Modelling tail risk with tempered stable distributions: an overview

Abstract

In this study, we investigate the performance of different parametric models with stable and tempered stable distributions for capturing the tail behaviour of log-returns (financial asset returns). First, we define and discuss the properties of stable and tempered stable random variables. We then show how to estimate their parameters and simulate them based on their characteristic functions. Finally, as an illustration, we conduct an empirical analysis to explore the performance of different models representing the distributions of log-returns for the S&P500 and DAX indexes.

Appendices

Appendix A: Lévy measure and characteristic function

In this section, we describe the Lévy measure and characteristic function for the stable and different classes of the tempered stable processes.

• The Lévy measure and characteristic function of a stable random variable $X_{S}$ are given respectively by

  $$\begin{aligned} \nu (dx)&=\left( \frac{C_+}{x^{1+\alpha }}1_{x>0} + \frac{C_-}{x^{1+\alpha }}1_{x<0}\right) dx.\\ \Phi (u;S)&=\mathbb {E}[e^{iuX_S}]\\&=\left\{ \begin{array}{ll} \exp \left( i\mu u - |\sigma u|^{\alpha }\left( 1- i\beta (\text {sign} ~ u)\tan \left( \frac{\pi \alpha }{2}\right) \right) \right) , &{}\quad \alpha \ne 1 \\ \exp \left( i\mu u - \sigma |u|\left( 1+ i\beta \frac{2}{\pi } (\text {sign} ~ u)\ln |u| \right) \right) , &{}\quad \alpha = 1 \end{array} \right. \end{aligned}$$

  where

  $$\begin{aligned} \text {sign} ~u = \left\{ \begin{array}{ll} 1,&{}\quad u>0\\ 0,&{} \quad u=0\\ -1,&{}\quad u<0, \end{array} \right. \end{aligned}$$

  $0<\alpha \le 2$, $\sigma \ge 0$, $-1\ge \beta \ge 1$, and $\mu \in \mathbb {R}$.

• The Lévy measure and characteristic function of a Classical Tempered Stable (CTS) random variable $X_{CTS}$ are given respectively by

  $$\begin{aligned} \nu (dx)&=\left( C\frac{e^{-\lambda _+ x}}{x^{1+\alpha }}1_{x>0} + C\frac{e^{-\lambda _- x}}{x^{1+\alpha }}1_{x<0}\right) dx.\\ \Phi (u;X_{CTS})&= {\mathbb {E}}[e^{iuX_{CTS}}] \\&= \int _{\mathbb {R}}e^{iux}f_X(x)dx\\&= exp\left( -iuC\Gamma (1-\alpha )(\lambda _+^{\alpha - 1} - \lambda _-^{\alpha - 1})\right. \\&\quad + C\Gamma (-\alpha )\left( (\lambda _+ - iu)^{\alpha } - \lambda _+^{\alpha } + (\lambda _- + iu)^{\alpha } - \lambda _-^{\alpha } \right) \Big ). \end{aligned}$$

• The Lévy measure and characteristic function of a Generalised Tempered Stable (GCTS) random variable are given respectively by

  $$\begin{aligned} \nu (dx)&=\left( C_+\frac{e^{-\lambda _+ x}}{x^{1+\alpha }}1_{x>0} + C_-\frac{e^{-\lambda _- x}}{x^{1+\alpha }}1_{x<0}\right) dx.\\ \Phi (u;X_{GCTS})&= {\mathbb {E}}[e^{iuX_{GCTS}}] \\&= \int _{\mathbb {R}}e^{iux}f_X(x)dx\\&= exp\left( -iu\Gamma (1-\alpha _+)(C_+\lambda _+^{\alpha _+- 1} - C_-\lambda _-^{\alpha _- - 1} )\right. \\&\quad + C_+\Gamma (-\alpha _+)\left( (\lambda _+ - iu)^{\alpha _+} - \lambda _+^{\alpha _+} )\right) \\&\quad + C_-\Gamma (-\alpha _-)\left( (\lambda _- + iu)^{\alpha _-} - \lambda _-^{\alpha _-} )\right) \end{aligned}$$

  where $\alpha _+, \alpha _-\in (0,1) \cup (1,2)$, $C_+, C_-, \lambda _+, \text {and}, \lambda _->0$.

• The Lévy measure and characteristic function of a Modified Tempered Stable (MTS)s random variable are given respectively by

  $$\begin{aligned} \nu (dx)&=C\left( \frac{\lambda _+^{\frac{\alpha +1}{2}}K_{\frac{\alpha +1}{2}}(\lambda _+x)}{x^{\frac{\alpha +1}{2}}}1_{x>0}+ \frac{\lambda _-^{\frac{\alpha +1}{2}}K_{\frac{\alpha +1}{2}}(\lambda _-x)}{x^{\frac{\alpha +1}{2}}}1_{x<0}\right) dx.\\ \Phi (u;X_{MTS})&= {\mathbb {E}}[e^{iuX_{MTS}}] \\&= \int _{\mathbb {R}}e^{iux}f_X(x)dx\\&= exp\Big (G_R(u;\alpha ,C,\lambda _+,\lambda _-)+G_I(u;\alpha ,C,\lambda _+,\lambda _-)\Big ) \end{aligned}$$

