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| '''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 [[normal distribution|normality]] cannot explain stylized phenomena such as [[skewness]], [[heavy tails]], and [[volatility clustering]] of the empirical asset returns in finance. In 1963, [[Benoît Mandelbrot|Benoit Mandelbrot]] first used the [[Stable distribution|stable (or <math>\alpha</math>-stable) distribution]] to model the empirical distributions which have the skewness and heavy-tail property. Since <math>\alpha</math>-stable distributions have infinite <math>p</math>-th moments for all <math>p>\alpha</math>, the tempered stable processes have been proposed for overcoming this limitation of the stable distribution. | | You mіght have mɑde the dеcision you're going to take steps relating to your nouгishment. It is a amazing decision. Regrettably, there's ɑ lot to discoѵer. Don't fear, the diet suggeѕtions found on this page might Һelp. The tіps fߋrward will be going to acqսire going on a diet pathway which will make you feel much healthier.<br><br> |
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| 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.
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| 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.
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| == Infinitely divisible distributions ==
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| A random variable <math>Y</math> is called ''[[infinite divisibility (probability)|infinitely divisible]]'' if,
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| for each <math>n=1,2,\dots</math>, there are [[independent and identically-distributed random variables]] | |
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| : <math>Y_{n,1}, Y_{n,2}, \dots, Y_{n,n} \, </math> | |
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| such that
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| : <math>Y\stackrel{\mathrm{d}}{=}\sum_{k=1}^n Y_{n,k}, \, </math>
| |
| | |
| where <math>\stackrel{\mathrm{d}}{=}</math> denotes equality in distribution.
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| A [[Borel measure]] <math>\nu</math> on <math>\mathbb{R}</math> is called a ''Lévy measure'' if <math>\nu({0})=0</math> and
| |
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| : <math>\int_\mathbb{R}(1\wedge|x^2|) \, \nu(dx) < \infty . </math>
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| If <math>Y</math> is infinitely divisible, then the [[Characteristic function (probability theory)|characteristic function]]
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| <math>\phi_Y(u)=E[e^{iuY}]</math> is given by
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| : <math>
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| \phi_Y(u) =\exp \left( i\gamma u- \frac{1}{2} \sigma^2 u^2 +
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| \int_{-\infty}^\infty
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| (e^{iux}-1-iux1_{|x|\le 1} ) \, \nu(dx) \right),
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| \sigma\ge0,~~\gamma\in\mathbb{R}
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| </math> | |
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| where <math>\sigma\ge0</math>, <math>\gamma\in\mathbb{R}</math> and <math>\nu</math> is a Lévy measure.
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| Here the triple <math>(\sigma^2, \nu, \gamma)</math> is called a ''Lévy triplet of'' <math>Y</math>. This triplet is unique. Conversely, for any choice <math>(\sigma^2, \nu, \gamma)</math> satisfying the conditions above, there exists an infinitely divisible random variable <math>Y</math> whose characteristic function is given as <math>\phi_Y</math>.
| |
| | |
| == ''α''-Stable distributions ==
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| An real-valued random variable <math>X</math> is said to have an
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| ''<math>\alpha</math>-stable distribution'' if for any <math>n\ge 2</math>, there
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| are a positive number <math>C_n</math> and a real number <math>D_n</math> such that
| |
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| : <math>
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| X_1+ \cdots + X_n \stackrel{\mathrm{d}}{=} C_n X + D_n, \,
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| </math>
| |
| | |
| where <math>X_1, X_2, \dots, X_n</math> are independent and have the same
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| distribution as that of <math>X</math>. All stable random variables are infinitely divisible. It is known that <math>C_n=n^{1/\alpha}</math> for some <math>0<\alpha\le 2</math>. A stable random
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| variable <math>X</math> with index <math>\alpha</math> is called an
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| ''<math>\alpha</math>-stable random variable''.
