Asymptotic theory for range-based estimation of integrated variance of a continuous semi-martingale

We provide a set of probabilistic laws for range-based estimation of integrated variance of a continuous semi-martingale. To accomplish this, we exploit the properties of the price range as a volatility proxy and suggest a new method for non-parametric measurement of return variation. Assuming the entire sample path realization of the log-price process is available - and given weak technical conditions - we prove that the high-low statistic converges in probability to the integrated variance. Moreover, with slightly stronger conditions, in particular a zero drift-term, we find an asymptotic distribution theory. To relax the mean-zero constraint, we modify the estimator using an adjusted range. A weak law of large numbers and central limit theorem is then derived under more general assumptions about drift. In practice, inference about integrated variance is drawn from discretely sampled data. Here, we split the sampling period into sub-intervals containing the same number of price recordings and estimate the true range. In this setting, we also prove consistency and asymptotic normality. Finally, we analyze our framework in the presence of microstructure noise.