Volatility is one of those things that everyone feels but no one can touch.
Like that feeling when you are alone but someone is there watching you.
Maliciously or benignly.
Vol does exist, but in our world it's just a ghost, operating in some odd in-between-space only seen in the odd mirror or out of the corner of one's eye.
If I take an option and back out the vol I can 'see' an implied vol, which is of course the volatility which produces the right price according to the Black Scholes' model.
This is what the Vix index is (or close, actually the modern Vix levels come from variance swaps).
You can also measure price movements and calculate the historical or realised volatility over a period of time.
Volatility is useful precisely because it is so amorphous, people can use it for multiple purposes, just as a priest might explain mishaps through god's mysterious plans.
But it's problematic when you actually want to pin it down.
Vol is otherworldly
Take the Vix for example. It is a vol forecast of the S&P 500 over the next 30 calendar days.
How do we check whether this forecasted number is correct?
We could take the realised vol over the next month - but we would have to find the volatility of daily returns and multiply them by
SQRT(~21) (assuming 21 business days in a month).
The square root of time works if each return is more or less independent of the other.
During apocalyptic periods we need to multiply by something larger than
SQRT(~21) as one bad day may well be followed by others.
The chart above shows the difference between monthly realised vol and implied.
Interestingly, realised vol only ever outweighs implied when, er, shit gets real.
Other than that, it seems that (a) there's a built in implied risk premium, or (b) our assumption of independent returns is not quite correct and the square root of time is leading us astray.
Maybe a bit of both.
If there is an implied risk premium - increases in the spread would reflect increasing nervousness - but I doubt that would capture much more than the skew index does.
Nevertheless another possible string to our bows.