IDEAS home Printed from https://ideas.repec.org/p/sbs/wpsefe/2004fe21.html
   My bibliography  Save this paper

A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales

Author

Listed:
  • Ole BARNDORFF-NIELSEN
  • Svend Erik GRAVERSEN
  • Jean JACOD
  • Mark PODOLSKIJ
  • Neil SHEPHARD

Abstract

Consider a semimartingale of the form Y_{t}=Y_0+\int _0^{t}a_{s}ds+\int _0^{t}_{s-} dW_{s}, where a is a locally bounded predictable process and (the "volatility") is an adapted right--continuous process with left limits and W is a Brownian motion. We define the realised bipower variation process V(Y;r,s)_{t}^n=n^{((r+s)/2)-1} \sum_{i=1}^{[nt]}|Y_{(i/n)}-Y_{((i-1)/n)}|^{r}|Y_{((i+1)/n)}-Y_{(i/n)}|^{s}, where r and s are nonnegative reals with r+s>0. We prove that V(Y;r,s)_{t}n converges locally uniformly in time, in probability, to a limiting process V(Y;r,s)_{t} (the "bipower variation process"). If further is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with W and by a Poisson random measure, we prove a central limit theorem, in the sense that \sqrt(n) (V(Y;r,s)^n-V(Y;r,s)) converges in law to a process which is the stochastic integral with respect to some other Brownian motion W', which is independent of the driving terms of Y and \sigma. We also provide a multivariate version of these results.

Suggested Citation

  • 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 2004fe21, Oxford Financial Research Centre.
  • Handle: RePEc:sbs:wpsefe:2004fe21
    as

    Download full text from publisher

    File URL: http://www.finance.ox.ac.uk/file_links/finecon_papers/2004fe21.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Ole E. Barndorff-Nielsen & Neil Shephard, 2006. "Econometrics of Testing for Jumps in Financial Economics Using Bipower Variation," Journal of Financial Econometrics, Oxford University Press, vol. 4(1), pages 1-30.
    2. Ole E. Barndorff-Nielsen & Neil Shephard, 2004. "Econometric Analysis of Realized Covariation: High Frequency Based Covariance, Regression, and Correlation in Financial Economics," Econometrica, Econometric Society, vol. 72(3), pages 885-925, May.
    3. Ole E. Barndorff‐Nielsen & Neil Shephard, 2002. "Econometric analysis of realized volatility and its use in estimating stochastic volatility models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 253-280, May.
    4. Ole E. Barndorff-Nielsen, 2004. "Power and Bipower Variation with Stochastic Volatility and Jumps," Journal of Financial Econometrics, Oxford University Press, vol. 2(1), pages 1-37.
    5. Ole E. Barndorff-Nielsen & Neil Shephard, 2001. "Econometric Analysis of Realised Covariation: High Frequency Covariance, Regression and Correlation in Financial Economics," Economics Papers 2002-W13, Economics Group, Nuffield College, University of Oxford, revised 18 Mar 2002.
    6. Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford.
    7. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 2003. "Modeling and Forecasting Realized Volatility," Econometrica, Econometric Society, vol. 71(2), pages 579-625, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bollerslev, Tim & Gibson, Michael & Zhou, Hao, 2011. "Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities," Journal of Econometrics, Elsevier, vol. 160(1), pages 235-245, January.
    2. Barndorff-Nielsen, Ole E. & Graversen, Svend Erik & Jacod, Jean & Shephard, Neil, 2006. "Limit Theorems For Bipower Variation In Financial Econometrics," Econometric Theory, Cambridge University Press, vol. 22(4), pages 677-719, August.
    3. Barndorff-Nielsen, Ole E. & Shephard, Neil & Winkel, Matthias, 2006. "Limit theorems for multipower variation in the presence of jumps," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 796-806, May.
    4. Tauchen, George & Zhou, Hao, 2011. "Realized jumps on financial markets and predicting credit spreads," Journal of Econometrics, Elsevier, vol. 160(1), pages 102-118, January.
    5. Ole E. Barndorff-Nielsen & Neil Shephard, 2006. "Econometrics of Testing for Jumps in Financial Economics Using Bipower Variation," Journal of Financial Econometrics, Oxford University Press, vol. 4(1), pages 1-30.
    6. Kim, Jihyun & Meddahi, Nour, 2020. "Volatility regressions with fat tails," Journal of Econometrics, Elsevier, vol. 218(2), pages 690-713.
    7. Ole E. Barndorff-Nielsen & Svend Erik Graversen & Neil Shephard, 2003. "Power variation & stochastic volatility: a review and some new results," Economics Papers 2003-W19, Economics Group, Nuffield College, University of Oxford.
    8. Bollerslev, Tim & Patton, Andrew J. & Quaedvlieg, Rogier, 2016. "Exploiting the errors: A simple approach for improved volatility forecasting," Journal of Econometrics, Elsevier, vol. 192(1), pages 1-18.
    9. Veiga, Helena, 2006. "Volatility forecasts: a continuous time model versus discrete time models," DES - Working Papers. Statistics and Econometrics. WS ws062509, Universidad Carlos III de Madrid. Departamento de Estadística.
    10. Jeremy Large, 2005. "Estimating Quadratic Variation When Quoted Prices Jump by a Constant Increment," Economics Series Working Papers 2005-FE-05, University of Oxford, Department of Economics.
    11. Chaboud, Alain P. & Chiquoine, Benjamin & Hjalmarsson, Erik & Loretan, Mico, 2010. "Frequency of observation and the estimation of integrated volatility in deep and liquid financial markets," Journal of Empirical Finance, Elsevier, vol. 17(2), pages 212-240, March.
    12. 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.
    13. Barndorff-Nielsen, Ole E. & Shephard, Neil, 2006. "Impact of jumps on returns and realised variances: econometric analysis of time-deformed Levy processes," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 217-252.
    14. Gregory Bauer & Keith Vorkink, 2007. "Multivariate Realized Stock Market Volatility," Staff Working Papers 07-20, Bank of Canada.
    15. Baruník, Jozef & Hlínková, Michaela, 2016. "Revisiting the long memory dynamics of the implied–realized volatility relationship: New evidence from the wavelet regression," Economic Modelling, Elsevier, vol. 54(C), pages 503-514.
    16. Denisa Georgiana Banulescu & Ferrara Laurent & Marsilli Clément, 2019. "Prévoir la volatilité d’un actif financier à l’aide d’un modèle à mélange de fréquences," Working Papers hal-03563168, HAL.
    17. Bauer, Gregory H. & Vorkink, Keith, 2011. "Forecasting multivariate realized stock market volatility," Journal of Econometrics, Elsevier, vol. 160(1), pages 93-101, January.
    18. Christophe Chorro & Florian Ielpo & Benoît Sévi, 2017. "The contribution of jumps to forecasting the density of returns," Post-Print halshs-01442618, HAL.
    19. Denisa Banulescu-Radu & Christophe Hurlin & Bertrand Candelon & Sébastien Laurent, 2016. "Do We Need High Frequency Data to Forecast Variances?," Annals of Economics and Statistics, GENES, issue 123-124, pages 135-174.
    20. Maneesoonthorn, Worapree & Martin, Gael M. & Forbes, Catherine S. & Grose, Simone D., 2012. "Probabilistic forecasts of volatility and its risk premia," Journal of Econometrics, Elsevier, vol. 171(2), pages 217-236.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:sbs:wpsefe:2004fe21. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Maxine Collett (email available below). General contact details of provider: https://edirc.repec.org/data/frcoxuk.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.