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Spectral estimation of the fractional order of a Lévy process

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  • Denis Belomestny

Abstract

We consider the problem of estimating the fractional order of a L´evy 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

  • Denis Belomestny, 2009. "Spectral estimation of the fractional order of a Lévy process," SFB 649 Discussion Papers SFB649DP2009-021, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    • Handle: RePEc:hum:wpaper: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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    Citations

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    Cited by:

    1. Roland Strausz, 2009. "The Political Economy of Regulatory Risk," SFB 649 Discussion Papers SFB649DP2009-040, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    2. Jakob Söhl, 2012. "Confidence sets in nonparametric calibration of exponential Lévy models," SFB 649 Discussion Papers SFB649DP2012-012, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    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. Reiß, Markus, 2013. "Testing the characteristics of a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2808-2828.
    5. 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.
    6. 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.
    7. Jakob Söhl, 2014. "Confidence sets in nonparametric calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 18(3), pages 617-649, July.
    8. Trabs, Mathias, 2015. "Quantile estimation for Lévy measures," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3484-3521.
    9. Barbara Choroś & Wolfgang Härdle & Ostap Okhrin, 2009. "CDO and HAC," SFB 649 Discussion Papers SFB649DP2009-038, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    10. Jakob Sohl, 2012. "Confidence sets in nonparametric calibration of exponential L\'evy models," Papers 1202.6611, arXiv.org, revised Sep 2013.
    11. Mathias Trabs, 2011. "Calibration of selfdecomposable Lévy models," SFB 649 Discussion Papers SFB649DP2011-073, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    12. 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.
    13. 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.
    14. Maria Grith & Wolfgang Härdle & Juhyun Park, 2009. "Shape invariant modelling pricing kernels and risk aversion," SFB 649 Discussion Papers SFB649DP2009-041, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

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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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