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Spectral estimation of the fractional order of a Lévy process

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  • Belomestny, Denis

Abstract

We consider the problem of estimating the fractional order of a Lévy process from low frequency historical and options data. An estimation methodology is developed which allows us to treat both estimation and calibration problems in a unified way. The corresponding procedure consists of two steps: the estimation of a conditional characteristic function and the weighted least squares estimation of the fractional order in spectral domain. While the second step is identical for both calibration and estimation, the first one depends on the problem at hand. Minimax rates of convergence for the fractional order estimate are derived, the asymptotic normality is proved and a data-driven algorithm based on aggregation is proposed. The performance of the estimator in both estimation and calibration setups is illustrated by a simulation study.

Suggested Citation

  • Belomestny, Denis, 2009. "Spectral estimation of the fractional order of a Lévy process," SFB 649 Discussion Papers 2009-021, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    • Handle: RePEc:zbw:sfb649:sfb649dp2009-021
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    References listed on IDEAS

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    1. Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 10(4), pages 449-474, December.
    2. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
    3. Svetlana I. Boyarchenko & Sergei Z. Levendorskiĭ, 2002. "Barrier options," World Scientific Book Chapters, in: Non-Gaussian Merton-Black-Scholes Theory, chapter 8, pages 185-198, World Scientific Publishing Co. Pte. Ltd..
    4. Akgiray, Vedat & Lamoureux, Christopher G, 1989. "Estimation of Stable-Law Parameters: A Comparative Study," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(1), pages 85-93, January.
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    Cited by:

    1. Michał Grajek & Lars-Hendrik Röller, 2012. "Regulation and Investment in Network Industries: Evidence from European Telecoms," Journal of Law and Economics, University of Chicago Press, vol. 55(1), pages 189-216.
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    3. Erdemlioglu, Deniz & Laurent, Sébastien & Neely, Christopher J., 2015. "Which continuous-time model is most appropriate for exchange rates?," Journal of Banking & Finance, Elsevier, vol. 61(S2), pages 256-268.
    4. Söhl, Jakob, 2012. "Confidence sets in nonparametric calibration of exponential Lévy models," SFB 649 Discussion Papers 2012-012, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    5. Reiß, Markus, 2013. "Testing the characteristics of a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2808-2828.
    6. Strausz, Roland, 2009. "The political economy of regulatory risk," SFB 649 Discussion Papers 2009-040, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    7. Yacine Aït-Sahalia & Jean Jacod, 2012. "Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data," Journal of Economic Literature, American Economic Association, vol. 50(4), pages 1007-1050, December.
    8. Trabs, Mathias, 2011. "Calibration of self-decomposable Lévy models," SFB 649 Discussion Papers 2011-073, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    9. repec:hum:wpaper:sfb649dp2011-073 is not listed on IDEAS
    10. Belomestny, Denis, 2011. "Spectral estimation of the Lévy density in partially observed affine models," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1217-1244, June.
    11. repec:hum:wpaper:sfb649dp2009-041 is not listed on IDEAS
    12. Grith, Maria & Härdle, Wolfgang Karl & Park, Juhyun, 2009. "Shape invariant modelling pricing kernels and risk aversion," SFB 649 Discussion Papers 2009-041, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    13. repec:hum:wpaper:sfb649dp2009-039 is not listed on IDEAS
    14. Choroś, Barbara & Härdle, Wolfgang Karl & Okhrin, Ostap, 2009. "CDO and HAC," SFB 649 Discussion Papers 2009-038, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    15. Jakob Söhl, 2014. "Confidence sets in nonparametric calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 18(3), pages 617-649, July.
    16. Trabs, Mathias, 2015. "Quantile estimation for Lévy measures," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3484-3521.
    17. repec:hum:wpaper:sfb649dp2009-038 is not listed on IDEAS
    18. Jakob Sohl, 2012. "Confidence sets in nonparametric calibration of exponential L\'evy models," Papers 1202.6611, arXiv.org, revised Sep 2013.
    19. repec:hum:wpaper:sfb649dp2009-040 is not listed on IDEAS
    20. Belomestny, Denis & Panov, Vladimir, 2013. "Abelian theorems for stochastic volatility models with application to the estimation of jump activity," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 15-44.

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    More about this item

    Keywords

    regular Lévy processes; Blumenthal-Getoor index; semiparametric estimation;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

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