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Spot volatility estimation using delta sequences

Author

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  • Cecilia Mancini
  • Vanessa Mattiussi
  • Roberto Renò

Abstract

We introduce a unifying class of nonparametric spot volatility estimators based on delta sequences and conceived to include many of the existing estimators in the field as special cases. The full limit theory is first derived when unevenly sampled observations under infill asymptotics and fixed time horizon are considered, and the state variable is assumed to follow a Brownian semimartingale. We then extend our class of estimators to include Poisson jumps or financial microstructure noise in the observed price process. This work makes different approaches (kernels, wavelets, Fourier) comparable. For example, we explicitly illustrate some drawbacks of the Fourier estimator. Specific delta sequences are applied to data from the S&P 500 stock index futures market. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Cecilia Mancini & Vanessa Mattiussi & Roberto Renò, 2015. "Spot volatility estimation using delta sequences," Finance and Stochastics, Springer, vol. 19(2), pages 261-293, April.
    • Handle: RePEc:spr:finsto:v:19:y:2015:i:2:p:261-293
    DOI: 10.1007/s00780-015-0255-1
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    Cited by:

    1. Maria Elvira Mancino & Tommaso Mariotti & Giacomo Toscano, 2022. "Asymptotic Normality for the Fourier spot volatility estimator in the presence of microstructure noise," Papers 2209.08967, arXiv.org.
    2. Rachele Foschi & Francesca Lilla & Cecilia Mancini, 2020. "Warnings about future jumps: properties of the exponential Hawkes model," Working Papers 13/2020, University of Verona, Department of Economics.
    3. Kanaya, Shin & Kristensen, Dennis, 2016. "Estimation Of Stochastic Volatility Models By Nonparametric Filtering," Econometric Theory, Cambridge University Press, vol. 32(4), pages 861-916, August.
    4. Park, Joon Y. & Wang, Bin, 2021. "Nonparametric estimation of jump diffusion models," Journal of Econometrics, Elsevier, vol. 222(1), pages 688-715.
    5. Liu, Qiang & Liu, Yiqi & Liu, Zhi & Wang, Li, 2018. "Estimation of spot volatility with superposed noisy data," The North American Journal of Economics and Finance, Elsevier, vol. 44(C), pages 62-79.
    6. Curato, Imma Valentina & Mancino, Maria Elvira & Recchioni, Maria Cristina, 2018. "Spot volatility estimation using the Laplace transform," Econometrics and Statistics, Elsevier, vol. 6(C), pages 22-43.
    7. Ghysels, Eric, 2014. "Factor Analysis with Large Panels of Volatility Proxies," CEPR Discussion Papers 10034, C.E.P.R. Discussion Papers.
    8. Christensen, Kim & Oomen, Roel & Renò, Roberto, 2022. "The drift burst hypothesis," Journal of Econometrics, Elsevier, vol. 227(2), pages 461-497.
    9. Jos'e E. Figueroa-L'opez & Cheng Li & Jeffrey Nisen, 2018. "Optimal Iterative Threshold-Kernel Estimation of Jump Diffusion Processes," Papers 1811.07499, arXiv.org, revised Apr 2020.
    10. Richard Y. Chen, 2019. "The Fourier Transform Method for Volatility Functional Inference by Asynchronous Observations," Papers 1911.02205, arXiv.org.
    11. Zu, Yang & Peter Boswijk, H., 2014. "Estimating spot volatility with high-frequency financial data," Journal of Econometrics, Elsevier, vol. 181(2), pages 117-135.
    12. Li, Yingying & Liu, Guangying & Zhang, Zhiyuan, 2022. "Volatility of volatility: Estimation and tests based on noisy high frequency data with jumps," Journal of Econometrics, Elsevier, vol. 229(2), pages 422-451.
    13. José E. Figueroa-López & Cheng Li & Jeffrey Nisen, 2020. "Optimal iterative threshold-kernel estimation of jump diffusion processes," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 517-552, October.
    14. Chao Yu & Yue Fang & Zeng Li & Bo Zhang & Xujie Zhao, 2014. "Non-Parametric Estimation Of High-Frequency Spot Volatility For Brownian Semimartingale With Jumps," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(6), pages 572-591, November.
    15. Mancini, Cecilia, 2023. "Drift burst test statistic in the presence of infinite variation jumps," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 535-591.
    16. Zhang, Congshan & Li, Jia & Bollerslev, Tim, 2022. "Occupation density estimation for noisy high-frequency data," Journal of Econometrics, Elsevier, vol. 227(1), pages 189-211.
    17. Liu, Qiang & Liu, Yiqi & Liu, Zhi, 2018. "Estimating spot volatility in the presence of infinite variation jumps," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 1958-1987.
    18. Figueroa-López, José E. & Li, Cheng, 2020. "Optimal kernel estimation of spot volatility of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4693-4720.
    19. Maria Elvira Mancino & Maria Cristina Recchioni, 2015. "Fourier Spot Volatility Estimator: Asymptotic Normality and Efficiency with Liquid and Illiquid High-Frequency Data," PLOS ONE, Public Library of Science, vol. 10(9), pages 1-33, September.

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    More about this item

    Keywords

    Spot volatility; High-frequency data; Microstructure noise; Dirac delta; Fourier estimator; 91G70; C13; C14; C22; G1;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • G1 - Financial Economics - - General Financial Markets

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