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 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) and Hofert (2011) 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.

Nonparametric Bayesian inference for Gamma-type Lévy subordinators

“…In order to extend the inferential approach presented in the current work to this new model, we need to be able to sample from the distribution of X conditional on X T = x T . The problem of sampling from tempered stable bridges has been recently studied in Kim et al (2016). Let us also mention the fact that the problem of estimating the stability index α is difficult from a Bayesian point of view due to singularity of the measures induced by two Lévy processes with different stability indices.…”

Sequential sampling for CGMY processes via decomposition of their time changes

“…The random number generation in sampling D N τ (F R ) is otherwise quite straightforward. The stopping rule (20) leads to the following distance bound which is exact rather than in the L 1 sense:…”

“…In pricing such path‐dependent options as lookback and barrier options, Kim and Kim () have recently developed a bridge sampling scheme (although only for the finite‐variation case) that can lead to savings of simulation costs when combined with adaptive sampling techniques and to variance reduction when combined with stratified sampling techniques. This bridge sampling method is based upon saddle‐point approximations for the related probability density functions and is otherwise comparable in costs and accuracy to the existing rejection sampling method when generating a fixed number of observations.…”

Exact Simulation for a Class of Tempered Stable and Related Distributions

“…The resulting tails are lighter than those of the stable but heavier than those of the Gaussian, so it is more suitable for modelling financial returns. Moreover, for those distributions that are obtained by tilting one-sided stable distributions 2 with ζ = 1, namely the positive TS distributions, they can be further adopted as the building blocks to construct various stochastic processes (such as Lévy TS subordinators and processes) that facilitate numerous applications in finance and many other 1 Here, we adopt the naming for the TS distribution similarly as Küchler and Tappe (2013) and Kim and Kim (2016). In fact, the class of TS distributions has been recently enlarged by Rosiński (2007) and Grabchak et al (2012) in a variety of specifications, so we consider them as generalised TS distributions.…”