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Calibration of selfdecomposable Lévy models

  • Mathias Trabs
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    We study the nonparametric calibration of exponential, self-decomposable Levy models whose jump density can be characterized by the k-function, which is typically nonsmooth at zero. On the one hand the estimation of the drift, the activity measure alpha:= k(0+) + k(0-) and analog parameters for the derivatives are considered and on the other hand we estimate the k-function outside of a neighborhood of zero. Minimax convergence rates are derived, which depend on . Therefore, we construct estimators adapting to this unknown parameter. Our estimation method is based on spectral representations of the observed option prices and on regularization by cutting off high frequencies. Finally, the procedure is applied to simulations and real data.

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    File URL: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2011-073.pdf
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    Paper provided by Sonderforschungsbereich 649, Humboldt University, Berlin, Germany in its series SFB 649 Discussion Papers with number SFB649DP2011-073.

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    Length: 50 pages
    Date of creation: Nov 2011
    Date of revision:
    Handle: RePEc:hum:wpaper:sfb649dp2011-073
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    1. Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy Models [2]," SFB 649 Discussion Papers SFB649DP2006-035, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    2. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-24, October.
    3. Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy Models [1]," SFB 649 Discussion Papers SFB649DP2006-034, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    4. 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.
    5. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    6. V. Chernozhukov & I. Fernández-Val & A. Galichon, 2009. "Improving point and interval estimators of monotone functions by rearrangement," Biometrika, Biometrika Trust, vol. 96(3), pages 559-575.
    7. Johanna Kappus & Markus Reiß, 2010. "Estimation of the characteristics of a Lévy process observed at arbitrary frequency," SFB 649 Discussion Papers SFB649DP2010-015, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    8. 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.
    9. Johanna Kappus & Markus Rei�, 2010. "Estimation of the characteristics of a Lévy process observed at arbitrary frequency," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(s1), pages 314-328.
    10. Yor, Marc & Madan, Dilip B. & Carr, Peter & Geman, Hélyette, 2007. "Self-decomposability and option pricing," Economics Papers from University Paris Dauphine 123456789/1380, Paris Dauphine University.
    11. 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.
    12. Ernst Eberlein & Dilip Madan, 2009. "Sato processes and the valuation of structured products," Quantitative Finance, Taylor & Francis Journals, vol. 9(1), pages 27-42.
    13. Denis Belomestny & John Schoenmakers, 2006. "A jump-diffusion Libor model and its robust calibration," SFB 649 Discussion Papers SFB649DP2006-037, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    14. Efromovich, Sam & Samarov, Alex, 1996. "Asymptotic equivalence of nonparametric regression and white noise model has its limits," Statistics & Probability Letters, Elsevier, vol. 28(2), pages 143-145, June.
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