Efficient Estimation of Stochastic Volatility Using Noisy Observations: A Multi-Scale Approach

With the availability of high frequency financial data, nonparametric estimation of volatility of an asset return process becomes feasible. A major problem is how to estimate the volatility consistently and efficiently, when the observed asset returns contain error or noise, for example, in the form of microstructure noise. The former (consistency) has been addressed heavily in the recent literature, however, the resulting estimator is not quite efficient. In Zhang, Mykland, and Aı̈t-Sahalia (2003), the best estimator converges to the true volatility only at the rate of n. In this paper, we propose an efficient estimator which converges to the true at the rate of n, which is the best attainable. The estimator remains valid when the observation noise is dependent. Some key words and phrases: consistency, dependent noise, discrete observation, efficiency, Ito process, microstructure noise, observation error, rate of convergence, realized volatility Lan Zhang is Assistant Professor, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 15213. E-mail: [email protected]. I gratefully acknowledge the support of the National Science Foundation under grant DMS-0204639. Multiple Scale Estimator for Volatility 1