Value-at-Risk formula when using skewed-t distribution

Question

I am trying to find a formula for the skewed-t VaR. For example the VaR formula for a t-distribution is

$$ \sqrt{\frac{df-2}{df}} \times \Sigma{t} \times \mbox{quantitle}(t-\mbox{dist}, 0.01) + \mu $$

(Please excuse the messy formula & the sigma(t) denotes a GARCH model)

However I am struggling to do the same for a skewed-t distribution

I am using the rugarch package in R and I am struggling to find out which version of the skewed-t distribution is being used. I went to the fGarch pdf and downloaded the reference ON BAYESIAN MODELLING OF FAT TAILS AND SKEWNESS by C. Fernandez et al., but my lack of Bayesian knowledge means the pdf it says is the Skew-Student is not helping perhaps as much as it should.

Any help would be much appreciated.

Answer 1

The answer can be found in the following paper (section 2.3 Distribution and quantile functions of a skewed distribution):

Lambert and Laurent, 2002 Lambert, P., Laurent, S., 2002. Modeling skewness dynamics in series of financial data using skewed location-scale distributions. Working Paper, Université Catholique de Louvain and Université de Liège.

And I summarize it in the following :

The quantile function $skst_{\alpha,v,\xi}$ of a non standardized skewed-Student density (Fernandez and Steel (1998)) is given by : $$ skst_{\alpha,v,\xi}= \frac{1}{\xi}st_{\alpha,v}\left[ \frac{\alpha}{2}(1+\xi^2)\right] \qquad \text{if} \qquad \alpha < \frac{1}{1+\xi^2} $$

or :

$$ skst_{\alpha,v,\xi}= -\xi st_{\alpha,v}\left[ \frac{1-\alpha}{2}(1+\xi^{-2})\right] \qquad \text{if} \qquad \alpha \geq \frac{1}{1+\xi^2} $$

where $\xi$ is the asymmetry coefficient, $v$ the degree of freedom and $\alpha $ is the quantile probability. $st_{\alpha,v}$ is the quantile function of the (unit variance) Student-t density.

Then the VaR for long position is given by $μ_{t} + > skst_{\alpha,v,\xi}\sigma_{t}$ with $skst_{\alpha,v,\xi}$ being the left quantile at $\alpha$% for the skewed-Student distribution with $v$ degrees of freedom and asymmetry coefficient $\xi$;$μ_{t}$ is the conditional mean process.

If you are interested in application, the g@rch package by S.Laurent in Ox programming language implements this computation (this answer is also based on its documentation).