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Spectral estimation of the fractional order of a Lévy process

  • Denis Belomestny

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.

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File URL: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2009-021.pdf
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Paper provided by Sonderforschungsbereich 649, Humboldt University, Berlin, Germany in its series SFB 649 Discussion Papers with number SFB649DP2009-021.

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Length: 38 pages
Date of creation: Apr 2009
Date of revision:
Handle: RePEc:hum:wpaper:sfb649dp2009-021
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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. Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy Models [1]," SFB 649 Discussion Papers SFB649DP2006-034, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  3. 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.
  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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