The introduction VIX options makes the concept of "volatility of volatility" a real life concept. The idea of "nested volatility" seems interesting, and I am wondering if there are any academic treatment on the subject. For example, we know that variances follow chi-square distributions, but what happens when you take the volatility n times?
To take a look at these things, I simulated a driftless Brownian motion time series $x(t+1) = x(t) + N(0,1)$ for 5000 days, then I calculated its 10 day volatility, and the 10 day volatility of the volatility, and so on. The process is repeated 8 times. A few interesting observations were made.
- The average value of the time series decrease by about 65% - 73% each time you take the volatility. The scale of this decrease also decreases over time.
- The correlation the between time series and its volatility ranges from 0.4 to 0.6, and this correlation increases each time you take the volatility. This is very strange as CBOE claims that historically VIX has little correlation with its volatility index VVIX.
It would be nice to find some theoretical explanations for these observations. I really hope there are something out there something similar to power series expansions, where we can break down an object into infinite number of ever-decreasing smaller objects.