A new look at variance estimation based on low, high and closing prices taking into account the drift

The joint distribution of low, high and closing prices of the arithmetic Brownian motion is used to evaluate the properties of the most popular estimators of the variance constructed on the basis of high, low and closing prices. The expected values and mean square errors of the Parkinson, Garman-Klass and Rogers-Satchell estimators for the process with a zero drift and a non-zero drift are derived. Moreover, new volatility estimators, more efficient in the majority of financial applications than the Rogers-Satchell estimator, are proposed. The considered estimators are applied to the estimation of the volatility of the Polish stock index WIG20.