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Large Deviations of Realized Volatility

This paper studies large and moderate deviation properties of a realized volatility statistic of high frequency financial data. We establish a large deviation principle for the realized volatility when the number of high frequency observations in a fixed time interval increases to infinity. Our large deviation result can be used to evaluate tail probabilities of the realized volatility. We also derive a moderate deviation rate function for a standardized realized volatility statistic. The moderate deviation result is useful for assessing the validity of normal approximations based on the central limit theorem. In particular, it clarifies that there exists a trade-off between the accuracy of the normal approximations and the path regularity of an underlying volatility process. Our large and moderate deviation results complement the existing asymptotic theory on high frequency data. In addition, the paper contributes to the literature of large deviation theory in that the theory is extended to a high frequency data environment.

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File URL: http://cowles.econ.yale.edu/P/cd/d17b/d1798.pdf
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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1798.

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Length: 37 pages
Date of creation: May 2011
Date of revision:
Handle: RePEc:cwl:cwldpp:1798
Contact details of provider: Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
Phone: (203) 432-3702
Fax: (203) 432-6167
Web page: http://cowles.econ.yale.edu/

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Order Information: Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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  1. 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.
  2. Ole E. Barndorff-Nielsen & Neil Shephard & Matthias Winkel, 2005. "Limit theorems for multipower variation in the presence of jumps," OFRC Working Papers Series 2005fe06, Oxford Financial Research Centre.
  3. Kim Christensen & Roel Oomen & Mark Podolskij, 2010. "Realised quantile-based estimation of the integrated variance," Post-Print hal-00732538, HAL.
  4. Ole E. Barndorff-Nielsen & Neil Shephard, 2004. "Econometrics of testing for jumps in financial economics using bipower variation ," OFRC Working Papers Series 2004fe01, Oxford Financial Research Centre.
  5. Zhang, Lan & Mykland, Per A. & Aït-Sahalia, Yacine, 2011. "Edgeworth expansions for realized volatility and related estimators," Journal of Econometrics, Elsevier, vol. 160(1), pages 190-203, January.
  6. Barndorff-Nielsen, Ole Eiler & Graversen, Svend Erik & Jacod, Jean & Podolskij, Mark, 2004. "A central limit theorem for realised power and bipower variations of continuous semimartingales," Technical Reports 2004,51, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
  7. Ole E. Barndorff-Nielsen & Sven Erik Graversen & Jean Jacod & Neil Shephard, 2005. "Limit theorems for bipower variation in financial econometrics," OFRC Working Papers Series 2005fe09, Oxford Financial Research Centre.
  8. Ole E. Barndorff-Nielsen & Neil Shephard, 2000. "Econometric analysis of realised volatility and its use in estimating stochastic volatility models," Economics Papers 2001-W4, Economics Group, Nuffield College, University of Oxford, revised 05 Jul 2001.
  9. Neil Shephard & Ole E. Barndorff-Nielsen, 2006. "Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise," Economics Series Working Papers 2006-W03, University of Oxford, Department of Economics.
  10. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 2001. "Modeling and Forecasting Realized Volatility," NBER Working Papers 8160, National Bureau of Economic Research, Inc.
  11. Neil Shephard & Ole E. Barndorff-Nielsen, 2003. "Power and bipower variation with stochastic volatility and jumps," Economics Series Working Papers 2003-W18, University of Oxford, Department of Economics.
  12. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
  13. Meddahi, Nour & Mykland, Per & Shephard, Neil, 2011. "Realized Volatility," Journal of Econometrics, Elsevier, vol. 160(1), pages 1-1, January.
  14. Sílvia Gonçalves & Nour Meddahi, 2009. "Bootstrapping Realized Volatility," Econometrica, Econometric Society, vol. 77(1), pages 283-306, 01.
  15. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous-time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323.
  16. F. M. Bandi & J. R. Russell, 2008. "Microstructure Noise, Realized Variance, and Optimal Sampling," Review of Economic Studies, Oxford University Press, vol. 75(2), pages 339-369.
  17. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
  18. Yacine Aït-Sahalia, 2005. "How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise," Review of Financial Studies, Society for Financial Studies, vol. 18(2), pages 351-416.
  19. Comte, F. & Renault, E., 1996. "Long memory continuous time models," Journal of Econometrics, Elsevier, vol. 73(1), pages 101-149, July.
  20. repec:oxf:wpaper:264 is not listed on IDEAS
  21. Ole E. Barndorff-Nielsen, 2004. "Power and Bipower Variation with Stochastic Volatility and Jumps," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 2(1), pages 1-37.
  22. Marc Romano & Nizar Touzi, 1997. "Contingent Claims and Market Completeness in a Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 399-412.
  23. Grama, Ion & Haeusler, Erich, 2000. "Large deviations for martingales via Cramér's method," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 279-293, February.
  24. repec:fth:inseep:9607 is not listed on IDEAS
  25. Torben G. Andersen & Tim Bollerslev & Nour Meddahi, 2005. "Correcting the Errors: Volatility Forecast Evaluation Using High-Frequency Data and Realized Volatilities," Econometrica, Econometric Society, vol. 73(1), pages 279-296, 01.
  26. 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, vol. 63(2), pages 167-241.
  27. Torben G. Andersen & Tim Bollerslev & Dobrislav Dobrev, 2007. "No-Arbitrage Semi-Martingale Restrictions for Continuous-Time Volatility Models subject to Leverage Effects, Jumps and i.i.d. Noise: Theory and Testable Distributional Implications," NBER Working Papers 12963, National Bureau of Economic Research, Inc.
  28. Lesigne, Emmanuel & Volný, Dalibor, 2001. "Large deviations for martingales," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 143-159, November.
  29. Todorov, Viktor, 2011. "Econometric analysis of jump-driven stochastic volatility models," Journal of Econometrics, Elsevier, vol. 160(1), pages 12-21, January.
  30. Neil Shephard, 2004. "A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales," Economics Series Working Papers 2004-FE-21, University of Oxford, Department of Economics.
  31. Bercu, B. & Gamboa, F. & Rouault, A., 1997. "Large deviations for quadratic forms of stationary Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 75-90, October.
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