Lower bounds for volatility estimation in microstructure noise models

TL;DR

This paper establishes optimal lower bounds for estimating instantaneous volatility in models with Gaussian microstructure noise, revealing increased ill-posedness due to noise.

Contribution

It derives the first minimax lower bounds for volatility estimation under Gaussian microstructure noise, highlighting the impact of noise on estimation difficulty.

Findings

Lower bounds are proven to be optimal.

Gaussian noise adds an extra layer of ill-posedness.

The results extend understanding of volatility estimation limits.

Abstract

In this paper we derive lower bounds in minimax sense for estimation of the instantaneous volatility if the diffusion type part cannot be observed directly but under some additional Gaussian noise. Three different models are considered. Our technique is based on a general inequality for Kullback-Leibler divergence of multivariate normal random variables and spectral analysis of the processes. The derived lower bounds are indeed optimal. Upper bounds can be found in Munk and Schmidt-Hieber [18]. Our major finding is that the Gaussian microstructure noise introduces an additional degree of ill-posedness for each model, respectively.