Historical volatility

The calculation has three steps. The return of each period in a chosen window is taken, commonly as the logarithm of one closing price divided by the one before it. The standard deviation of those returns is calculated, which gives dispersion for a period of that length. That figure is then multiplied by the square root of the number of periods in a year to restate it as an annual rate, which is the form nearly every published number takes.

What results is a single measure of dispersion and nothing else. It is blind to direction, so a steady climb and a steady fall of the same magnitude produce the same figure, and it is blind to path, so two very different-looking charts can share a number. Its value also depends entirely on the window: a short window and a long window over the same instrument are both correct and can differ by a multiple, which is why a quoted figure without its window attached cannot be compared with another one. Implied volatility is the forward-looking counterpart, extracted from option prices rather than from past closes, and the two are routinely different.

The assumption inside the annualisation is where the trouble sits. Scaling by the square root of time treats returns as independent and identically distributed, and financial returns are neither: volatility clusters, so calm follows calm and turbulence follows turbulence, and very large moves occur far more often than a normal distribution allows for. Practitioners also disagree on the period count used for scaling, since a market that trades on business days and one that quotes almost continuously imply different conventions, and two figures computed on different conventions are not comparable however precisely each is stated.