Financial models with long-tailed distributions and volatility clustering

α-Stable distributions

A real-valued random variable ${\displaystyle X}$ is said to have an ${\displaystyle \alpha }$-stable distribution if for any ${\displaystyle n\ge 2}$, there are a positive number ${\displaystyle C_n}$ and a real number ${\displaystyle D_n}$ such that

${\displaystyle X_1+\cdots +X_n\stackrel{\mathrm{d}}{=}{C}_{n}X+D_n,}$

where ${\displaystyle X_1,X_2,\dots ,X_n}$ are independent and have the same distribution as that of ${\displaystyle X}$. All stable random variables are infinitely divisible. It is known that ${\displaystyle C_n=n^{1/\alpha }}$ for some ${\displaystyle 0<\alpha \le 2}$. A stable random variable ${\displaystyle X}$ with index ${\displaystyle \alpha }$ is called an ${\displaystyle \alpha }$-stable random variable.

Let ${\displaystyle X}$ be an ${\displaystyle \alpha }$-stable random variable. Then the characteristic function ${\displaystyle {\varphi }_X}$ of ${\displaystyle X}$ is given by

${\displaystyle {\varphi }_X(u)=\{\begin{array}{ll}\exp \left(i\mu u-{\sigma }^{\alpha }|u{|}^{\alpha }\left(1-i\beta \mathrm{sgn}(u)\tan \left(\frac{\pi \alpha }{2}\right)\right)\right)& \text{if }\alpha \in (0,1)\cup (1,2)\\ \exp \left(i\mu u-\sigma |u|\left(1+i\beta \mathrm{sgn}(u)\left(\frac{2}{\pi }\right)\ln (|u|)\right)\right)& \text{if }\alpha =1\\ \exp \left(i\mu u-\frac{1}{2}{\sigma }^2{u}^2\right)& \text{if }\alpha =2\end{array}}$

for some ${\displaystyle \mu \in \mathbb{R}}$, ${\displaystyle \sigma >0}$ and ${\displaystyle \beta \in [-1,1]}$.

Tempered stable distributions

An infinitely divisible distribution is called a classical tempered stable (CTS) distribution with parameter ${\displaystyle (C_1,C_2,{\lambda }_{+},{\lambda }_{-},\alpha )}$, if its Lévy triplet ${\displaystyle ({\sigma }^2,\nu ,\gamma )}$ is given by ${\displaystyle \sigma =0}$, ${\displaystyle \gamma \in \mathbb{R}}$ and

${\displaystyle \nu (dx)=\left(\frac{C_1{e}^{-{\lambda }_{+}x}}{x^{1+\alpha }}{1}_{x>0}+\frac{C_2{e}^{-{\lambda }_{-}|x|}}{|x{|}^{1+\alpha }}{1}_{x<0}\right)dx,}$

where ${\displaystyle C_1,C_2,{\lambda }_{+},{\lambda }_{-}>0}$ and ${\displaystyle \alpha <2}$.

This distribution was first introduced by under the name of Truncated Lévy Flights and 'exponentially truncated stable distribution'. It was subsequently called the tempered stable or the KoBoL distribution. In particular, if ${\displaystyle C_1=C_2=C>0}$, then this distribution is called the CGMY distribution.

The characteristic function ${\displaystyle {\varphi }_{CTS}}$ for a tempered stable distribution is given by

${\displaystyle {\varphi }_{CTS}(u)=\exp \left(iu\mu +C_1\mathrm{\Gamma }(-\alpha )(({\lambda }_{+}-iu{)}^{\alpha }-{\lambda }_{+}^{\alpha })+C_2\mathrm{\Gamma }(-\alpha )(({\lambda }_{-}+iu{)}^{\alpha }-{\lambda }_{-}^{\alpha })\right),}$

for some ${\displaystyle \mu \in \mathbb{R}}$. Moreover, ${\displaystyle {\varphi }_{CTS}}$ can be extended to the region ${\displaystyle \{z\in \mathbb{C}:\mathrm{Im}(z)\in (-{\lambda }_{-},{\lambda }_{+})\}}$.

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 ${\displaystyle (C,{\lambda }_{+},{\lambda }_{-},\alpha )}$, if its Lévy triplet ${\displaystyle ({\sigma }^2,\nu ,\gamma )}$ is given by ${\displaystyle \sigma =0}$, ${\displaystyle \gamma \in \mathbb{R}}$ and

${\displaystyle \nu (dx)=C\left(\frac{q_{\alpha }({\lambda }_{+}|x|)}{x^{\alpha +1}}{1}_{x>0}+\frac{q_{\alpha }({\lambda }_{-}|x|)}{|x{|}^{\alpha +1}}{1}_{x<0}\right)dx,}$

where ${\displaystyle C,{\lambda }_{+},{\lambda }_{-}>0,\alpha <2}$ and

${\displaystyle q_{\alpha }(x)=x^{\frac{\alpha +1}{2}}{K}_{\frac{\alpha +1}{2}}(x).}$

Here ${\displaystyle K_p(x)}$ 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.

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 (${\displaystyle {ϵ}_t}$) is assumed that ${\displaystyle {ϵ}_t={\sigma }_t{z}_t}$, where ${\displaystyle z_t\sim iidN(0,1)}$ and where the series ${\displaystyle {\sigma }_t^2}$ are modeled by

${\displaystyle {\sigma }_t^2={\alpha }_0+{\alpha }_1{ϵ}_{t-1}^2+\cdots +{\alpha }_q{ϵ}_{t-q}^2={\alpha }_0+\sum _{i=1}^q{\alpha }_i{ϵ}_{t-i}^2}$

and where ${\displaystyle {\alpha }_0>0}$ and ${\displaystyle {\alpha }_i\ge 0,i>0}$.

However, the assumption of ${\displaystyle z_t\sim iidN(0,1)}$ is often rejected empirically. For that reason, new GARCH models with stable or tempered stable distributed innovation have been developed. GARCH models with ${\displaystyle \alpha }$-stable innovations have been introduced. Subsequently, GARCH Models with tempered stable innovations have been developed.

Objections against the use of stable distributions in Financial models are given in