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Nonparametric Stochastic Volatility

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  • Federico M. Bandi
  • Roberto Reno

Abstract

Using recent advances in the nonparametric estimation of continuous-time processes under mild statistical assumptions as well as recent developments on nonparametric volatility estimation by virtue of market microstructure noise-contaminated high-frequency asset price data, we provide (i) a theory of spot variance estimation and (ii) functional methods for stochastic volatility modelling. Our methods allow for the joint evaluation of return and volatility dynamics with nonlinear drift and diffusion functions, nonlinear leverage effects, jumps in returns and volatility with possibly state-dependent jump intensities, as well as nonlinear risk-return trade-offs. Our identification approach and asymptotic results apply under weak recurrence assumptions and, hence, accommodate the persistence properties of variance in finite samples. Functional estimation of a generalized (i.e., nonlinear) version of the square-root stochastic variance model with jumps in both volatility and returns for the S&P500 index suggests the need for richer variance dynamics than in existing work. We find a linear specification for the variance's diffusive variance to be misspecified (and inferior to a more flexible CEV specification) even when allowing for jumps in the variance dynamics.

Suggested Citation

  • Federico M. Bandi & Roberto Reno, 2009. "Nonparametric Stochastic Volatility," Global COE Hi-Stat Discussion Paper Series gd08-035, Institute of Economic Research, Hitotsubashi University.
  • Handle: RePEc:hst:ghsdps:gd08-035
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    File URL: http://gcoe.ier.hit-u.ac.jp/research/discussion/2008/pdf/gd08-035.pdf
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    Cited by:

    1. Markus Bibinger & Nikolaus Hautsch & Peter Malec & Markus Reiss, 2019. "Estimating the Spot Covariation of Asset Prices—Statistical Theory and Empirical Evidence," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(3), pages 419-435, July.
    2. 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.
    3. repec:hal:journl:peer-00796745 is not listed on IDEAS
    4. Bandi, F.M. & Renò, R., 2016. "Price and volatility co-jumps," Journal of Financial Economics, Elsevier, vol. 119(1), pages 107-146.
    5. Charles S. Bos & Paweł Janus & Siem Jan Koopman, 2012. "Spot Variance Path Estimation and Its Application to High-Frequency Jump Testing," The Journal of Financial Econometrics, Society for Financial Econometrics, vol. 10(2), pages 354-389, 2012 06.
    6. Corradi, Valentina & Swanson, Norman R., 2011. "Predictive density construction and accuracy testing with multiple possibly misspecified diffusion models," Journal of Econometrics, Elsevier, vol. 161(2), pages 304-324, April.
    7. Park, Joon Y. & Wang, Bin, 2021. "Nonparametric estimation of jump diffusion models," Journal of Econometrics, Elsevier, vol. 222(1), pages 688-715.
    8. 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.
    9. Markus Bibinger & Markus Reiss & Nikolaus Hautsch & Peter Malec, 2014. "Estimating the Spot Covariation of Asset Prices – Statistical Theory and Empirical Evidence," SFB 649 Discussion Papers SFB649DP2014-055, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    10. Cecilia Mancini & Vanessa Mattiussi & Roberto Renò, 2015. "Spot volatility estimation using delta sequences," Finance and Stochastics, Springer, vol. 19(2), pages 261-293, April.
    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. Anzarut, Michelle & Mena, Ramsés H., 2019. "A Harris process to model stochastic volatility," Econometrics and Statistics, Elsevier, vol. 10(C), pages 151-169.

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    Keywords

    Spot variance; stochastic volatility; jumps in returns; jumps in volatility; leverage effects; risk-return trade-offs; kernel methods; recurrence; market microstructure noise.;
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