Realized or Historical Volatility

Realized volatility measures the actual movement of an asset’s price over a specific period using historical data. It analyzes price changes over predetermined periods, often based on the change from one closing price to the next, though it can also consider intraday changes. This metric, also known as historical volatility (HV), typically spans from 10 to 180 trading days, depending on the specific needs of the analysis.

Unlike implied volatility, which is forward-looking, HV is backward-looking and describes how the security has actually behaved. An increase in HV indicates that a security’s price is moving more than usual, suggesting potential changes or events affecting the asset. Decrease in HV implies a return to normalcy, as uncertainties are perceived to have been resolved.

Backward-looking does not mean useless as a forecast. Volatility clusters — quiet periods follow quiet periods and violent ones follow violent ones — which makes recent realized volatility a genuinely informative predictor of near-term future volatility, in sharp contrast to recent returns, which predict nothing. Andersen, Bollerslev, Diebold and Labys established this formally.⁵⁴ It is the one place in this book where extrapolating the recent past is defensible, and the reason is that you are extrapolating the magnitude of moves, not their direction.

Realized volatility is calculated using historical price data — the recipe below is the same computation the worked example earlier in this section performed on twelve monthly prices, stated once in general form. The most common method is to use daily returns:

Daily Returns

Calculate the log returns for each period (e.g., daily returns if using daily data) $r_t$:

$$r_t=\text{ln}<!--\; FUNCTION\; APPLICATION\; -->\left(\frac{P_t}{P_{t-1}}\right)$$

where $P_t$ is the price at time $t$.

Variance

Compute the variance of the returns over $n$ periods:

$${\sigma }^2=\frac{1}{n-1}\sum _{t=1}^n{(r_t-\overline{r})}^2$$

where $\overline{r}$ is the mean of the log returns.

Annualize (optional)

To annualize the volatility, multiply the standard deviation by the square root of the number of periods in a year. For daily data:

$${\sigma }_{\text{annual}}=\sqrt{{\sigma }^2\times 252}$$

where 252 is the approximate number of trading days in a year.