Lower bounds for volatility estimation in microstructure noise models

Abstract

In this paper minimax lower bounds are derived for the estimation of the instantaneous volatility in three related high-frequency statistical models. These bounds are based on new upper bounds for the Kullback-Leibler divergence between two multivariate normal random variables along with a spectral analysis of the processes. A comparison with known upper bounds shows that these lower bounds are optimal. Our major finding is that the Gaussian microstructure noise introduces an additional degree of ill-posedness for each model, respectively.