QQ 102, 2 min read

Volatility

Volatility measures how much returns vary. The usual measure is the standard deviation of returns.

mean       = sum of returns / n
variance   = sum of (return - mean) squared / (n - 1)
volatility = square root of variance

Dividing by n - 1 instead of n is the sample standard deviation. It is the right one when you are estimating from a sample, which is always the case with market data.

Annualising

Volatility measured on daily returns is a daily number. To compare with anything else it is scaled to a year:

annual volatility = daily volatility x square root of 252

252 is roughly the number of trading days in a year. For hourly bars or weekly bars the count is different, and choosing it wrongly changes the answer.

The assumption inside the square root

The square root comes from assuming each period's return is independent of the last. Then variances add, so standard deviations grow with the square root of time.

Real returns are not independent. Volatility clusters, so calm days tend to follow calm days and wild days follow wild days. The square root rule is still used because it makes numbers comparable, and it should be read as a convention.

What volatility does not see

Standard deviation treats a surprise gain and a surprise loss the same. It also describes the ordinary day far better than the rare one. Market returns have fat tails: very large moves happen more often than a normal distribution predicts. A strategy can show low volatility for years and still be exposed to one of those moves.

Further reading

• Mandelbrot, B. (1963). "The Variation of Certain Speculative Prices." Journal of Business 36(4). The original evidence for fat tails.
• Engle, R. F. (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation." Econometrica 50(4). The paper that modelled volatility clustering.