The joint probability density function of random variables minimum, maximum and finish values returns of the arithmetic Brownian motion is presented in the paper. The application of this density allowed to analytically evaluate main properties of the most popular estimators of the variance constructed on the basis of low, high and closing prices of the arithmetic Brownian motion. In particular, the unbiasedness of the Parkinson and Garman-Klass estimators for the process with a zero drift and of the Rogers-Satchell estimator for any drift is proved. Thus the main results from the papers of the mentioned authors are confirmed. Moreover the expected values of the Parkinson and Garman-Klass estimators for the arithmetic Brownian motion with a non-zero drift are derived. The mean square errors of the Parkinson, Garman-Klass and Rogers-Satchell estimators for the process with a non-zero drift are also formulated. According to our knowledge those characteristics have not been published yet. 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. It is shown that volatility estimates based on low, high and closing prices are more accurate than ones formulated on the basis of the GARCH model. The estimators based on low, high and closing prices can be applied in the future to the construction of the GARCH models, so that it will be possible to obtain even more accurate volatility estimates.

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