Range-Based Volatility Estimators

Judge a river's daily flow by comparing the water level at 8am to the water level at 6pm and you might completely miss that it flooded at noon and receded by evening. Look instead at the high-water mark and the low-water mark left on the bank and you capture the true extent of the day's swing, using data that was sitting there the whole time. Range-based volatility estimators apply exactly this idea to prices: instead of using only the day's open and close, they use the day's high and low, extracting far more signal from data that's already routinely recorded.

The Parkinson estimator, defined

$$\hat{\sigma}^2_{\text{Park}} = \frac{1}{4\ln 2} \left(\ln \frac{H}{L}\right)^2$$

In words: $H$ and $L$ are the day's high and low prices. $\ln(H/L)$ is the log-range — how far the price traveled from its low to its high over the day, in log terms. Squaring that captures the day's spread, and dividing by the constant $4\ln 2 \approx 2.77$ rescales it to be an unbiased estimator of variance under the assumption that price moves like a continuous random walk with no drift. In plain terms: the day's high-low spread alone, appropriately scaled, is a legitimate stand-in for a whole day's worth of variance — and because it uses the full extreme range rather than just two snapshot points, it's dramatically more statistically efficient than close-to-close volatility, using the same one observation per day.

Worked example 1 — Parkinson vs close-to-close on the same day

A stock opens at $100, closes at $101, but during the day trades as high as $104 and as low as $98. Close-to-close return: $\ln(101/100) = 0.00995$, squared: $0.0000990$ — that's the entire close-to-close variance estimate for the day, suggesting a very quiet session. Parkinson instead uses the range: $\ln(104/98) = \ln(1.0612) = 0.0594$, squared: $0.003529$, divided by $4\ln 2 = 2.7726$: $\hat{\sigma}^2_{\text{Park}} = 0.001273$. That's nearly 13 times larger than the close-to-close variance estimate for the identical day — because the close-to-close number completely missed the $6 round trip the stock actually took between its high and low, which the range-based estimator captures directly.

Worked example 2 — efficiency in a multi-day average

Averaging over 5 days of Parkinson estimates — daily variances 0.00080, 0.00120, 0.00095, 0.00140, 0.00105 — gives a mean of $0.00108$, annualized: $0.00108 \times 252 = 0.272$, so $\hat{\sigma}_{\text{ann}} = \sqrt{0.272} \approx$ 52.1%. In simulation studies, Parkinson needs roughly 5 times fewer daily observations than close-to-close RV for the same statistical precision, extracting the equivalent of several intraday observations from a single high and low most data feeds already report for free.

What this means in practice

Because high and low prices are standard fields in almost every historical OHLC data set, range-based estimators are cheap to compute for long histories where true tick-level intraday data either doesn't exist or is expensive, making them a common choice for backtesting volatility strategies over years of daily bars. Refinements like the Garman-Klass and Rogers-Satchell estimators extend Parkinson by also using the open and close, correcting for known biases the pure high-low version has.

Practice

1. A stock's daily high is $52 and low is $49. Compute the Parkinson variance estimate for that day.
2. Why would a stock that gaps up $5 overnight (no trading in between) but otherwise moves only $0.50 intraday be a case where the Parkinson estimator, applied naively to that day's high-low, gives a misleading read on "intraday" volatility?