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A central limit theorem for the functional estimation of the spot volatility

Author

Listed:
  • Ngo Hoang-Long

    (Department of Mathematical Sciences, Ritsumeikan University, Shiga, Japan. Email: gr041086@ed.ritsumei.ac.jp)

  • Ogawa Shigeyoshi

    (Department of Mathematical Sciences, Ritsumeikan University, Shiga, Japan. Email: ogawa-s@se.ritsumei.ac.jp)

Abstract

In this paper we introduce a class of statistics for the functional estimation of the spot volatility in the setting of frequency observed diffusion processes which may be disturbed by microstructure noise. We show that the limit theorems for the estimation of the spot volatility and the cross spot volatility of the statistics are still valid even if we add jump processes of finite or infinite activity to the underlying diffusion process. These statistics extend the quadratic variational approach and are related to the concept of multipower variation, which is used in the problem of estimating the integrated volatility.

Suggested Citation

  • Ngo Hoang-Long & Ogawa Shigeyoshi, 2009. "A central limit theorem for the functional estimation of the spot volatility," Monte Carlo Methods and Applications, De Gruyter, vol. 15(4), pages 353-380, January.
    • Handle: RePEc:bpj:mcmeap:v:15:y:2009:i:4:p:353-380:n:4
    DOI: 10.1515/MCMA.2009.019
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    References listed on IDEAS

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    1. Podolskij, Mark & Vetter, Mathias, 2009. "Bipower-type estimation in a noisy diffusion setting," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2803-2831, September.
    2. Kinnebrock, Silja & Podolskij, Mark, 2008. "A note on the central limit theorem for bipower variation of general functions," Stochastic Processes and their Applications, Elsevier, vol. 118(6), pages 1056-1070, June.
    3. Zhang, Lan & Mykland, Per A. & Ait-Sahalia, Yacine, 2005. "A Tale of Two Time Scales: Determining Integrated Volatility With Noisy High-Frequency Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1394-1411, December.
    4. Ole E. Barndorff-Nielsen & Peter Reinhard Hansen & Asger Lunde & Neil Shephard, 2008. "Designing Realized Kernels to Measure the ex post Variation of Equity Prices in the Presence of Noise," Econometrica, Econometric Society, vol. 76(6), pages 1481-1536, November.
    5. Zhou, Bin, 1996. "High-Frequency Data and Volatility in Foreign-Exchange Rates," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(1), pages 45-52, January.
    6. Mancino, M.E. & Sanfelici, S., 2008. "Robustness of Fourier estimator of integrated volatility in the presence of microstructure noise," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 2966-2989, February.
    7. Maria Elvira Mancino & Paul Malliavin, 2002. "Fourier series method for measurement of multivariate volatilities," Finance and Stochastics, Springer, vol. 6(1), pages 49-61.
    8. Ole E. Barndorff-Nielsen & Peter Reinhard Hansen & Asger Lunde & Neil Shephard, 2008. "Designing Realized Kernels to Measure the ex post Variation of Equity Prices in the Presence of Noise," Econometrica, Econometric Society, vol. 76(6), pages 1481-1536, November.
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    Cited by:

    1. Nien-Lin Liu & Hoang-Long Ngo, 2014. "Approximation of eigenvalues of spot cross volatility matrix with a view toward principal component analysis," Papers 1409.2214, arXiv.org.
    2. Cecilia Mancini & Vanessa Mattiussi & Roberto Renò, 2015. "Spot volatility estimation using delta sequences," Finance and Stochastics, Springer, vol. 19(2), pages 261-293, April.

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