Nonparametric estimation of the volatility under microstructure noise: wavelet adaptation

Hoffmann, Marc and Munk, Axel and Schmidt-Hieber, Johannes (2010): Nonparametric estimation of the volatility under microstructure noise: wavelet adaptation.

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Adaptive estimation; diffusion processes; high-frequency data; microstructure noise; minimax estimation; semimartingales; wavelets.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C0 - General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

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