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Summary: A Tale of Two Time Scales: Determining Integrated
Volatility With Noisy High-Frequency Data
Lan ZHANG, Per A. MYKLAND, and Yacine AÏT-SAHALIA
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 continuous-time modeling and discrete-time samples. We propose an estimation approach that takes
advantage of the rich sources in tick-by-tick data while preserving the continuous-time 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 "two-scales
estimator," works for any size of the noise.
KEY WORDS: Bias-correction; Market microstructure; Martingale; Measurement error; Realized volatility; Subsampling.
1. INTRODUCTION
1.1 High-Frequency Financial Data With Noise
In the analysis of high-frequency 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
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