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Erschienen in:
2015 | OriginalPaper | Buchkapitel
Estimation of Volatility Functionals: The Case of a \(\sqrt{n}\) Window
verfasst von : Jean Jacod, Mathieu Rosenbaum
Erschienen in: Large Deviations and Asymptotic Methods in Finance
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Abstract
We consider a multidimensional Itô semimartingale regularly sampled on [0, t] at high frequency \(1/\Delta _n\), with \(\Delta _n\) going to zero. The goal of this paper is to provide an estimator for the integral over [0, t] of a given function of the volatility matrix, with the optimal rate \(1/\sqrt{\Delta _n}\) and minimal asymptotic variance. To achieve this, we use spot volatility estimators based on observations within time intervals of length \(k_n\Delta _n\). In [5], this was done with \(k_n\rightarrow \infty \) and \(k_n \sqrt{\Delta _n}\rightarrow 0\), and a central limit theorem was given after suitable de-biasing. Here we do the same with the choice \(k_n\asymp 1/\sqrt{\Delta _n}\). This results in a smaller bias, although more difficult to eliminate.