In earlier posts we studied the relationship between the VIX Index and GARCH Models of the daily volatility of the S&P 500 Index. We found a quadratic relationship between the log of the VIX and the log of the GARCH model.

However, we also found considerable support for completely asymmetric response between empirical volatility and market returns. When we properly incorporate this structure, we find the curvature in the relationship between the VIX Index and the GARCH model is removed, as is illustrated in the above chart.

We have previously exhibited comparisons of the VIX index and a simple GARCH(1,1) model for the daily volatility of the S&P 500 Share Index. Here, we present a comparison with a completely asymmetric model, CAGARCH(1,1) — this is a process for which the index volatility only increases on down days.

The above analysis is based on a completely in-sample fit of the volatility process. The reason why I have included this metric is to guarantee that the variance process has been chosen to give a best possible match to the experienced variance over the past decade. Thus the systematic offset between the value of the VIX Index and the modeled process cannot be ascribed to scaling error due to sampling — it represents a real systematic premium of the VIX over the actual volatility of the index.