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(2011 / vol.13 / no.1)
THE TEMPERED STABLE MODELS
DONG MYUNG CHUNG
Pages. 83-93   



In recent years the tempered stable processes have been popular models for asset returns because they can explain the observed reality of financial market in more accurate way than the Brownian motion model. Such processes are pure jump Levy processes of which the Levy density is obtained by multiplying the Levy density of the stable processes with appropriate tempering functions. In this article, we will focus on two different tempered stable models for asset returns: the exponentially tempered stable (ETS) model and the modified tempered stable (MTS) model. The first one is obtained by using a exponential tempering function and the second one is defined by using a Bessel tempering function.

We first briefly review some results on 6-parametric ETS process, and then introduce a new 6-parametric family of MTS processes, which contain VG processes as a special subclass. We secondly show that the class of MTS processes is totally disjoint with that of ETS processes, but share many nice structural and analytical properties with the ETS processes. We thirdly compare the MTS processes with the ETS processes in terms of the decay rate of jump sizes. We finally discuss that every pure jump processes with infinity activity (in particular MTS process) can be approximated as a jump-diffusion processes.



1. Introduction
2. Infinitely divisible distributions and Levy processes
3. The Exponentially Tempered Stable (ETS) Model
4. The Modified Tempered Stable (MTS) Model
5. Comparison with the CGMY process
6. Approximation of small jumps by Brownian motion
7. Conclusion