The Garman-Klass volatility estimator revisited

The Garman–Klass unbiased estimator of the variance per unit time of a zero–drift Brownian Motion (B), based on the usual financial data that reports for time windows of equal length the open (OPEN), minimum (MIN), maximum (MAX) and close (CLOSE) values, is quadratic in the statistic S1 = (CLOSE − OPEN, OPEN − MIN,MAX − OPEN). This estimator, with efficiency 7.4 with respect to the classical estimator (CLOSE − OPEN), is widely believed to be of minimal variance. The current report disproves this belief by exhibiting an unbiased estimator with slightly but strictly higher efficiency 7.7322. The essence of the improvement lies in the observation that the data should be compressed to the statistic S2 defined on W (t) = B(0) + [B(t) − B(0)]sign((B(1) − B(0)) as S1 was defined on the Brownian path B(t). The best S2–based quadratic unbiased estimator will be presented explicitly. The Cramér–Rao upper bound for the efficiency of unbiased estimators, corresponding to the efficiency of largesample Maximum Likelihood estimators, is 8.471. This bound cannot be attained because the distribution is not of exponential type.