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)^2, 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 is presented explicitly. The Cramer-Rao upper bound for the efficiency of unbiased estimators, corresponding to the efficiency of large-sample Maximum Likelihood estimators, is 8.471. This bound cannot be attained because the distribution is not of exponential type. Regression-fitted quadratic functions of S2 (with mean 1) markedly out-perform those of S1 when applied to random walks with heavy-tail-distributed increments. Performance is empirically studied in terms of the tail parameter.

💡 Research Summary

The paper revisits the widely‑used Garman‑Klass estimator of variance for a zero‑drift Brownian motion, which traditionally relies on the three‑dimensional statistic S1 = (CLOSE‑OPEN, OPEN‑MIN, MAX‑OPEN). This quadratic, unbiased estimator achieves an efficiency of 7.4 relative to the naïve (CLOSE‑OPEN)² estimator and has long been considered optimal. The authors challenge this belief by introducing a data‑compression transformation that flips the Brownian path according to the sign of its net movement: W(t) = B(0) +