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Unbiased truncated quadratic variation for volatility estimation in jump diffusion processes

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  • Amorino, Chiara
  • Gloter, Arnaud

Abstract

The problem of integrated volatility estimation for an Ito semimartingale is considered under discrete high-frequency observations in short time horizon. We provide an asymptotic expansion for the integrated volatility that gives us, in detail, the contribution deriving from the jump part. The knowledge of such a contribution allows us to build an unbiased version of the truncated quadratic variation, in which the bias is visibly reduced. In earlier results to have the original truncated realized volatility well-performed the condition β>12(2−α) on β (that is such that (1n)β is the threshold of the truncated quadratic variation) and on the degree of jump activity α was needed (see Mancini, 2011; Jacod, 2008). In this paper we theoretically relax this condition and we show that our unbiased estimator achieves excellent numerical results for any couple (α, β).

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  • Amorino, Chiara & Gloter, Arnaud, 2020. "Unbiased truncated quadratic variation for volatility estimation in jump diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 5888-5939.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:10:p:5888-5939
    DOI: 10.1016/j.spa.2020.04.010
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    References listed on IDEAS

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    1. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    2. Mancini, Cecilia, 2011. "The speed of convergence of the Threshold estimator of integrated variance," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 845-855, April.
    3. Barndorff-Nielsen, Ole E. & Shephard, Neil & Winkel, Matthias, 2006. "Limit theorems for multipower variation in the presence of jumps," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 796-806, May.
    4. Ole E. Barndorff-Nielsen, 2004. "Power and Bipower Variation with Stochastic Volatility and Jumps," Journal of Financial Econometrics, Oxford University Press, vol. 2(1), pages 1-37.
    5. Jacod, Jean, 2008. "Asymptotic properties of realized power variations and related functionals of semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 517-559, April.
    6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    7. Yasutaka Shimizu & Nakahiro Yoshida, 2006. "Estimation of Parameters for Diffusion Processes with Jumps from Discrete Observations," Statistical Inference for Stochastic Processes, Springer, vol. 9(3), pages 227-277, October.
    8. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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    Cited by:

    1. B. Cooper Boniece & Jos'e E. Figueroa-L'opez & Yuchen Han, 2022. "Efficient Volatility Estimation for L\'evy Processes with Jumps of Unbounded Variation," Papers 2202.00877, arXiv.org.
    2. Milan Kumar Das & Anindya Goswami & Sharan Rajani, 2023. "Inference of Binary Regime Models with Jump Discontinuities," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 49-86, May.
    3. Chiara Amorino & Arnaud Gloter, 2021. "Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function," Statistical Inference for Stochastic Processes, Springer, vol. 24(1), pages 61-148, April.
    4. B. Cooper Boniece & Jos'e E. Figueroa-L'opez & Yuchen Han, 2022. "Efficient Integrated Volatility Estimation in the Presence of Infinite Variation Jumps via Debiased Truncated Realized Variations," Papers 2209.10128, arXiv.org, revised Apr 2024.

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