In the prior posts we have noted that the VIX index itself is extraordinarily volatile. Going back to the series for the start of this month, we see that the VIX increases from around 20 points on 05/03/2010 to over 40 points on 05/07/2010 — doubling in the space of a week — and then drops by more than 30% to then end of 05/10/2010. Even fundamentally leptokurtotic GARCH has difficulties explaining these moves, as exhibited by the χ²⁄d.o.f. in our histogram of the homogenized innovations of the process taking the value of 193⁄97.
One model that is often discussed for financial time series is the Asymmetric GARCH process, such as that defined by:
\;\mathrm{where}\;u_t=\frac{d\mathrm{VIX}_t}{\mathrm{VIX}_{t-1}}\;\textrm{and}\;h_t=\textrm{Var}(u_t^2|t-1))
Here I is an indicator function that is zero if its argument is non-negative and unity otherwise. This is the model of Glosten, et al., known as GJR. This specifies as piecewise quadratic response of the variance to the innovation, conditioned on it's sign. This particular specification is non-degenerate with respect to B when D is fixed, which allows us to use the maximum likelihood ratio test to examine the significance of a non-zero parameter.
Performing this regression, we find a m.l.r. statistic of χ²(1) = 11.670634 with significance level 0.00063494, indicating strong desired of the data to include this term. The analysis chart does not look much different from the prior one, so I won't include a redundant copy, but the χ²⁄d.o.f. for the histogram of homogenized innovations decreases to 165⁄97. The regression results are:
GARCH Model - Estimation by BFGS
Convergence in 26 Iterations. Final criterion was 0.0000000 <= 0.0000100
Daily(5) Data From 2001:01:02 To 2010:05:17
Usable Observations 2172
Log Likelihood 3205.25327235
Variable Coeff Std Error T-Stat Signif
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1. Mean -0.002781284 0.001104296 -2.51860 0.01178209
2. C 0.000208902 0.000046653 4.47779 0.00000754
3. A 0.131878668 0.022583630 5.83957 0.00000001
4. B 0.860413599 0.021960624 39.17983 0.00000000
5. D -0.122355877 0.034771870 -3.51882 0.00043347
6. Shape 5.361237381 0.572901025 9.35805 0.00000000
Here the estimated D almost exactly cancels the A, indicating that the volatility of the VIX increases when the value of the index increases, but that it decreases when the index decreases. This is an extreme asymmetry — a severe negative shock does not increase VOLVIX at all and provides the conditions for the runaway trend followed by an abrupt reversal we found in the recent data.