Risk management under time varying volatility and Pareto-stable distributions

ABSTRACT

Risk measures based on Gaussian return distributions are simple but inaccurate while such measures based on alternative methodologies are known to be more precise but complex. In this context, practitioners seem biased towards simplicity and tend to choose the inaccurate Gaussian measures, leading to unsuspected losses in the event of a negative episode. This article proposes generalized autoregressive conditional heteroskedasticity (GARCH) family models with stable Paretian innovations in measuring the value-at-risk, expected shortfall and spectral risk measures that promise a markedly improved performance while maintaining simplicity.

KEYWORDS:

• GARCH models
• stable processes
• risk measures
• stable Paretian distribution

JEL CLASSIFICATION:

• G15
• G10

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ This has been demonstrated by Mittnik, Paolella and Rachev (1998), Liu and Brorsen (1995), Panorska, Mittnik and Rachev (1995) and Mittnik and Paolella (2003).

² The S&P500 is an American stock market index based on the market capitalizations of 500 large companies having common stock listed on the NYSE or NASDAQ. The FTSE100 Index is a share index of the 100 companies listed on the London Stock Exchange with the highest market capitalization. These 100 companies represent around 80% of the entire market capitalization of the London Stock Exchange. The DAX is German a blue-chip stock market index consisting of the 30 major German companies trading on the Frankfurt Stock Exchange. DAX measures the performance of the 30 largest German companies in terms of order book volume and market capitalization.

³ Additional choices for risk aversion functions are discussed in Dowd and Cotter (2006).

⁴ Notice that ${\eta }_t$ iid $\sim S_{\alpha ,\beta }$. $S_{\alpha ,\beta }$ represents the standard asymmetric stable Paretian distribution with stable index $\alpha \in \left(0,2\right]$, skewness parameter $\beta \in \left[-1,1\right]$, zero location parameter and unit scale parameter. Following Samorodnitsky and Taqqu (1994) and Rachev and Mittnik (2000), the characteristic function corresponding to this model is given by

$\mathbb{E}\left(e^{itX}\right)=\left\{\begin{array}{c}exp\left\{it-{\left|t\right|}^{\alpha }\left[1-i\beta tan\frac{\pi \alpha }{2}sign\left(t\right)\right]\right\}if\alpha \ne 0\\ \\ exp\left\{it-\left|t\right|\left[1+i\beta \frac{2}{\pi }sign\left(t\right)\mathrm{l}\mathrm{n}\left|t\right|\right]\right\}if\alpha =0\end{array}\right.$

where the characterization yields a distribution which is symmetric about the zero location parameter if $\beta =0$ and the characteristic exponent α determines the total probability in the extreme tails of the distribution. The effective feature of this characterization is that as α decreases from 2 to 0, the tail areas of the stable distribution become increasingly fatter than the Normal. For $\alpha <2$, the kth absolute moment exists only for $k<\alpha$ and, except for the normal case, the stable Paretian distribution has infinite variance. Thus, the analogue of the variance in this kind of distributions is the scale parameter $h_t$. In this article, we assume that $\alpha >1$, the necessary condition for the existence of the mean.

⁵ We follow this approach because randomisation of the integral often fails to converge under numerical schemes such as one used by MATLAB function ‘quadl’.