Simulation of Tempered Stable Lévy Bridges and Its Applications

Abstract

We consider tempered stable Lévy subordinators and develop a bridge sampling method. An approximate conditional probability density function (PDF) given the terminal values is derived with stable index less than one, using the double saddlepoint approximation. We then propose an acceptance-rejection algorithm based on the existing gamma bridge and the inverse Gaussian bridge as proposal densities. Its performance is comparable to existing sequential sampling methods such as Devroye (2009) [Devroye L (2009) Random variate generation for exponentially and ploynomially tilted stable distributions. ACM Trans. Modeling Comput. Simulation 19(4):18:1–20.] and Hofert (2011) [Hofert M (2011) Sampling exponentially tilted stable distributions. ACM Trans. Modeling Comput. Simulation 22(1):3:1–11.] when generating a fixed number of observations. As applications, we consider option pricing problems in Lévy models. First, we demonstrate the effectiveness of bridge sampling when combined with adaptive sampling under finite-variance CGMY processes. Second, further efficiency gain is achieved in terms of variance reduction via stratified sampling.