Repeated Volatility Measures and Their Limits in Real Markets
Summary
The note explores repeated volatility calculations, asking how the distribution and relationships change when volatility is measured on volatility again. It describes a simulation of a driftless Brownian motion, computes rolling ten-day volatility, and repeats that operation eight times. The author reports that the series’ average level falls at each stage and that correlations between successive series range from roughly 0.4 to 0.6, increasing with each iteration.
The response cautions against carrying those Gaussian simulation observations over to markets. It contrasts the setup with equity and volatility index behavior, noting a strongly negative historical relationship between the S&P 500 and VIX, and describes volatility as prone to jumps followed by fading. A Hawkes process is offered as one possible way to model that behavior. The note raises useful questions but provides no derivation or evidence establishing general properties of nested volatility; the simulation is stylized and its findings depend on its construction.
Key ideas
- The document studies repeated rolling volatility transformations on simulated driftless Brownian motion.
- The simulated series’ average level declines at each successive volatility calculation.
- Correlations between successive transformed series rise in the reported simulation.
- Real market volatility may jump and fade, making simple Gaussian simulation results difficult to generalize.
- A Hawkes process is mentioned as a possible model for clustered volatility moves.
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Full text
# Nested volatility # Nested volatility 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. ## Answer by Richi Wa (score 2) https://quant.stackexchange.com/a/14508 It is a stylized feature that the correlation between SPX and VIX is negative (around -0.7) thus we have an interesting process here. Nothing as simple as driftless Brownian motion. The volatility process jumps up and then fades out which is often modelled as Hawkes process (see e.g. here). Thus conclusions from the simple Gaussian model (with Chi-quared volatiltiy) to the real world are difficult.
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