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Estimation of Tempered Stable Lévy Models of Infinite Variation

Author

Listed:
  • José E. Figueroa-López

    (Washington University in St. Louis)

  • Ruoting Gong

    (Illinois Institute of Technology)

  • Yuchen Han

    (Washington University in St. Louis)

Abstract

Truncated realized quadratic variations (TRQV) are among the most widely used high-frequency-based nonparametric methods to estimate the volatility of a process in the presence of jumps. Nevertheless, the truncation level is known to critically affect its performance, especially in the presence of infinite variation jumps. In this paper, we study the optimal truncation level, in the mean-square error sense, for a semiparametric tempered stable Lévy model. We obtain a novel closed-form 2nd-order approximation of the optimal threshold in a high-frequency setting. As an application, we propose a new estimation method, which combines iteratively an approximate semiparametric method of moment estimator and TRQVs with the newly found small-time approximation for the optimal threshold. The method is tested via simulations to estimate the volatility and the Blumenthal-Getoor index of a generalized CGMY model and, via a localization technique, to estimate the integrated volatility of a Heston type model with CGMY jumps. Our method is found to outperform other alternatives proposed in the literature when working with a Lévy process (i.e., the volatility is constant), or when the index of jump intensity Y is larger than 3/2 in the presence of stochastic volatility.

Suggested Citation

  • José E. Figueroa-López & Ruoting Gong & Yuchen Han, 2022. "Estimation of Tempered Stable Lévy Models of Infinite Variation," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 713-747, June.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:2:d:10.1007_s11009-022-09940-7
    DOI: 10.1007/s11009-022-09940-7
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    References listed on IDEAS

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    1. Zhang, Lan & Mykland, Per A. & Ait-Sahalia, Yacine, 2005. "A Tale of Two Time Scales: Determining Integrated Volatility With Noisy High-Frequency Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1394-1411, December.
    2. José E. Figueroa-López & Ruoting Gong & Christian Houdré, 2016. "High-Order Short-Time Expansions For Atm Option Prices Of Exponential Lévy Models," Mathematical Finance, Wiley Blackwell, vol. 26(3), pages 516-557, July.
    3. Reiß, Markus, 2013. "Testing the characteristics of a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2808-2828.
    4. Figueroa-López, José E. & Mancini, Cecilia, 2019. "Optimum thresholding using mean and conditional mean squared error," Journal of Econometrics, Elsevier, vol. 208(1), pages 179-210.
    5. 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.
    6. José E. Figueroa-López & Ruoting Gong & Christian Houdré, 2017. "Third-order short-time expansions for close-to-the-money option prices under the CGMY model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 24(6), pages 547-574, November.
    7. Jos'e E. Figueroa-L'opez & Ruoting Gong & Christian Houdr'e, 2013. "Third-Order Short-Time Expansions for Close-to-the-Money Option Prices under the CGMY Model," Papers 1305.4719, arXiv.org, revised Nov 2017.
    8. Adam D. Bull, 2014. "Near-optimal estimation of jump activity in semimartingales," Papers 1409.8150, arXiv.org, revised Jan 2016.
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