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| Let <math>X</math> be an <math>\alpha</math>-stable random variable. Then the
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| characteristic function <math>\phi_X</math> of <math>X</math> is given by
| |
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| : <math>
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| \phi_X(u) =
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| \begin{cases}
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| \exp\left( i\mu u - \sigma^\alpha |u|^\alpha \left( 1-i\beta
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| \operatorname{sgn}(u) \tan\left(
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| \frac{\pi\alpha}{2}\right)\right)\right)
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| & \text{if } \alpha \in(0,1)\cup(1,2) \\
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| \exp\left( i\mu u - \sigma |u| \left( 1+i\beta \operatorname{sgn}(u)
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| \left(
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| \frac{2}{\pi}\right)\ln(|u|)\right)\right)
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| & \text{if } \alpha = 1 \\
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| \exp\left( i\mu u - \frac{1}{2} \sigma^2 u^2\right)
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| & \text{if }\alpha = 2
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| \end{cases}
| |
| </math>
| |
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| for some <math>\mu\in\mathbb{R}</math>, <math>\sigma>0</math> and <math>\beta\in[-1,1]</math>. | |
| | |
| == Tempered stable distributions==
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| An infinitely divisible distribution is called a ''classical tempered
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| stable (CTS) distribution'' with parameter
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| <math>(C_1,C_2,\lambda_+,\lambda_-,\alpha)</math>,
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| if its Lévy triplet <math>(\sigma^2, \nu, \gamma)</math> is given by
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| <math>\sigma=0</math>, <math>\gamma\in\mathbb{R}</math> and
| |
| :<math>\nu(dx) = \left( \frac{C_1e^{-\lambda_+x}}{x^{1+\alpha}}1_{x>0} + | |
| \frac{C_2e^{-\lambda_-|x|}}{|x|^{1+\alpha}}1_{x<0}\right) \, dx,
| |
| </math>
| |
| where <math>C_1, C_2, \lambda_+, \lambda_->0</math> and <math>\alpha<2</math>.
| |
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| This distribution was first introduced by under
| |
| the name of ''Truncated Lévy Flights''<ref name=sb5/> and has been called the ''tempered stable'' or the ''KoBoL'' distribution.<ref name=sb1/> In particular, if
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| <math>C_1=C_2=C>0</math>, then this distribution is called the CGMY
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| distribution which has been used for
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| financial modeling.<ref name=sb2/>
| |
| | |
| The characteristic function <math>\phi_{CTS}</math> for a tempered stable
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| distribution is given by
| |
| | |
| : <math>\phi_{CTS}(u) = \exp\left( iu\mu
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| +C_1\Gamma(-\alpha)((\lambda_+-iu)^\alpha-\lambda_+^\alpha)
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| +C_2\Gamma(-\alpha)((\lambda_-+iu)^\alpha-\lambda_-^\alpha)
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| \right),</math>
| |
| | |
| for some <math>\mu\in\mathbb{R}</math>. Moreover, <math>\phi_{CTS}</math> can be extended to the
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| region <math>\{z\in\mathbb{C}: \operatorname{Im}(z)\in(-\lambda_-,\lambda_+)\}</math>.
