VIX and ATM SP500 Vol

As practitioners in the field of finance, we have run into the very famous VIX index calculation as the result of a formula using the inverse strike (squared) weighted prices of out of the money strips of options. The celebrated formula is almost as famous as the Black Scholes equation. In this brief note, we make no attempt to replicate the formula. There are many aspects of the calculation which are complicated and reflect technical conditions (dealing with strikes where there is no trade, just wide bid/ask prices, for example). We take the index as a given. The thrust of this note is to illustrate some interesting dynamics that it exhibits and that the ES (EMini SP500) at the money vol exhibits, we treat the strike closest to 0.5 as the ATM strike.

The thrusts of this note are:

• Describe the simple AR(1) dynamics of the two series.
• Look for relationships across several other products/variables.
• Look for the dynamic relationship that should exist between the ATM vol and the VIX Index.

The primary tool for our analysis is the rolling regression. We’d like to investigate the relationships as a researcher or trader might encounter them in conducting their work. Let’s begin by looking at changes, that is simple differences day to day (Chg) of the vol associated with the 50 delta option. The front contract rolled 15 days prior to expiry.

There are several things to notice about the regression results. In the top and top left table we show the regression results for the entire period and then below we show the last 30 day’s regression. The plots show the parameters over the period, the R-squared and other summary/goodness of fit information. What do the regressions show? First, the regressions show a weak association between changes (today’s vol – yesterday’s vol) in the 50 Delta versus its own lagged realizations. On average, over the entire period, there is a strong reversal. Positive increases in vols seem to sell off at the 1 and 5 day lags. The results for the most recent 30 days are yet weaker with only the one day lag showing significance. The plot of the parameters of the rolling regressions show wide variation in their realizations. The one day reversal looks somewhat persistent, though its magnitude fluctuates.

We turn our attention to the percent changes in the VIX index. This is a little different from the analysis of the vol levels. Here are the results.

We see the same pattern of reversal. In the case of VIX, the reversal is rather quick. The reader will also notice that the Rsq is a bit higher, moving from 5% to 8%. There is a similar pattern, so perhaps we can use one to help improve the predictions of the other. In econometrics, there is a concept called Granger causality. Clive Granger posits a theory that if X “causes” Y, then the regression of Y on its own realizations, lagged, with X’s realization will fit better than the regression of X on its history and Y. If X causes Y, then there can’t be any extra explanatory power from Y in X’s regression. We use these insights in an admittedly simpler framework to see we can determine any hint of causality. Is it the SP which is moving the VIX or vice-a-versa?

The change in the 50Delta Vol is mildly improved by the inclusion of the changes in VIX. The broad pattern is the same as in the very first regression. There is strong reversal at the 1 day and 5 day mark. The most recent regression shows that the VIX’s effect is quite significant. The pattern is much more sophisticated, with yesterday’s VIX return contributing the reversal effect, but there is a 5 day trend which is quite significant.

We run a similar regression panel for the % changes in the VIX index.

We see some interesting things. The overall regression shows that ES plays some small, but significant role in the returns of the VIX index. Note that the Rsq improves slightly as well. We can say there is some weak evidence that suggests causality runs from the SP500 ATM implied vol to the index. That means the vols at the other SP500 strikes are very much influenced by the ATM strike. There are many reasons why VIX can move. The entire vol curve can be bid up, or alternatively, the wings of the curve can move leaving the ATM strike and its closest neighbors unchanged. Our regressions suggest that the ATM strike weakly “causes” the rest of the vol curve to move.

What this exposition is saves us from needing to run the following types of regressions (see last two panels). We show that, again, the 50 Delta is slightly more useful in explaining the 40 Delta’s vol changes than the opposite. This short research piece will not win Noble prizes, it does illustrate that there are some discernible pulls in the market. It does also illustrate that the ATM strike moves with the curve, so changes in VIX are probably preceded by moves in the ATM vol. Moreover, there is a predictable pattern of reversal in both the VIX and a couple of points on the SP500 skew.

We provide these as a reference for what's happened over the past year.

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