 
Summary: A Tale of Two Time Scales: Determining Integrated
Volatility With Noisy HighFrequency Data
Lan ZHANG, Per A. MYKLAND, and Yacine AÏTSAHALIA
It is a common practice in finance to estimate volatility from the sum of frequently sampled squared returns. However, market microstructure
poses challenges to this estimation approach, as evidenced by recent empirical studies in finance. The present work attempts to lay out
theoretical grounds that reconcile continuoustime modeling and discretetime samples. We propose an estimation approach that takes
advantage of the rich sources in tickbytick data while preserving the continuoustime assumption on the underlying returns. Under our
framework, it becomes clear why and where the "usual" volatility estimator fails when the returns are sampled at the highest frequencies.
If the noise is asymptotically small, our work provides a way of finding the optimal sampling frequency. A better approach, the "twoscales
estimator," works for any size of the noise.
KEY WORDS: Biascorrection; Market microstructure; Martingale; Measurement error; Realized volatility; Subsampling.
1. INTRODUCTION
1.1 HighFrequency Financial Data With Noise
In the analysis of highfrequency financial data, a ma
jor problem concerns the nonparametric determination of the
volatility of an asset return process. A common practice is
to estimate volatility from the sum of the frequently sampled
squared returns. Although this approach is justified under the
assumption of a continuous stochastic model in an idealized
world, it runs into the challenge from market microstructure
