Realized Volatility Estimators

Estimate a runner's average speed from just their start and finish times, and you get one number that hides everything — did they sprint and coast, or run steadily? Attach a GPS tracker recording position every second and you get the whole story of how fast they moved throughout. Realized volatility is the GPS-tracker approach applied to price moves: instead of inferring volatility indirectly from an option price (implied vol) or a rough close-to-close estimate, it adds up every observed intraday return, squared, to measure how much actually happened.

The formula, defined

$$RV_t = \sum_{i=1}^{n} r_{i}^2$$

In words: split the day (or any period) into $n$ small intervals, compute the log return $r_i$ in each one, square each return (so up moves and down moves both add rather than cancel), and sum them all up. $RV_t$ is the realized variance for that period; taking its square root gives realized volatility on that period's own time scale, and multiplying by an annualization factor (like $\sqrt{252}$ for daily data) puts it on a standard annualized footing for comparison across assets and horizons.

Worked example 1 — computing RV from five daily returns

A stock's daily log returns over a week are: +1.2%, -0.8%, +0.3%, -1.5%, +0.6%. Squaring each: $0.012^2 = 0.000144$, $0.008^2 = 0.000064$, $0.003^2 = 0.000009$, $0.015^2 = 0.000225$, $0.006^2 = 0.000036$. Summing: $RV = 0.000478$. That's the realized variance for the 5-day window; realized vol for that window is $\sqrt{0.000478} \approx 2.19\%$. Annualized (scaling to 252 trading days): $\sigma_{\text{ann}} = \sqrt{0.000478 / 5 \times 252} \approx$ 15.5%.

Worked example 2 — sampling more frequently sharpens the estimate

Suppose instead of 1 daily return, the same day is split into 8 half-hour intraday returns: +0.3%, -0.2%, +0.1%, -0.4%, +0.2%, -0.1%, +0.3%, -0.1%. Squaring and summing: $0.000009 + 0.000004 + 0.000001 + 0.000016 + 0.000004 + 0.000001 + 0.000009 + 0.000001 = 0.000045$. Compare that day's full-day close-to-close return alone: roughly $0.3-0.2+0.1-0.4+0.2-0.1+0.3-0.1 = 0.1\%$, squared to just $0.000001$ — the single close-to-close return massively understates that day's actual activity, because offsetting up-and-down moves during the day cancel out in a single overnight-to-close difference but don't cancel when each interval is squared and summed separately. This is the core reason RV built from finer sampling is more informative than a single daily return: it captures the path, not just the endpoint.

What this means in practice

Realized volatility computed from high-frequency data is the standard against which implied volatility is compared (was the option "worth" what it cost, given what actually happened?), and it's the target variable that forecasting models like HAR-RV (see HAR-RV model) try to predict. It's also the basic building block that more sophisticated estimators — range-based, two-scale, bipower — are built to improve on for specific weaknesses.

Practice

1. A stock has four half-hour returns: +0.5%, -0.3%, +0.2%, -0.4%. Compute the realized variance and realized volatility for that two-hour window.
2. Why does computing RV from 1-second returns typically give a worse (more biased) estimate of true daily volatility than computing it from 5-minute returns, despite using far more data points?