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Nonparametric estimation of the volatility under microstructure noise: wavelet adaptation

Author

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  • Hoffmann, Marc
  • Munk, Axel
  • Schmidt-Hieber, Johannes

Abstract

We study nonparametric estimation of the volatility function of a diffusion process from discrete data, when the data are blurred by additional noise. This noise can be white or correlated, and serves as a model for microstructure effects in financial modeling, when the data are given on an intra-day scale. By developing pre-averaging techniques combined with wavelet thresholding, we construct adaptive estimators that achieve a nearly optimal rate within a large scale of smoothness constraints of Besov type. Since the underlying signal (the volatility) is genuinely random, we propose a new criterion to assess the quality of estimation; we retrieve the usual minimax theory when this approach is restricted to deterministic volatility.

Suggested Citation

  • Hoffmann, Marc & Munk, Axel & Schmidt-Hieber, Johannes, 2010. "Nonparametric estimation of the volatility under microstructure noise: wavelet adaptation," MPRA Paper 24562, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:24562
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    Citations

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    Cited by:

    1. Zu, Yang & Peter Boswijk, H., 2014. "Estimating spot volatility with high-frequency financial data," Journal of Econometrics, Elsevier, vol. 181(2), pages 117-135.
    2. Yu, Chao & Fang, Yue & Zhao, Xujie & Zhang, Bo, 2013. "Kernel filtering of spot volatility in presence of Lévy jumps and market microstructure noise," MPRA Paper 63293, University Library of Munich, Germany, revised 10 Mar 2014.
    3. E. Bacry & S. Delattre & M. Hoffmann & J. F. Muzy, 2013. "Modelling microstructure noise with mutually exciting point processes," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 65-77, January.

    More about this item

    Keywords

    Adaptive estimation; diffusion processes; high-frequency data; microstructure noise; minimax estimation; semimartingales; wavelets.;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C0 - Mathematical and Quantitative Methods - - General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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