IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2304.12450.html
   My bibliography  Save this paper

Asymptotic Expansions for High-Frequency Option Data

Author

Listed:
  • Carsten H. Chong
  • Viktor Todorov

Abstract

We derive a nonparametric higher-order asymptotic expansion for small-time changes of conditional characteristic functions of It\^o semimartingale increments. The asymptotics setup is of joint type: both the length of the time interval of the increment of the underlying process and the time gap between evaluating the conditional characteristic function are shrinking. The spot semimartingale characteristics of the underlying process as well as their spot semimartingale characteristics appear as leading terms in the derived asymptotic expansions. The analysis applies to a general class of It\^o semimartingales that includes in particular L\'evy-driven SDEs and time-changed L\'evy processes. The asymptotic expansion results are of direct use for constructing nonparametric estimates pertaining to the stochastic volatility dynamics of an asset from high-frequency data of options written on the underlying asset.

Suggested Citation

  • Carsten H. Chong & Viktor Todorov, 2023. "Asymptotic Expansions for High-Frequency Option Data," Papers 2304.12450, arXiv.org.
    • Handle: RePEc:arx:papers:2304.12450
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2304.12450
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ilze Kalnina & Dacheng Xiu, 2017. "Nonparametric Estimation of the Leverage Effect: A Trade-Off Between Robustness and Efficiency," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 384-396, January.
    2. Carsten H. Chong & Viktor Todorov, 2023. "Volatility of Volatility and Leverage Effect from Options," Papers 2305.04137, arXiv.org, revised Jan 2024.
    3. Simona Sanfelici & Imma Valentina Curato & Maria Elvira Mancino, 2015. "High-frequency volatility of volatility estimation free from spot volatility estimates," Quantitative Finance, Taylor & Francis Journals, vol. 15(8), pages 1331-1345, August.
    4. Yacine Aït-Sahalia & Jean Jacod, 2014. "High-Frequency Financial Econometrics," Economics Books, Princeton University Press, edition 1, number 10261.
    5. Simon Clinet & Yoann Potiron, 2021. "Estimation for high-frequency data under parametric market microstructure noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(4), pages 649-669, August.
    6. P. Carr & D. Madan, 2001. "Optimal positioning in derivative securities," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 19-37.
    7. Johannes Muhle-Karbe & Marcel Nutz, 2010. "Small-Time Asymptotics of Option Prices and First Absolute Moments," Papers 1006.2294, arXiv.org, revised Jun 2011.
    8. Giacomo Toscano & Giulia Livieri & Maria Elvira Mancino & Stefano Marmi, 2021. "Volatility of volatility estimation: central limit theorems for the Fourier transform estimator and empirical study of the daily time series stylized facts," Papers 2112.14529, arXiv.org, revised Sep 2022.
    9. Carsten Chong & Viktor Todorov, 2022. "Short-time expansion of characteristic functions in a rough volatility setting with applications," Papers 2208.00830, arXiv.org.
    10. Todorov, Viktor, 2021. "Higher-order small time asymptotic expansion of Itô semimartingale characteristic function with application to estimation of leverage from options," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 671-705.
    11. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    12. Yacine Aït-Sahalia & Jianqing Fan & Roger J. A. Laeven & Christina Dan Wang & Xiye Yang, 2017. "Estimation of the Continuous and Discontinuous Leverage Effects," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(520), pages 1744-1758, October.
    13. Li, Yingying & Liu, Guangying & Zhang, Zhiyuan, 2022. "Volatility of volatility: Estimation and tests based on noisy high frequency data with jumps," Journal of Econometrics, Elsevier, vol. 229(2), pages 422-451.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Carsten H. Chong & Viktor Todorov, 2023. "Volatility of Volatility and Leverage Effect from Options," Papers 2305.04137, arXiv.org, revised Jan 2024.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Carsten H. Chong & Viktor Todorov, 2023. "Volatility of Volatility and Leverage Effect from Options," Papers 2305.04137, arXiv.org, revised Jan 2024.
