  • [18] Madan, D. and Seneta, E. (1990). The variance gamma (V.G.) model for share market returns. Journal of Business 63 511–524.
  • [19] Schoutens, W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives. Wiley, Chichester.
  • [20] Schoutens, W. and Cariboni, J. (2009). Lévy Processes in Credit Risk. Wiley, Chichester.
  • [21] Song, R. and Vondraček, Z. (2008). On suprema of Lévy processes and application in risk theory. Ann. Inst. H. Poincaré Probab. Statist. 44 977–986.
  • [22] Vigon, V. (2002). Simplifiez vos Lévy en titillant la factorisation de Wiener–Hopf. Thèse. Laboratoire de Mathématiques de L’INSA de Rouen.