On the other hand, GARCH models have been developed to explain the volatility clustering. In the GARCH model, the innovation (or residual) distributions are assumed to be a standard normal distribution, despite the fact that this assumption is often rejected empirically. For this reason, GARCH models with non-normal innovation distribution have been developed.
Many financial models with the stable and the tempered stable distributions together with volatility clustering have been developed and applied to risk management, option pricing, and portfolio selection.
A random variable is called infinitely divisible if, for each , there are i.i.d random variables such that
where denotes equality in distribution.
A Borel measure on is called a Levy measure if and
If is infinitely divisible, then the characteristic function is given by
where and .
This distribution was first introduced by Koponen  under the name of Truncated Levy Flights and has been called the tempered stable or the KoBoL distribution . In particular, if , then this distribution is called the CGMY distribution which has been used in Carr et al.  for financial modeling.
The characteristic function for a tempered stable distribution is given by
for some . Moreover, can be extended to the
Rosiński  generalized the CTS distribution under the name of the tempered stable distribution. The KR distribution, which is a subclass of the Rosiński's generalized tempered stable distributions, is used in finance .
An infinitely divisible distribution is called a modified tempered stable (MTS) distribution with parameter , if its Levy triplet is given by , and where and . is the modified Bessel function of the second kind. The MTS distribution is not included in the class of Rosiński's generalized tempered stable distributions .
 S. I. Boyarchenko, S. Z. Levendorskiǐ, Option pricing for truncated Levy processes, International Journal of Theoretical and Applied Finance, 3 (2000), 3, pp. 549-552.
 P. Carr, H. Geman, D. Madan, M. Yor, The Fine Structure of Asset Returns: An Empirical Investigation, Journal of Business, 75 (2002), 2, pp. 305-332.
 Y.S. Kim, D. M. Chung, Svetlozar. T. Rachev, M. L. Bianchi, The modified tempered stable distribution, GARCH models and option pricing, Probability and Mathematical Statistics, to appear
 Y.S. Kim, Svetlozar. T. Rachev, M. L. Bianchi, Frank J. Fabozzi, A New Tempered Stable Distribution and Its Application to Finance, Georg Bol, Svetlozar T. Rachev, and Reinold Wuerth (Eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer 2007.
 I. Koponen, Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process, Physical Review E, 52 (1995), pp. 1197-1199.
 J. Rosiński, Tempering Stable Processes, Stochastic Processes and their Applications, 117 (2007), 6, pp. 677-707.
In order to describe the volatility clustering effect of the return process of an asset, the GARCH model can be used. In the GARCH model, innovation () is assumed that , where and where the series are modeled by
and where and .
However, the assumption of is often rejected empirically. For that reason, new GARCH models with stable or tempered stable distributed innovation have been developed. GARCH models with -stable innovations are introduced by Menn and Rachev [3,4,5]. Then the GARCH Models with tempered stable innovations have been developed in  and .
 Y.S. Kim, Svetlozar. T. Rachev, D.M. Chung, Michele L. Bianchi, The modified tempered stable distribution, GARCH models and option pricing, Probability and Mathematical Statistics, to appear.
 Y.S. Kim, Svetlozar. T. Rachev, Michele L. Bianchi, Frank J. Fabozzi, Financial market models with Levy processes and time-varying volatility Journal of Banking & Finance, 32, 7, 1363-1378, 2008.
 C. Menn, Svetlozar. T. Rachev, A GARCH Option Pricing Model with -stable Innovations, European Journal of Operational Research, 163, 201-209, 2005.
 C. Menn, Svetlozar. T. Rachev, Smoothly Truncated Stable Distributions, GARCH-Models, and Option Pricing, Technical report: Chair of Econometrics, Statistics and Math- ematical Finance School of Economics and Business Engineering University of Karlsruhe, 2005.
 Svetlozar. T. Rachev, C. Menn, Frank J. Fabozzi, Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio selection, and Option Pricing, John Wiley & Sons, 2005.