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Activity Signature Functions for High-Frequency Data Analysis

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  • George Tauchen
  • Viktor Todorov

Abstract

We define a new concept termed the activity signature function, which is constructed from discrete observations of a process evolving continuously in time. Under quite general regularity conditions, we derive the asymptotic properties of the function as the sampling frequency increases and show that it is a useful device for making inferences about the activity level of an Ito semimartingale. Monte Carlo work confirms the theoretical results. One empirical application is from finance. It indicates that the classical model comprised of a continuous component plus jumps is more plausible than a pure-jump model for the spot $/DM exchange rate over 1986-1999. A second application pertains to internet traffic data at NASA servers. We find that a pure-jump model with no continuous component and paths of infinite variation is appropriate for modeling this data set. In both cases the evidence obtained from the signature functions is quite convincing, and these two very disparate empirical outcomes illustrate the discriminatory power of the methodology.

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Bibliographic Info

Paper provided by Duke University, Department of Economics in its series Working Papers with number 10-08.

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Length: 30
Date of creation: 2010
Date of revision:
Handle: RePEc:duk:dukeec:10-08

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Postal: Department of Economics Duke University 213 Social Sciences Building Box 90097 Durham, NC 27708-0097
Phone: (919) 660-1800
Fax: (919) 684-8974
Web page: http://econ.duke.edu/

Related research

Keywords: activity index; Blumenthal-Getoor index; jumps; Levy process; realized power variation;

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References

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  1. Carr, Peter & Wu, Liuren, 2007. "Stochastic skew in currency options," Journal of Financial Economics, Elsevier, vol. 86(1), pages 213-247, October.
  2. S. Z. Levendorski, 2004. "Early exercise boundary and option prices in Levy driven models," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 525-547.
  3. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
  4. Jacod, Jean, 2008. "Asymptotic properties of realized power variations and related functionals of semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 517-559, April.
  5. Ballotta, Laura, 2005. "A Lévy process-based framework for the fair valuation of participating life insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 173-196, October.
  6. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382.
  7. 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.
  8. Elton A. Daal & Dilip B. Madan, 2005. "An Empirical Examination of the Variance-Gamma Model for Foreign Currency Options," The Journal of Business, University of Chicago Press, vol. 78(6), pages 2121-2152, November.
  9. 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.
  10. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
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Citations

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Cited by:
  1. Viktor Todorov & Iaryna Grynkiv & George Tauchen, 2010. "Realized Laplace Transforms for Estimation of Jump Diffusive Volatility Models," Working Papers 10-75, Duke University, Department of Economics.
  2. Viktor Todorov & George Tauchen & Iaryna Grynkiv, 2011. "Volatility Activity: Specification and Estimation," Working Papers 11-23, Duke University, Department of Economics.
  3. Torben G. Andersen & Oleg Bondarenko & Maria T. Gonzalez-Perez, 2011. "Coherent Model-Free Implied Volatility: A Corridor Fix for High-Frequency VIX," CREATES Research Papers 2011-49, School of Economics and Management, University of Aarhus.
  4. Tim Bollerslev & Viktor Todorov, 2010. "Estimation of Jump Tails," CREATES Research Papers 2010-16, School of Economics and Management, University of Aarhus.
  5. Deniz Erdemlioglu & Sébastien Laurent & Christopher J. Neely, 2013. "Which continuous-time model is most appropriate for exchange rates?," Working Papers 2013-024, Federal Reserve Bank of St. Louis.
  6. Julien Chevallier & Benoît Sévi, 2014. "On the Stochastic Properties of Carbon Futures Prices," Environmental & Resource Economics, European Association of Environmental and Resource Economists, vol. 58(1), pages 127-153, May.
  7. Julien Chevallier & Benoît Sévi, 2012. "On the Stochastic Properties of Carbon Futures Prices," Working Papers halshs-00720166, HAL.
  8. Benoît Sévi & César Baena, 2011. "Brownian motion vs. pure-jump processes for individual stocks," Economics Bulletin, AccessEcon, vol. 31(4), pages 3138-3152.
  9. Torben G. Andersen & Oleg Bondarenko & Viktor Todorov & George Tauchen, 2013. "The Fine Structure of Equity-Index Option Dynamics," CREATES Research Papers 2013-52, School of Economics and Management, University of Aarhus.

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