For a while now I've been publishing daily volatility estimates for the VIX index as a page on this blog. This is built from a simple GARCH(1,1) model based on the relative changes. i.e.

In addition to the conditional variance structure above, we use a fundamentally leptokurtotic, but symmetric, innovation so that

where GED represents the Generalized Error Distribution and Student represents Student's t Distribution. In both cases the ν parameter controls the kurtosis of the distribution. This model tracks the conditional variance of the VIX index reasonably well and, for the cast of the Student's t Distribution is exhibited in the model summary chart below.

Now one of the recurring themes on this blog is that introducing simple GARCH models and fundamentally leptokurtotic innovations goes a very long way towards restoring the analytical tractability of financial data. However, the innovations exhibited above appear to the eye to be much less well described by the given p.d.f. than for regular series, such as the S&P 500. One only has to examine the series of annualized volatilities to find some extraordinary values. At the time of writing the estimate computed on Monday was over 270% on an annualized basis. (A snippet of this data is tabulated below.) This would imply a catastrophic drawdown of the VIX to zero should be pretty likely — yet this is clearly forbidden by the true dynamics of the series. There must be more complicated behaviour embedded in this series. We shall examine some of these extensions in future posts.