Volatility estimation with price quanta

Volatility estimators based on high, low, opening and closing prices have been developed, and perform well on simulated data, but on real data they frequently give lower values for volatility than the simple open-close estimator. This may be due to the fact that for real data, the maximum (or minimum) price is often at the beginning or end of the day. While this could not happen if the observed process was log Brownian, it could happen if the observed process were log Brownian, but observed only to the nearest penny. We develop the theory of such approximations to derive the corrected versions of the basic estimators.