A Tale of Two Time Scales

TLDR

In finance, volatility is typically estimated from the sum of squared returns, but market microstructure effects complicate this approach. This study aims to reconcile continuous‑time modeling with discrete‑time sampling by proposing a new estimation approach that leverages tick‑by‑tick data while maintaining continuous‑time assumptions. The proposed method uses tick‑by‑tick data to construct an estimator that preserves the continuous‑time framework for underlying returns. The framework explains why the standard volatility estimator fails at high sampling frequencies, identifies an optimal sampling frequency when noise is small, and introduces a two‑scales estimator that performs well regardless of noise magnitude.

Abstract

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.