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Inverse Realized Laplace Transforms for Nonparametric Volatility Estimation in Jump-Diffusions


  • Viktor Todorov
  • George Tauchen


We develop a nonparametric estimator of the stochastic volatility density of a discretely-observed Ito semimartingale in the setting of an increasing time span and finer mesh of the observation grid. There are two steps. The first is aggregating the high-frequency increments into the realized Laplace transform, which is a robust nonparametric estimate of the underlying volatility Laplace transform. The second step is using a regularized kernel to invert the realized Laplace transform. The two steps are relatively quick and easy to compute, so the nonparametric estimator is practicable. We derive bounds for the mean squared error of the estimator. The regularity conditions are sufficiently general to cover empirically important cases such as level jumps and possible dependencies between volatility moves and either diffusive or jump moves in the semimartingale. Monte Carlo work indicates that the nonparametric estimator is reliable and reasonably accurate in realistic estimation contexts. An empirical application to 5-minute data for three large-cap stocks, 1997-2010, reveals the importance of big short-term volatility spikes in generating high levels of stock price variability over and above that induced by price jumps. The application also shows how to trace out the dynamic response of the volatility density to both positive and negative jumps in the stock price.

Suggested Citation

  • Viktor Todorov & George Tauchen, 2011. "Inverse Realized Laplace Transforms for Nonparametric Volatility Estimation in Jump-Diffusions," Working Papers 11-21, Duke University, Department of Economics.
  • Handle: RePEc:duk:dukeec:11-21

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    References listed on IDEAS

    1. Todorov, Viktor & Tauchen, George, 2010. "Activity signature functions for high-frequency data analysis," Journal of Econometrics, Elsevier, vol. 154(2), pages 125-138, February.
    2. Torben G. Andersen & Luca Benzoni & Jesper Lund, 2002. "An Empirical Investigation of Continuous-Time Equity Return Models," Journal of Finance, American Finance Association, vol. 57(3), pages 1239-1284, June.
    3. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    4. Viktor Todorov & George Tauchen, 2011. "Volatility Jumps," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(3), pages 356-371, July.
    5. Carrasco, Marine & Chernov, Mikhail & Florens, Jean-Pierre & Ghysels, Eric, 2007. "Efficient estimation of general dynamic models with a continuum of moment conditions," Journal of Econometrics, Elsevier, vol. 140(2), pages 529-573, October.
    6. Viktor Todorov & George Tauchen, 2012. "The Realized Laplace Transform of Volatility," Econometrica, Econometric Society, vol. 80(3), pages 1105-1127, May.
    7. Chernov, Mikhail & Ronald Gallant, A. & Ghysels, Eric & Tauchen, George, 2003. "Alternative models for stock price dynamics," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 225-257.
    8. 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.
    9. Todorov, Viktor & Tauchen, George & Grynkiv, Iaryna, 2011. "Realized Laplace transforms for estimation of jump diffusive volatility models," Journal of Econometrics, Elsevier, vol. 164(2), pages 367-381, October.
    10. Cecilia Mancini, 2009. "Non-parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 270-296.
    11. 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.
    12. 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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    Cited by:

    1. Li, Gang & Zhang, Chu, 2016. "On the relationship between conditional jump intensity and diffusive volatility," Journal of Empirical Finance, Elsevier, vol. 37(C), pages 196-213.
    2. Jia Li & Andrew J. Patton, 2013. "Asymptotic Inference about Predictive Accuracy Using High Frequency Data," Working Papers 13-27, Duke University, Department of Economics.
    3. Clements, A.E. & Hurn, A.S. & Volkov, V.V., 2016. "Common trends in global volatility," Journal of International Money and Finance, Elsevier, vol. 67(C), pages 194-214.

    More about this item


    Laplace transform; stochastic volatility; ill-posed problems; regularization; nonparametric density estimation; high-frequency data;

    JEL classification:

    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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