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Stochastic volatility of volatility in continuous time

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Author Info

  • Ole E. Barndorff-Nielsen

    ()
    (The T.N. Thiele Centre for Mathematics in Natural Science, Department of Mathematical Sciences, & CREATES, Aarhus University)

  • Almut E. D. Veraart

    ()
    (School of Economics and Management, Aarhus University and CREATES)

Abstract

This paper introduces the concept of stochastic volatility of volatility in continuous time and, hence, extends standard stochastic volatility (SV) models to allow for an additional source of randomness associated with greater variability in the data. We discuss how stochastic volatility of volatility can be defined both non–parametrically, where we link it to the quadratic variation of the stochastic variance process, and parametrically, where we propose two new SV models which allow for stochastic volatility of volatility. In addition, we show that volatility of volatility can be estimated by a novel estimator called pre–estimated spot variance based realised variance.

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Bibliographic Info

Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2009-25.

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Length: 36
Date of creation: 06 Jul 2009
Date of revision:
Handle: RePEc:aah:create:2009-25

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Web page: http://www.econ.au.dk/afn/

Related research

Keywords: Stochastic volatility; volatility of volatility; non-Gaussian Ornstein–Uhlenbeck process; superposition; leverage effect; Lévy processes.;

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References

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  1. Ole BARNDORFF-NIELSEN & Svend Erik GRAVERSEN & Jean JACOD & Mark PODOLSKIJ & Neil SHEPHARD, 2004. "A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales," OFRC Working Papers Series, Oxford Financial Research Centre 2004fe21, Oxford Financial Research Centre.
  2. Almut E. D. Veraart & Luitgard A. M. Veraart, 2009. "Stochastic volatility and stochastic leverage," CREATES Research Papers, School of Economics and Management, University of Aarhus 2009-20, School of Economics and Management, University of Aarhus.
  3. Veraart, Almut E.D., 2010. "Inference For The Jump Part Of Quadratic Variation Of Itô Semimartingales," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 26(02), pages 331-368, April.
  4. Tim Bollerslev & Hao Zhou, 2007. "Expected Stock Returns and Variance Risk Premia," CREATES Research Papers, School of Economics and Management, University of Aarhus 2007-17, School of Economics and Management, University of Aarhus.
  5. Mark Podolskij & Mathias Vetter, 2007. "Estimation of Volatility Functionals in the Simultaneous Presence of Microstructure Noise and Jumps," CREATES Research Papers, School of Economics and Management, University of Aarhus 2007-27, School of Economics and Management, University of Aarhus.
  6. Harvey, Andrew C & Shephard, Neil, 1996. "Estimation of an Asymmetric Stochastic Volatility Model for Asset Returns," Journal of Business & Economic Statistics, American Statistical Association, American Statistical Association, vol. 14(4), pages 429-34, October.
  7. Ole E. Barndorff-Nielsen & Neil Shephard, 2001. "Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, Royal Statistical Society, vol. 63(2), pages 167-241.
  8. Andersen, Torben G. & Bollerslev, Tim & Diebold, Francis X. & Ebens, Heiko, 2001. "The distribution of realized stock return volatility," Journal of Financial Economics, Elsevier, Elsevier, vol. 61(1), pages 43-76, July.
  9. Hansen, Peter R. & Lunde, Asger, 2006. "Realized Variance and Market Microstructure Noise," Journal of Business & Economic Statistics, American Statistical Association, American Statistical Association, vol. 24, pages 127-161, April.
  10. Neil Shephard & Ole Barndorff-Nielsen, 2003. "Econometrics of testing for jumps in financial economics using bipower variation," Economics Series Working Papers, University of Oxford, Department of Economics 2004-FE-01, University of Oxford, Department of Economics.
  11. Silja Kinnebrock & Mark Podolskij, 2007. "A Note on the Central Limit Theorem for Bipower Variation of General Functions," OFRC Working Papers Series, Oxford Financial Research Centre 2007fe03, Oxford Financial Research Centre.
  12. Jacod, Jean, 2008. "Asymptotic properties of realized power variations and related functionals of semimartingales," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 118(4), pages 517-559, April.
  13. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, Econometric Society, vol. 59(2), pages 347-70, March.
  14. Jeannette H.C. Woerner, 2003. "Purely discontinuous Levy processes and power variation: inference for integrated volatility and the scale parameter," OFRC Working Papers Series, Oxford Financial Research Centre 2003mf08, Oxford Financial Research Centre.
  15. Jun Yu, 2004. "On leverage in a stochastic volatility model," Econometric Society 2004 Far Eastern Meetings, Econometric Society 497, Econometric Society.
  16. repec:oxf:wpaper:264 is not listed on IDEAS
  17. Suzanne S. Lee & Per A. Mykland, 2008. "Jumps in Financial Markets: A New Nonparametric Test and Jump Dynamics," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 21(6), pages 2535-2563, November.
  18. Fulvio Corsi & Stefan Mittnik & Christian Pigorsch & Uta Pigorsch, 2008. "The Volatility of Realized Volatility," Econometric Reviews, Taylor & Francis Journals, Taylor & Francis Journals, vol. 27(1-3), pages 46-78.
  19. Granger, C. W. J., 1981. "Some properties of time series data and their use in econometric model specification," Journal of Econometrics, Elsevier, Elsevier, vol. 16(1), pages 121-130, May.
  20. Elisa Nicolato & Emmanouil Venardos, 2003. "Option Pricing in Stochastic Volatility Models of the Ornstein-Uhlenbeck type," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 13(4), pages 445-466.
  21. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, Econometric Society, vol. 53(2), pages 385-407, March.
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Citations

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Cited by:
  1. Imma Valentina Curato, 2012. "Asymptotics for the Fourier estimators of the volatility of volatility and the leverage," Working Papers - Mathematical Economics, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa 2012-11, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa.
  2. Almut Veraart & Luitgard Veraart, 2012. "Stochastic volatility and stochastic leverage," Annals of Finance, Springer, Springer, vol. 8(2), pages 205-233, May.

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