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Quantile estimation for Lévy measures

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  • Trabs, Mathias

Abstract

Generalizing the concept of quantiles to the jump measure of a Lévy process, the generalized quantiles qτ±>0, for τ>0, are given by the smallest values such that a jump larger than qτ+ or a negative jump smaller than −qτ−, respectively, is expected only once in 1/τ time units. Nonparametric estimators of the generalized quantiles are constructed using either discrete observations of the process or using option prices in an exponential Lévy model of asset prices. In both models minimax convergence rates are shown. Applying Lepski’s approach, we derive adaptive quantile estimators. The performance of the estimation method is illustrated in simulations and with real data.

Suggested Citation

  • Trabs, Mathias, 2015. "Quantile estimation for Lévy measures," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3484-3521.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:9:p:3484-3521
    DOI: 10.1016/j.spa.2015.04.004
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    References listed on IDEAS

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    1. Trabs, Mathias, 2014. "On infinitely divisible distributions with polynomially decaying characteristic functions," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 56-62.
    2. repec:hum:wpaper:sfb649dp2009-021 is not listed on IDEAS
    3. Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 10(4), pages 449-474, December.
    4. Johanna Kappus & Markus Reiß, 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.
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    6. Kappus, Johanna, 2014. "Adaptive nonparametric estimation for Lévy processes observed at low frequency," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 730-758.
    7. repec:hum:wpaper:sfb649dp2012-003 is not listed on IDEAS
    8. 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.
    9. repec:hum:wpaper:sfb649dp2006-034 is not listed on IDEAS
    10. Reiß, Markus, 2013. "Testing the characteristics of a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2808-2828.
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    12. Hans FÃllmer & Peter Leukert, 1999. "Quantile hedging," Finance and Stochastics, Springer, vol. 3(3), pages 251-273.
    13. 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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    1. Kato, Kengo & Kurisu, Daisuke, 2020. "Bootstrap confidence bands for spectral estimation of Lévy densities under high-frequency observations," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1159-1205.
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    3. Céline Duval & Jalal Taher & Ester Mariucci, 2025. "Nonparametric density estimation for the small jumps of Lévy processes," Statistical Inference for Stochastic Processes, Springer, vol. 28(3), pages 1-26, December.
    4. Todorov, Viktor, 2022. "Nonparametric jump variation measures from options," Journal of Econometrics, Elsevier, vol. 230(2), pages 255-280.

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