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

Abstract

We consider the models Yi,n=∫0i/nσ(s)dWs+τ(i/n)εi,n, and Ỹi,n=σ(i/n)Wi/n+τ(i/n)εi,n, i=1,…,n, where (Wt)t∈[0,1] denotes a standard Brownian motion and εi,n are centered i.i.d. random variables with E (εi,n2)=1 and finite fourth moment. Furthermore, σ and τ are unknown deterministic functions and (Wt)t∈[0,1] and (ε1,n,…,εn,n) are assumed to be independent processes. Based on a spectral decomposition of the covariance structures we derive series estimators for σ2 and τ2 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. A major finding is that the microstructure noise εi,n introduces an additionally degree of ill-posedness of 1/2; irrespectively of the tail behavior of εi,n. The performance of the estimates is illustrated by a small numerical study.

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