Financial models with long-tailed distributions and volatility clustering

A random variable ${\displaystyle Y}$ is called infinitely divisible if, for each ${\displaystyle n=1,2,\dots }$, there are independent and identically-distributed random variables

${\displaystyle Y_{n,1},Y_{n,2},\dots ,Y_{n,n}\,}$

such that

${\displaystyle Y{\stackrel {\mathrm {d} }{=}}\sum _{k=1}^{n}Y_{n,k},\,}$

where ${\displaystyle {\stackrel {\mathrm {d} }{=}}}$ denotes equality in distribution.

A Borel measure ${\displaystyle \nu }$ on ${\displaystyle \mathbb {R} }$ is called a Lévy measure if ${\displaystyle \nu ({0})=0}$ and

${\displaystyle \int _{\mathbb {R} }(1\wedge |x^{2}|)\,\nu (dx)<\infty .}$

If ${\displaystyle Y}$ is infinitely divisible, then the characteristic function ${\displaystyle \phi _{Y}(u)=E[e^{iuY}]}$ is given by

${\displaystyle \phi _{Y}(u)=\exp \left(i\gamma u-{\frac {1}{2}}\sigma ^{2}u^{2}+\int _{-\infty }^{\infty }(e^{iux}-1-iux1_{|x|\leq 1})\,\nu (dx)\right),\sigma \geq 0,~~\gamma \in \mathbb {R} }$

where ${\displaystyle \sigma \geq 0}$, ${\displaystyle \gamma \in \mathbb {R} }$ and ${\displaystyle \nu }$ is a Lévy measure. Here the triple ${\displaystyle (\sigma ^{2},\nu ,\gamma )}$ is called a Lévy triplet of ${\displaystyle Y}$. This triplet is unique. Conversely, for any choice ${\displaystyle (\sigma ^{2},\nu ,\gamma )}$ satisfying the conditions above, there exists an infinitely divisible random variable ${\displaystyle Y}$ whose characteristic function is given as ${\displaystyle \phi _{Y}}$.

A real-valued random variable ${\displaystyle X}$ is said to have an ${\displaystyle \alpha }$-stable distribution if for any ${\displaystyle n\geq 2}$, 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 ${\displaystyle 0<\alpha \leq 2}$. 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 ${\displaystyle \phi _{X}}$ of ${\displaystyle X}$ is given by

${\displaystyle \phi _{X}(u)={\begin{cases}\exp \left(i\mu u-\sigma ^{\alpha }|u|^{\alpha }\left(1-i\beta \operatorname {sgn} (u)\tan \left({\frac {\pi \alpha }{2}}\right)\right)\right)&{\text{if }}\alpha \in (0,1)\cup (1,2)\\\exp \left(i\mu u-\sigma |u|\left(1+i\beta \operatorname {sgn} (u)\left({\frac {2}{\pi }}\right)\ln(|u|)\right)\right)&{\text{if }}\alpha =1\\\exp \left(i\mu u-{\frac {1}{2}}\sigma ^{2}u^{2}\right)&{\text{if }}\alpha =2\end{cases}}}$

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

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[1] and has been called the tempered stable or the KoBoL distribution.[2] In particular, if ${\displaystyle C_{1}=C_{2}=C>0}$, then this distribution is called the CGMY distribution which has been used for financial modeling.[3]

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

${\displaystyle \phi _{CTS}(u)=\exp \left(iu\mu +C_{1}\Gamma (-\alpha )((\lambda _{+}-iu)^{\alpha }-\lambda _{+}^{\alpha })+C_{2}\Gamma (-\alpha )((\lambda _{-}+iu)^{\alpha }-\lambda _{-}^{\alpha })\right),}$

for some ${\displaystyle \mu \in \mathbb {R} }$. Moreover, ${\displaystyle \phi _{CTS}}$ can be extended to the region ${\displaystyle \{z\in \mathbb {C}$ :\operatorname {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.[4]

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.[5]

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 (${\displaystyle ~\epsilon _{t}~}$) is assumed that ${\displaystyle ~\epsilon _{t}=\sigma _{t}z_{t}~}$, where ${\displaystyle z_{t}\sim iid~N(0,1)}$ and where the series ${\displaystyle \sigma _{t}^{2}}$ are modeled by

${\displaystyle \sigma _{t}^{2}=\alpha _{0}+\alpha _{1}\epsilon _{t-1}^{2}+\cdots +\alpha _{q}\epsilon _{t-q}^{2}=\alpha _{0}+\sum _{i=1}^{q}\alpha _{i}\epsilon _{t-i}^{2}}$

and where ${\displaystyle ~\alpha _{0}>0~}$ and ${\displaystyle \alpha _{i}\geq 0,~i>0}$.

However, the assumption of ${\displaystyle z_{t}\sim iid~N(0,1)}$ 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.[6][7][8] Subsequently, GARCH Models with tempered stable innovations have been developed.[5][9]

Objections against the use of stable distributions in Financial models are given in [10][11]

• B. B. Mandelbrot (1963) "New Methods in Statistical Economics", Journal of Political Economy, 71, 421-440
• Svetlozar T. Rachev, Stefan Mittnik (2000) Stable Paretian Models in Finance, Wiley
• G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall/CRC.
• S. I. Boyarchenko, S. Z. Levendorskiǐ (2000) "Option pricing for truncated Lévy processes", International Journal of Theoretical and Applied Finance, 3 (3), 549–552.
• J. Rosiński (2007) "Tempering Stable Processes", Stochastic Processes and their Applications, 117 (6), 677–707.