Abstracts

A Lévy bridge is a process conditioning a Lévy process on its endpoints, generalizing the notion of a Brownian bridge. Diffusion bridge and squared Bessel bridge are well-known. A gamma bridge (C. Ribero and N. Webber. Valuing Path Dependent Options in the Variance-Gamma Model by Monte Carlo with a Gamma bridge. Journal of Computational Finance, 7(2):81–100, 2004) and an inverse Gaussian bridge (C. Ribero and N. Webber. A Monte Carlo Method for the NIG Option Valuation Model using an Inverse Gaussian Bridge. Working paper, 2003) are examples of Lévy bridges, based on their closed forms of PDFs. However, since the PDF of a Lévy process is hardly known, only few approximation schemes have been developed, e.g., (P. Glasserman and K. Kim. Beta Approximations for Bridge Sampling. Proceedings of the 2008 Winter Simulation Conference, 2008).
In this work, we consider a tempered stable Lévy subordinator, which is an exponentially tempered version of a stable process, with a stable index less than one. The purpose of this paper is to investigate the bridge distribution of the process, which we call a tempered stable Lévy bridge, and also provide an efficient simulation method. The tempered stable Lévy bridge contains gamma and inverse Gaussian bridge as its special case, and this is in fact the same as the bridge of a stable process.
An approximate conditional PDF, that is, the PDF of the tempered stable Lévy bridge, is derived using the double saddle-point approximation. This not only provides the exact PDFs of gamma and inverse Gaussian bridges, but also shows high accuracy in approximation over a wide range of parameters. Then an acceptance-rejection algorithm is proposed using the PDFs of the known bridges as its proposal densities, depending on a stable index of the tempered stable process.
The proposed sampling method is applied to the pricing of path-dependent options under subordinated Brownian motion models. We illustrate accuracy and bias on numerical examples for Asian, look-back, and barrier options, compared to the existing sampling method. Moreover, the difference-gamma bridge sampling method in the paper (A. N. Avramidis and P. L’Ecuyer. Efficient Monte Carlo and Quasi-Monte Carlo Option Pricing Under the Variance Gamma Model. Management Science, 52(12):1930–1944, 2006) is extended to two sided tempered stable processes, e.g., finite variation CGMY processes.