Determining the Volatility of a Price Process in the Presence of Rounding Errors

Let S denote the price process of a security, and suppose that S follow a geometric Brownian motion with volatility σ. We consider the case when the observations at the discrete time points 0, 1/n, 2/n, · · · , 1 are the rounded-off values Sn i/n = αnbSi/n/αnc (i = 0, · · · , n), where αn > 0 is the round-off level corresponding to the sample frequency n. We investigate the asymptotic behavior of the “Realized Volatility” V n = ∑n i=1(log(S (αn) i/n ) − log(S (αn) (i−1)/n)) , which is commonly used as an estimator of the volatility σ. We prove the convergence of V n or scaled V n under different conditions on αn. A bias corrected estimator of the volatility is proposed and an associated central limit theorem is shown. Simulation results show that improvement in statistical properties can be substantial.