Financial models with long-tailed distributions and volatility clustering

Since Black and Scholes introduced the pricing and hedging theory for the option market, their model has been the most popular model for option pricing. However, the model which assumes homoskedasticity and lognormality, cannot explain stylized phenomena such as skewness, heavy tails, and volatility clustering of the stock returns, which are observed in stock prices. To explain the stylized phenomena, Mandelbrot was the first to use a the Levy table (or $\alpha$-stable) distribution to model the empirical distribution of asset prices as an asset price process. However, $\alpha$-stable distributions have infinite $p$-th moments for all $p>\alpha$. Tempered stable processes are presented for overcoming the limitation of the stable distribution.

Many scientists including S.T. Rachev have developed financial models with $\alpha$-stable and tempered stable distributions and applied them to market and credit risk management, option pricing, and portfolio selection as will as discussing the major attacks on the $\alpha$-stable models.

Infinitely divisible distributions

A random variable $Y$ is called infinitely divisible if, for each $n\in\N$, there are i.i.d random variables $Y_{n,1},Y_{n,2},\cdots ,Y_{n,n}$ such that

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

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

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

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

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

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

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

$\alpha$-Stable distributions

An real valued random variable $X$ is said to have a $\alpha$-stable distribution if for any $n\geq 2$, there are a positive number $C_{n}$ and a real number $D_{n}$ such that

$X_{1}+\cdots +X_{n}{\stackrel {\mathrm {d} }{=}}C_{n}X+D_{n},$

where $X_{1},X_{2},\cdots ,X_{n}$ are independent and have the same distribution as that of $X$.All stable random variables are infinitely divisible. It is known that $C_{n}=n^{1/\alpha }$ for some $0<\alpha \leq 2$. A stable random variable $X$ with index $\alpha$ is called $\alpha$-stable random variable.

Let $X$ be an $\alpha$-stable random variable. Then the characteristic function $\phi _{X}$ of $X$ is given by
$\phi _{X}(u)=\left\{{\begin{array}{ll}\exp \left(i\mu u-\sigma ^{\alpha }|u|^{\alpha }\left(1-i\beta {\text{sgn}}(u)\tan \left({\frac {\pi \alpha }{2}}\right)\right)\right)&{\text{if}}~~\alpha \in (0,1)\cup (1,2)\\\exp \left(i\mu u-\sigma |u|\left(1+i\beta {\text{sgn}}(u)\left({\frac {2}{\pi }}\right)\ln(|u|)\right)\right)&{\text{if}}~~\alpha =1\\\exp \left(i\mu u-{\frac {1}{2}}\sigma ^{2}u^{2}\right)&{\text{if}}~~\alpha =2\end{array}}\right.$
for some $\mu \in \mathbb {R}$, $\sigma >0$ and $\beta \in [-1,1]$.

References
G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Chapman \& Hall/CRC.

Tempered stable distributions

An infinitely divisible distribution is called a classical tempered stable (CTS) distribution with parameter $(C_{1},C_{2},\lambda _{+},\lambda _{-},\alpha )$, if its Levy triplet $(\sigma ^{2},\nu ,\gamma )$ is given by $\sigma =0$, $\gamma \in \mathbb {R}$ and $\nu (dx)=\left({\frac {C_{1}e^{-\lambda _{+}x}}{x^{1+\alpha }}}1_{x>0}+{\frac {C_{2}e^{-\lambda _{-}|x|}}{|x|^{1+\alpha }}}1_{x<0}\right)dx,$ where $C_{1},C_{2},\lambda _{+},\lambda _{-}>0$ and $\alpha <2$.

This distribution was first introduced by Koponen [5] under the name of Truncated Levy Flights and has been called the tempered stable or the KoBoL distribution [1]. In particular, if $C_{1}=C_{2}=C>0$, then this distribution is called the CGMY distribution which has been used in Carr et al. [2] for financial modeling.

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

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

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

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 (See [4]).

An infinitely divisible distribution is called a modified tempered stable (MTS) distribution with parameter $(C,\lambda _{+},\lambda _{-},\alpha )$, if its Levy triplet $(\sigma ^{2},\nu ,\gamma )$ is given by $\sigma =0$, $\gamma \in \mathbb {R}$ and $\nu (dx)=C({\frac {q_{\alpha }(\lambda _{+}|x|)}{x^{\alpha +1}}}1_{x>0}+{\frac {q_{\alpha }(\lambda _{-}|x|)}{|x|^{\alpha +1}}}1_{x<0})dx,$ where $C,\lambda _{+},\lambda _{-}>0,\alpha <2$ and $q_{\alpha }(x)=x^{\frac {\alpha +1}{2}}K_{\frac {\alpha +1}{2}}(x).$ $K_{p}(x)$ 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. (See [3])

References
[1] 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.
[2] 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.
[3] Y. S. Kim, D. M. Chung and S. T. Rachev, The modified tempered stable distribution, GARCH models and option pricing, Probability and Mathematical Statistic, to appear
[4] Y. S. Kim, D. M. Chung and S. T. Rachev, F. 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. [5] 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.
[6] J. Rosiński, Tempering Stable Processes, Stochastic Processes and their Applications, 117 (2007), 6, pp. 677--707.

See also

Financial Application

Volatility clustering with stable and tempered stable innovation : GARCH(1,1) model

References

Y.S. Kim, Svetlozar. T. Rachev, D.M. Chung, Michele L. Bianchi: A Modifed Tempered Stable Distribution with Volatility Clustering, 2008
Y.S. Kim, Svetlozar. T. Rachev, Michele L. Bianchi, Frank J. Fabozzi: "Financial Market Models with Levy Processes and Time-Varying Volatility", 2007

See Also

Aleksander Weron
Svetlozar. T. Rachev
G. Samorodnitsky