| |
| | |
| 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.<ref name=sb4/>
| |
| | |
| An infinitely divisible distribution is called a ''modified tempered stable (MTS) distribution'' with parameter <math>(C,\lambda_+,\lambda_-,\alpha)</math>,
| |
| if its Lévy triplet <math>(\sigma^2, \nu, \gamma)</math> is given by
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| <math>\sigma=0</math>, <math>\gamma\in\mathbb{R}</math> and
| |
| :<math>\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,
| |
| </math>
| |
| where <math>C, \lambda_+, \lambda_->0, \alpha<2</math> and
| |
| :<math>q_\alpha(x)=x^{\frac{\alpha+1}{2}}K_{\frac{\alpha+1}{2}}(x).</math>
| |
| Here <math>K_p(x)</math> 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.<ref name=sb3/>
| |
| | |
| == 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 (<math> ~\epsilon_t~ </math>) is assumed that <math> ~\epsilon_t=\sigma_t z_t ~</math>, where
| |
| <math> z_t\sim iid~ N(0,1) </math> and where
| |
| the series <math> \sigma_t^2 </math> are modeled by
| |
| | |
| : <math> \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 </math>
| |
| | |
| and where <math> ~\alpha_0>0~ </math> and <math> \alpha_i\ge 0,~i>0</math>. | |
| | |
| However, the assumption of <math> z_t\sim iid~ N(0,1) </math> is often rejected empirically. For that reason, new GARCH models with stable or tempered stable distributed innovation have been developed. GARCH models with <math>\alpha</math>-stable innovations have been introduced.<ref name=sc3/><ref name=sc4/><ref name=sc5/> Subsequently, GARCH Models with tempered stable innovations have been developed.<ref name=sb3/><ref name=sc2/>
| |
| | |
| ==Notes==
| |
| {{reflist |refs=
| |
| <ref name=sb1>S. I. Boyarchenko, S. Z. Levendorskiǐ (2000) "Option pricing for truncated Lévy processes", ''International Journal of Theoretical and Applied Finance'', 3 (3), 549–552</ref>
| |
| | |
| <ref name=sb2>P. Carr, H. Geman, D. Madan, M. Yor (2002) "The Fine Structure of Asset Returns: An Empirical Investigation", ''Journal of Business'', 75 (2), 305–332.</ref>
| |
| | |
| <ref name=sb3>Kim, Y.S., Chung, D. M. , Rachev, Svetlozar. T.; M. L. Bianchi, The modified tempered stable distribution, GARCH models and option pricing, ''Probability and Mathematical Statistics'', to appear</ref>
| |
| | |
| <ref name=sb4>Kim, Y.S.; Rachev, Svetlozar. T.;, Bianchi, M.L.; [[Frank J. Fabozzi|Fabozzi, F.J.]] (2007) "A New Tempered Stable Distribution and Its Application to Finance". In: Georg Bol, Svetlozar T. Rachev, and Reinold Wuerth (Eds.), ''Risk Assessment: Decisions in Banking and Finance'', Physika Verlag, Springer</ref> | |
| | |
| <ref name=sb5>Koponen, I. (1995) "Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process", ''Physical Review E'', 52, 1197–1199.</ref>
| |
| | |
| <ref name=sc2> Kim, Y.S.; Rachev, Svetlozar. T.; Michele L. Bianchi, [[Frank J. Fabozzi|Fabozzi, F.J.]] (2008) "Financial market models with Lévy processes and time-varying volatility", ''Journal of Banking & Finance'', 32 (7), 1363–1378 {{doi|10.1016/j.jbankfin.2007.11.004 }}</ref> | |
| | |
| <ref name=sc3>C. Menn, Svetlozar. T. Rachev (2005) "A GARCH Option Pricing Model with <math>\alpha</math>-stable Innovations", ''European Journal of Operational Research'', 163, 201–209</ref> | |
| | |
| <ref name=sc4>C. Menn, Svetlozar. T. Rachev (2005) [http://www.statistik.uni-karlsruhe.de/download/tr_smoothly_truncated.pdf "Smoothly Truncated Stable Distributions, GARCH-Models, and Option Pricing"], Technical report. Statistics and Mathematical Finance School of Economics and Business Engineering, University of Karlsruh</ref>
| |
| | |
| <ref name=sc5>Svetlozar. T. Rachev, C. Menn, [[Frank J. Fabozzi]] (2005) ''Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio selection, and Option Pricing'', Wiley</ref> | |
| | |
| }}
| |
| | |
| ==References==
| |
| * B. B. Mandelbrot (1963) "New Methods in Statistical Economics", ''Journal of Political Economy'', 71, 421-440
| |
| * Svetlozar. T. Rachev, S. Mitnik (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.
| |
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| [[Category:Actuarial science]]
| |
| [[Category:Stochastic models]]
| |
| [[Category:Mathematical finance]]
| |
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