Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise

TL;DR

This paper develops optimal nonparametric estimators for the volatility and noise functions in high-frequency financial models corrupted by microstructure noise, revealing an inherent ill-posedness degree of 1/2.

Contribution

It introduces spectral series estimators for volatility and noise functions, proving their minimax optimality and analyzing their convergence rates under smoothness assumptions.

Findings

Estimators achieve minimax optimal convergence rates.

Microstructure noise causes an ill-posedness degree of 1/2.

Method is validated through numerical simulations.

Abstract

We consider the models Y_{i,n}=\int_0^{i/n} \sigma(s)dW_s+\tau(i/n)\epsilon_{i,n}, and \tilde Y_{i,n}=\sigma(i/n)W_{i/n}+\tau(i/n)\epsilon_{i,n}, i=1,...,n, where W_t denotes a standard Brownian motion and \epsilon_{i,n} are centered i.i.d. random variables with E(\epsilon_{i,n}^2)=1 and finite fourth moment. Furthermore, \sigma and \tau are unknown deterministic functions and W_t and (\epsilon_{1,n},...,\epsilon_{n,n}) are assumed to be independent processes. Based on a spectral decomposition of the covariance structures we derive series estimators for \sigma^2 and \tau^2 and investigate their rate of convergence of the MISE in dependence of their smoothness. To this end specific basis functions and their corresponding Sobolev ellipsoids are introduced and we show that our estimators are optimal in minimax sense. Our work is motivated by microstructure noise models. Our major finding is that the microstructure noise \epsilon_{i,n} introduces an additionally degree of ill-posedness of 1/2; irrespectively of the tail behavior of \epsilon_{i,n}. The method is illustrated by a small numerical study.