A Wiener–Hopf Monte Carlo simulation technique for Lévy processes

Abstract

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.

On the density of exponential functionals of Lévy processes

“…The process ξ with such characteristics is a killed Lamperti subordinator with parameters (1/Γ(1−a), 1+a−β, 1/a, a), see Section 3.2 in Kuznetsov et al [15] for a proper definition. From Theorem 1.3 the density of I eq satisfies the equation…”

Pricing of contingent convertibles under smile conform models

“…Indeed for this class of Lévy processes, the law of the infimum at an independent and exponentially distributed random time can be written down in terms of the roots and poles of its characteristic exponent; all of which are easily found within regularly spaced intervals along one of the axes of the complex plane. Combining these results together with a recently suggested Monte-Carlo technique, due to Kuznetsov et al [13], which capitalises on the randomised law of the infimum we show the efficient and effective numerical pricing of CoCos. We perform our analysis using a special class of β-processes, known as β-VG, which have similar characteristics to the classical Variance-Gamma model.…”

“…More precisely, it is possible to calculate in semi-closed form the joint distribution of the process and its running minimum (maximum) processes. Using Wiener-Hopf theory as described in Kuznetsov et al [13] allows us to simulate the process and its running minimum at a series of time points in a very efficient way. This ability of fast Monte-Carlo simulation makes the process extremely well suited to price CoCos under an equity derivative approach.…”

Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

“…These remarks are based on the representation of the WH factors for the continuous-monitoring case as double integrals [15,Chapter 11.3]. With reference to financial applications, attempts to compute the WH factors have been done by Boyarchenko and Levendorskii [8] and Kuznetsov et al [41], among others.…”