    2. Carsten Chong & Marc Hoffmann & Yanghui Liu & Mathieu Rosenbaum & Gr'egoire Szymanski, 2022. "Statistical inference for rough volatility: Central limit theorems," Papers 2210.01216, arXiv.org, revised Jul 2023.
    3. Giacomo Toscano & Giulia Livieri & Maria Elvira Mancino & Stefano Marmi, 2021. "Volatility of volatility estimation: central limit theorems for the Fourier transform estimator and empirical study of the daily time series stylized facts," Papers 2112.14529, arXiv.org, revised Sep 2022.
    4. Li, Yingying & Liu, Guangying & Zhang, Zhiyuan, 2022. "Volatility of volatility: Estimation and tests based on noisy high frequency data with jumps," Journal of Econometrics, Elsevier, vol. 229(2), pages 422-451.
    5. Curato, Imma Valentina & Sanfelici, Simona, 2022. "Stochastic leverage effect in high-frequency data: a Fourier based analysis," Econometrics and Statistics, Elsevier, vol. 23(C), pages 53-82.
    6. Giacomo Toscano & Maria Cristina Recchioni, 2022. "Bias-optimal vol-of-vol estimation: the role of window overlapping," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 137-185, June.
    7. Todorov, Viktor, 2021. "Higher-order small time asymptotic expansion of Itô semimartingale characteristic function with application to estimation of leverage from options," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 671-705.
    8. Alessio Brini & Giacomo Toscano, 2024. "SpotV2Net: Multivariate Intraday Spot Volatility Forecasting via Vol-of-Vol-Informed Graph Attention Networks," Papers 2401.06249, arXiv.org.
    9. Giacomo Toscano & Maria Cristina Recchioni, 2020. "Bias optimal vol-of-vol estimation: the role of window overlapping," Papers 2004.04013, arXiv.org, revised Jul 2021.
    10. Bibinger, Markus & Neely, Christopher & Winkelmann, Lars, 2019. "Estimation of the discontinuous leverage effect: Evidence from the NASDAQ order book," Journal of Econometrics, Elsevier, vol. 209(2), pages 158-184.
    11. Maria Elvira Mancino & Tommaso Mariotti & Giacomo Toscano, 2022. "Asymptotic Normality for the Fourier spot volatility estimator in the presence of microstructure noise," Papers 2209.08967, arXiv.org.
    12. Carsten Chong & Viktor Todorov, 2022. "Short-time expansion of characteristic functions in a rough volatility setting with applications," Papers 2208.00830, arXiv.org.
    13. Curato, Imma Valentina, 2019. "Estimation of the stochastic leverage effect using the Fourier transform method," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3207-3238.
    14. Todorov, Viktor & Zhang, Yang, 2023. "Bias reduction in spot volatility estimation from options," Journal of Econometrics, Elsevier, vol. 234(1), pages 53-81.
    15. Masaaki Fukasawa & Tetsuya Takabatake & Rebecca Westphal, 2019. "Is Volatility Rough ?," Papers 1905.04852, arXiv.org, revised May 2019.
    16. Benth, Fred Espen & Schroers, Dennis & Veraart, Almut E.D., 2022. "A weak law of large numbers for realised covariation in a Hilbert space setting," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 241-268.
    17. Christensen, Kim & Thyrsgaard, Martin & Veliyev, Bezirgen, 2019. "The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing," Journal of Econometrics, Elsevier, vol. 212(2), pages 556-583.
    18. Li, Z. Merrick & Laeven, Roger J.A. & Vellekoop, Michel H., 2020. "Dependent microstructure noise and integrated volatility estimation from high-frequency data," Journal of Econometrics, Elsevier, vol. 215(2), pages 536-558.
    19. Aït-Sahalia, Yacine & Kalnina, Ilze & Xiu, Dacheng, 2020. "High-frequency factor models and regressions," Journal of Econometrics, Elsevier, vol. 216(1), pages 86-105.
    20. M.E. Mancino & S. Scotti & G. Toscano, 2020. "Is the Variance Swap Rate Affine in the Spot Variance? Evidence from S&P500 Data," Applied Mathematical Finance, Taylor & Francis Journals, vol. 27(4), pages 288-316, July.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2304.12450. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.