Realized Volatility When Sampling Times

When estimating integrated volatilities based on high-frequency data, simplifying assumptions are usually imposed on the relationship between the observation times and the price process. In this paper, we establish a central limit theorem for the Realized Volatility in a general endogenous time setting. We ∗We are grateful to Andrew Patton and Neil Shephard, and the participants of the Stevanovich Center CREATES 2009 conference and the Fourth Annual SoFiE Conference for their comments and suggestions. Financial support from the Research Grants Council of Hong Kong under grants GRF 602710 and GRF 606811 (Li and Zheng), and the National Science Foundation under grants DMS 06-04758, SES 06-31605, and SES 11-24526 (Mykland and Zhang) is also gratefully acknowledged. Address correspondence to: Xinghua Zheng, Department of ISOM, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; (852) 2358 7750 or email: [email protected]. 1 also establish a central limit theorem for the tricity under the hypothesis that there is no endogeneity, based on which we propose a test and document that this endogeneity is present in financial data.