Financial models with long-tailed distributions and volatility clustering have been introduced to overcome problems with the realism of classical financial models. These classical models of financial time series typically assume homoskedasticity and normality cannot explain stylized phenomena such as skewness, heavy tails, and volatility clustering of the empirical asset returns in finance. In 1963, Benoit Mandelbrot first used the stable (or
Contents
- Infinitely divisible distributions
- Stable distributions
- Tempered stable distributions
- Volatility clustering with stable and tempered stable innovation
- References
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 stable and tempered stable distributions together with volatility clustering have been developed and applied to risk management, option pricing, and portfolio selection.
Infinitely divisible distributions
A random variable
such that
where
A Borel measure
If
where
α-Stable distributions
An real-valued random variable
where
Let
for some
Tempered stable distributions
An infinitely divisible distribution is called a classical tempered stable (CTS) distribution with parameter
where
This distribution was first introduced by under the name of Truncated Lévy Flights and has been called the tempered stable or the KoBoL distribution. In particular, if
The characteristic function
for some
Rosiński [6] 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
where
Here
Volatility clustering with stable and tempered stable innovation
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 (
and where
However, the assumption of
Objections against the use of stable distributions in Financial models are given in