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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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    1. Federico M. Bandi & Roberto Reno, 2009. "Nonparametric Stochastic Volatility," Global COE Hi-Stat Discussion Paper Series gd08-035, Institute of Economic Research, Hitotsubashi University.
    2. Corsi, Fulvio & Pirino, Davide & Renò, Roberto, 2010. "Threshold bipower variation and the impact of jumps on volatility forecasting," Journal of Econometrics, Elsevier, vol. 159(2), pages 276-288, December.
    3. 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.
    4. Elena Andreou, 2004. "The Impact of Sampling Frequency and Volatility Estimators on Change-Point Tests," Journal of Financial Econometrics, Oxford University Press, vol. 2(2), pages 290-318.
    5. Jacod, Jean & Li, Yingying & Mykland, Per A. & Podolskij, Mark & Vetter, Mathias, 2009. "Microstructure noise in the continuous case: The pre-averaging approach," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2249-2276, July.
    6. Blundell,Richard & Newey,Whitney K. & Persson,Torsten (ed.), 2007. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521871532, September.
    7. repec:hal:journl:peer-00741630 is not listed on IDEAS
    8. Renò, Roberto, 2008. "Nonparametric Estimation Of The Diffusion Coefficient Of Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 24(5), pages 1174-1206, October.
    9. 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.
    10. Veraart, Almut E.D., 2010. "Inference For The Jump Part Of Quadratic Variation Of Itô Semimartingales," Econometric Theory, Cambridge University Press, vol. 26(2), pages 331-368, April.
    11. Federico M. Bandi & Peter C. B. Phillips, 2003. "Fully Nonparametric Estimation of Scalar Diffusion Models," Econometrica, Econometric Society, vol. 71(1), pages 241-283, January.
    12. Hoffmann, Marc, 2002. "Rate of convergence for parametric estimation in a stochastic volatility model," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 147-170, January.
    13. 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.
    14. Andersen, Torben G. & Bollerslev, Tim, 1997. "Intraday periodicity and volatility persistence in financial markets," Journal of Empirical Finance, Elsevier, vol. 4(2-3), pages 115-158, June.
    15. repec:ebl:ecbull:v:3:y:2004:i:36:p:1-8 is not listed on IDEAS
    16. Jean Jacod & Viktor Todorov, 2010. "Do price and volatility jump together?," Papers 1010.4990, arXiv.org.
    17. Blundell,Richard & Newey,Whitney & Persson,Torsten (ed.), 2007. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521692106, September.
    18. Shigeyoshi Ogawa & Simona Sanfelici, 2011. "An Improved Two‐step Regularization Scheme for Spot Volatility Estimation," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 40(3), pages 105-132, November.
    19. Blundell,Richard & Newey,Whitney K. & Persson,Torsten (ed.), 2007. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521692090, September.
    20. Mancini, Cecilia & Renò, Roberto, 2011. "Threshold estimation of Markov models with jumps and interest rate modeling," Journal of Econometrics, Elsevier, vol. 160(1), pages 77-92, January.
    21. Christophe Croux & Sébastien Laurent, 2011. "Outlyingness Weighted Covariation," Journal of Financial Econometrics, Oxford University Press, vol. 9(4), pages 657-684.
    22. Maria Elvira Mancino & Simona Sanfelici, 2011. "Estimating Covariance via Fourier Method in the Presence of Asynchronous Trading and Microstructure Noise," Journal of Financial Econometrics, Oxford University Press, vol. 9(2), pages 367-408, Spring.
    23. Cecilia Mancini, 2009. "Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 270-296, June.
    24. Foster, Dean P & Nelson, Daniel B, 1996. "Continuous Record Asymptotics for Rolling Sample Variance Estimators," Econometrica, Econometric Society, vol. 64(1), pages 139-174, January.
    25. Zu, Yang & Peter Boswijk, H., 2014. "Estimating spot volatility with high-frequency financial data," Journal of Econometrics, Elsevier, vol. 181(2), pages 117-135.
    26. repec:bla:jfinan:v:59:y:2004:i:1:p:227-260 is not listed on IDEAS
    27. Clément, Emmanuelle & Gloter, Arnaud, 2011. "Limit theorems in the Fourier transform method for the estimation of multivariate volatility," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 1097-1124, May.
    28. Blundell,Richard & Newey,Whitney & Persson,Torsten (ed.), 2007. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521871549, September.
    29. Taro Kanatani, 2004. "Integrated volatility measuring from unevenly sampled observations," Economics Bulletin, AccessEcon, vol. 3(36), pages 1-8.
    30. Bandi, Federico M. & Nguyen, Thong H., 2003. "On the functional estimation of jump-diffusion models," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 293-328.
    31. 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.
    32. Kristensen, Dennis, 2010. "Nonparametric Filtering Of The Realized Spot Volatility: A Kernel-Based Approach," Econometric Theory, Cambridge University Press, vol. 26(1), pages 60-93, February.
    33. 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.
    34. Alexander Alvarez & Fabien Panloup & Monique Pontier & Nicolas Savy, 2012. "Estimation of the instantaneous volatility," Statistical Inference for Stochastic Processes, Springer, vol. 15(1), pages 27-59, April.
    35. Maria Elvira Mancino & Paul Malliavin, 2002. "Fourier series method for measurement of multivariate volatilities," Finance and Stochastics, Springer, vol. 6(1), pages 49-61.
    Full references (including those not matched with items on IDEAS)

    Citations

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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. 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.
    5. Christensen, Kim & Oomen, Roel & Renò, Roberto, 2022. "The drift burst hypothesis," Journal of Econometrics, Elsevier, vol. 227(2), pages 461-497.
    6. 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.
    7. 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.
    8. 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.
    9. Park, Joon Y. & Wang, Bin, 2021. "Nonparametric estimation of jump diffusion models," Journal of Econometrics, Elsevier, vol. 222(1), pages 688-715.
    10. Zu, Yang & Peter Boswijk, H., 2014. "Estimating spot volatility with high-frequency financial data," Journal of Econometrics, Elsevier, vol. 181(2), pages 117-135.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. Ghysels, Eric, 2014. "Factor Analysis with Large Panels of Volatility Proxies," CEPR Discussion Papers 10034, C.E.P.R. Discussion Papers.
    19. Richard Y. Chen, 2019. "The Fourier Transform Method for Volatility Functional Inference by Asynchronous Observations," Papers 1911.02205, arXiv.org.

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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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