Does anyone have references for the following observed empirical volatility behavior?

Take daily index or share price returns. From this data calculate the volatility over a period T. This gives a volatility time series with steps of T. Also calculate the T-period return for the same data and scale with the square root of T. This is another time series.

If $$x_i=\ln(S_i/S_{i-1})$$ are the daily returns, then the first time series is
\[Vol_k = \sqrt{\frac{252}{T}\sum_{i=k-T}^k (x_i-\bar{x})^2},\quad\bar{x}=\sum_{i=k-T}^k x_i\]
and the second is
\[
z_k = \frac{1}{\sqrt{T}}\ln(S_{k}/S_{k-T}).
\]
Vol and z are measured over the same period.

Now plot Vol_k against z_k. For example, with T being 20 days: The black line is the empirical average of the volatility. The red line is
\[
Vol=\sqrt{\sigma^2+\frac{z^2}{2}}.
\]
I'm trying to find a reference for any relationship like this. Once you've scaled with T the relationship seems very stable.

No one? This empirical behavior is for stock returns and actual volatility. It’s not about implied volatility, maybe that’s why I can’t find anything in the quant literature.

By definition, [$]z_k = \frac{1}{\sqrt{T}} \sum_{i=1}^T x_i[$]. If the [$]x_i[$] were IID draws from something then the variance of the z's would equal the variance of the x's. If not, there are the auto-covariance cross terms. I would try to relate z to the cross terms, perhaps by a regression.

p.s. Or maybe the variance of the z's does equal (more or less) the variance of the x's, and both of your terms under the sqrt are simply noisy proxies for (half) of the variance of the z's.

I have plotted the "term structure" of the variance for SPX. I think it's pretty flat.

Not sure what you mean by the equation involving two noisy proxies for volatility? The equation is deterministic so for example if z=0 then the volatility is sigma. Also would you have any plots to illustrate your stats argument?

As another view, here are some plots showing volatility versus z for the S&P 500, DJIA, and FTSE indexes, along with artificial data from a lognormal time series for comparison. Black lines show the volatility computed from the mean variance, red line is the equation. The heat map in the background shows the density of points. The results are computed using periods T of one to fifty weeks.

I think it may simply be a natural result of [$]z_k[$] being a (scaled) sum of the x's, where the x's are (close to) IID draws from any distribution whatsoever. You can see it worked for your artificial data with [$]x \sim[$] normal. Instead of the normal, pick any distribution you like with (symmetric) support on [$]x \in (-\infty,\infty)[$]. Try some things that are definitely not log-stock-return distributions. I'll guess you get the same general picture. If so, it's not a property of stock returns but a property of having a sum of IID variates.

As for "noisy proxy", I just mean that each subsample of length [$]T[$] will have a realized volatility that differs from the (true) population volatility parameter. Those subsamples [$]k[$] that have a larger [$]z_k^2[$] will tend to have a larger realized volatility, so that's the noisy proxy. But if [$]z_k[$] is small, it's a poor proxy, so the constant term [$]\sigma^2[$] compensates.   


   

As for plots, here's a page from a book I'm working on that shows the (realized, scaled) US market volatility for various horizons [$]T[$] for a broad-market index.
The blue dots are the (real-world) realized vols, but I also show risk-neutral vols. The latter has an upward-sloping term structure, but the former is essentially flat. This supports the notion that the (real-world) log-returns over various horizons are close to a sum of independent draws from "something".


 


 

It is easy to produce the behaviour for a single period T using for example a jump-diffusion model, however there are three things:
1. The equation has no parameter other than a minimum volatility, so for a given period T you have to calibrate the model in a certain way to match the behaviour. 
2. The property has to hold for all other periods T as well. This requirement would seem to rule out any process based on versions of a random walk where the price change distribution becomes more normal in time.
3. The effect is not related to term structure, because it concerns not overall volatility but the dependence with z.

Here is an example using a jump-diffusion model, for T= 5, 10, 20, 40, 80 and 160 days. The model is calibrated to work at T=5 days but the volatility vs z curve becomes progressively flatter and the effect disappears for periods longer than a few months. 

I think you might be talking at cross purposes. I’m not sure it’s quite a term-structure thing. It feels like it is possibly a stats feature. I’m interested in Alan’s proxy idea. But if it was just “when vol is high, so is z” then I would expect a parameter in there…vol goes up and comes down, and there will be a timescale for this, and so a parameter, instead of the “universal” 0.5?

Let me turn this around – if this was a generic stats feature, it should be easy to produce a time series which has the property for all periods T!