Spectral Estimation of Covolatility from Noisy Observations Using Local Weights

Abstract

ABSTRACT. We propose localized spectral estimators for the quadratic covariation and the spot covolatility of diffusion processes which are observed discretely with additive observation noise. The eligibility of this approach to lead to an appropriate estimation for time-varying volatilities stems from an asymptotic equivalence of the underlying statistical model to a white noise model with correlation and volatility processes being constant over small intervals. The asymptotic equivalence of the continuous-time and the discrete-time experiments are proved by a construction with linear interpolation in one direction and local means for the other. The new estimator outperforms earlier nonparametric approaches in the considered model. We investigate its finite sample size characteristics in simulations and draw a comparison between the various proposed methods.

Estimating the quadratic covariation matrix from noisy observations: Local method of moments and efficiency

“…To ensure efficiency, the oracle and adaptive choice of the weight matrices W jk are based on Fisher information calculus; see Section 4 below. Let us mention that scalar weights for each matrix estimator entry as in Bibinger and Reiss [8] will not be sufficient to achieve (asymptotic) efficiency and the W jk will be densely populated.…”

“…To reduce variability in the estimate, a coarser grid of r −1 equidistant intervals, r/h ∈ N is employed forŴ jk . As derived in Bibinger and Reiss [8] for supremum norm loss and extended to L 1 -loss and Besov regularity using the L 1 -modulus of continuity as in the case of wavelet estimators (Corollary 3.3.1 in Cohen [11]), a preliminary estimator Σ(t) of the instantaneous co-volatility matrix Σ(t) exists with 1]). For block k with kh ∈ [mr, (m + 1)r), we set…”

“…We simulate n 1 = n 2 = 30,000 synchronous observations on [0, 1]. For estimating σ 2 1 and σ 12 = ρ, Figure 2 as well as the adaptive spectral estimator (SPEC ad ) by Bibinger and Reiss [8]. The latter relies on the same spectral approach, but uses only scalar weighting instead of the full information matrix approach.…”

Change-point inference on volatility in noisy Itô semimartingales

“…It can be obtained in a similar (in fact easier) way as for the term A n in (34). Such a bound is already given in Bibinger and Reiß (2013) on page 10 for the estimators in (15) with J = 1, and readily extends to the case J > 1. Since α n ∝ h −2a 2a+1 n is the rate-optimal choice, we get with the upper bound from Bibinger and Reiß (2013)…”

“…The third and fourth steps (c) and (d) are analogues of Propositions 6.3 and 6.4. Since the upper bound, which is presented in Bibinger and Reiß (2013), is not affected for overlapping big blocks, we omit the details. Concerning (e) let us only mention, that an additional tool which is necessary, is Lévy's modulus of continuity theorem in order to control the discretization error.…”