  where for $u\in \mathbb {R}$

  $$\begin{aligned} G_R(u;\alpha ,C,\lambda _+,\lambda _-)=\sqrt{\pi }2^{-\frac{\alpha }{2}-\frac{3}{2}}C\Gamma (-\frac{\alpha }{2})\left( (\lambda _+^2+u^2)^{\frac{\alpha }{2}} - \lambda _+^\alpha + (\lambda _-^2+u^2)^{\frac{\alpha }{2}}-\alpha _-^\alpha \right) \end{aligned}$$

  and

  $$\begin{aligned} G_I(u;\alpha ,C,\lambda _+,\lambda _-)&=\frac{iuC\Gamma (\frac{1-\alpha }{2})}{2^{\frac{\alpha +1}{2}}}\left( \lambda _+^{\alpha -1}F\left( 1,\frac{1-\alpha }{2};\frac{3}{2};-\frac{u^2}{\lambda _+^2}\right) \right. \\&\left. \quad - \lambda _-^{\alpha -1}F\left( 1,\frac{1-\alpha }{2};\frac{3}{2};-\frac{u^2}{\lambda _-^2}\right) \right) \end{aligned}$$

  where F is the hypergeometric function.⁴⁰

• The Lévy measure and characteristic function of a Rapidly Decreasing Tempered Stable (RDTS random) variable are given respectively by

  $$\begin{aligned} \nu (dx)&=\left( C\frac{e^{-\frac{\lambda _+^2x^2}{2}}}{x^{\alpha +1}}1_{x>0}+C\frac{e^{-\frac{\lambda _-^2x^2}{2}}}{x^{\alpha +1}}1_{|x|<0}\right) dx.\\ \Phi (u;X_{RDTS})&= {\mathbb {E}}[e^{iuX_{RDTS}}] \\&= \int _{\mathbb {R}}e^{iux}f_X(x)dx\\&= exp\Big (C(G(iu;\alpha ,\lambda _+) + G(-iu;\alpha ,\lambda _-))\Big ) \end{aligned}$$

  where

  $$\begin{aligned} G(x;\alpha ,\lambda )&=2^{-\frac{\alpha }{2}-1}\lambda ^{\alpha }\Gamma \left( -\frac{\alpha }{2} \right) \left( M\left( -\frac{\alpha }{2},\frac{1}{2};\frac{x^2}{2\lambda ^2}\right) -1 \right) \\&\quad +2^{-\frac{\alpha }{2}-\frac{1}{2}}\lambda ^{\alpha -1}\Gamma \left( \frac{1-\alpha }{2} \right) \left( M\left( \frac{1-\alpha }{2},\frac{3}{2};\frac{x^2}{2\lambda ^2}\right) -1 \right) \end{aligned}$$

  and M is the confluent hypergeometric function.

• The Lévy measure and characteristic function of a Normal tempered stable (NTS random) variable are given respectively by

  $$\begin{aligned} \nu (dx)&=\left( \dfrac{\sqrt{2}\theta ^{1-\frac{\alpha }{2}}(\beta ^2+2\gamma \lambda )^{\frac{\alpha +1}{4}}}{\sqrt{\pi }\gamma \Gamma (1-\frac{\alpha }{2})}\exp \left( \dfrac{x\gamma }{\gamma ^2}\right) \dfrac{K_{\frac{\alpha +1}{2}}\left( \frac{|x|\sqrt{\beta ^2+2\sigma ^2\lambda }}{\gamma ^2}\right) }{|x|^{\frac{\alpha +1}{2}}} \right) dx.\\ \Phi (u;X_t)&= exp\left( iu(\mu - \beta )t -\frac{2\theta ^{1-\frac{\alpha }{2}}}{\alpha }\Big (\big (\theta - i\beta u +\frac{\sigma ^2u^2}{2}\big )^{\frac{\alpha }{2}} - \theta ^{\frac{\alpha }{2}}\Big )t\right) . \end{aligned}$$

• The characteristic function of an exponential tilting stable (ETS random) variable is given by

  $$\begin{aligned} \Phi (u;X_t) = exp\left( iu(\mu - \beta )t -\frac{2\theta ^{1-\frac{\alpha }{2}}}{\alpha }\Big (\big (\theta - i\beta u +\frac{\sigma ^2u^2}{2}\big )^{\frac{\alpha }{2}} - \theta ^{\frac{\alpha }{2}}\Big )t\right) . \end{aligned}$$

  The Lévy measure of the ETS random variable is not available in a closed-form (see, Fallahgoul et al. 2018b).

Appendix B: Modified Bessel function of the second kind

The modified Bessel function is the solution of a differential equation called modified Bessel’s equation. It is given by

$$\begin{aligned} K_{\nu }(z)=\left( \dfrac{\pi }{2}\right) \dfrac{I_{-\nu }(z) - I_{\nu }(z)}{\sin (\nu \pi )} \end{aligned}$$

where $I_{-\nu }(z)$ and $I_{\nu }(z)$ are the fundamental solutions of the modifies Bessel’s equation. $I_{\nu }(z)$ is given by

$$\begin{aligned} I_{\nu }(z) =\left( \dfrac{z}{2}\right) ^{\nu }\sum _{k=0}^{\infty }\dfrac{\left( \dfrac{z^2}{4}\right) ^{k}}{k\!\Gamma (\nu +k+1)} \end{aligned}$$

where $\Gamma (a)$ is the Gamma function.