Asymptotic equivalence for inference on the volatility from noisy observations

TL;DR

This paper demonstrates that high-frequency financial data with microstructure noise can be approximated by a Gaussian model, enabling the development of optimal volatility estimators.

Contribution

It establishes asymptotic equivalence between the noisy observation model and a Gaussian shift experiment, leading to new efficient estimation methods.

Findings

Asymptotic equivalence to Gaussian shift model

Rate-optimal volatility estimators constructed

Efficient estimators for integrated volatility developed

Abstract

We consider discrete-time observations of a continuous martingale under measurement error. This serves as a fundamental model for high-frequency data in finance, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function $\sigma$ and a nonstandard noise level. As an application, new rate-optimal estimators of the volatility function and simple efficient estimators of the integrated volatility are constructed.