Measuring what happened

Realised, or historical, volatility is the standard deviation of log returns, scaled to a year. With daily closes \(P_i\) and \(n\) returns \(r_i = \ln(P_i / P_{i-1})\),

The 252 is the number of trading days in a year, and appears because variance grows linearly with time, so volatility grows with its square root. A daily standard deviation of 1.26% is an annual volatility of 20%. A rolling window, 21 days say, shows how volatility moves: it clusters, with calm months and violent ones.

What the market is saying

Implied volatility is the number that makes Black-Scholes reproduce an option's price. It is a forecast of realised volatility over the option's life, plus a premium. Over long samples implied volatility has exceeded realised volatility most of the time: option sellers are paid for bearing the risk that the forecast is wrong, just as insurers are. The ratio of implied to realised is a first look at whether options are dear or cheap.

Fat tails

Black-Scholes assumes log returns are normal. Real returns have negative skewness and excess kurtosis: a crash of five standard deviations, impossible in a normal world, happens every few years. The histogram on the data page makes this visible, and it is the empirical root of the smile. Every model in this app beyond Black-Scholes, from Heston to jumps, is an attempt to give the distribution the tails it needs.

Other estimators

Close-to-close volatility is the simplest estimator, but throws away information. Parkinson's estimator uses the daily high and low; Garman-Klass adds the open and close; both are several times more efficient. Realised variance from intraday returns is more efficient still and underlies variance swaps, contracts that pay the difference between realised and implied variance directly.