IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2401.02819.html
   My bibliography  Save this paper

Roughness Signature Functions

Author

Listed:
  • Peter Christensen

Abstract

Inspired by the activity signature introduced by Todorov and Tauchen (2010), which was used to measure the activity of a semimartingale, this paper introduces the roughness signature function. The paper illustrates how it can be used to determine whether a discretely observed process is generated by a continuous process that is rougher than a Brownian motion, a pure-jump process, or a combination of the two. Further, if a continuous rough process is present, the function gives an estimate of the roughness index. This is done through an extensive simulation study, where we find that the roughness signature function works as expected on rough processes. We further derive some asymptotic properties of this new signature function. The function is applied empirically to three different volatility measures for the S&P500 index. The three measures are realized volatility, the VIX, and the option-extracted volatility estimator of Todorov (2019). The realized volatility and option-extracted volatility show signs of roughness, with the option-extracted volatility appearing smoother than the realized volatility, while the VIX appears to be driven by a continuous martingale with jumps.

Suggested Citation

  • Peter Christensen, 2024. "Roughness Signature Functions," Papers 2401.02819, arXiv.org.
    • Handle: RePEc:arx:papers:2401.02819
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2401.02819
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Todorov, Viktor & Tauchen, George, 2010. "Activity signature functions for high-frequency data analysis," Journal of Econometrics, Elsevier, vol. 154(2), pages 125-138, February.
    2. Anine E. Bolko & Kim Christensen & Mikko S. Pakkanen & Bezirgen Veliyev, 2020. "A GMM approach to estimate the roughness of stochastic volatility," Papers 2010.04610, arXiv.org, revised Apr 2022.
    3. Wang, Xiaohu & Xiao, Weilin & Yu, Jun, 2023. "Modeling and forecasting realized volatility with the fractional Ornstein–Uhlenbeck process," Journal of Econometrics, Elsevier, vol. 232(2), pages 389-415.
    4. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    5. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2016. "Decoupling the short- and long-term behavior of stochastic volatility," Papers 1610.00332, arXiv.org, revised Jan 2021.
    6. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    7. Ole E. Barndorff-Nielsen & Peter Reinhard Hansen & Asger Lunde & Neil Shephard, 2008. "Designing Realized Kernels to Measure the ex post Variation of Equity Prices in the Presence of Noise," Econometrica, Econometric Society, vol. 76(6), pages 1481-1536, November.
    8. Giulia Livieri & Saad Mouti & Andrea Pallavicini & Mathieu Rosenbaum, 2018. "Rough volatility: Evidence from option prices," IISE Transactions, Taylor & Francis Journals, vol. 50(9), pages 767-776, September.
    9. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    10. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2015. "Hybrid scheme for Brownian semistationary processes," Papers 1507.03004, arXiv.org, revised May 2017.
    11. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2017. "Hybrid scheme for Brownian semistationary processes," Finance and Stochastics, Springer, vol. 21(4), pages 931-965, October.
    12. 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.
    13. Mikkel Bennedsen, 2020. "Semiparametric estimation and inference on the fractal index of Gaussian and conditionally Gaussian time series data," Econometric Reviews, Taylor & Francis Journals, vol. 39(9), pages 875-903, October.
    Full references (including those not matched with items on IDEAS)

    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. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2017. "Decoupling the short- and long-term behavior of stochastic volatility," CREATES Research Papers 2017-26, Department of Economics and Business Economics, Aarhus University.
    2. 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.
    3. Florian Bourgey & Stefano De Marco & Peter K. Friz & Paolo Pigato, 2023. "Local volatility under rough volatility," Mathematical Finance, Wiley Blackwell, vol. 33(4), pages 1119-1145, October.
    4. Blanka Horvath & Antoine Jacquier & Aitor Muguruza & Andreas Sojmark, 2017. "Functional central limit theorems for rough volatility," Papers 1711.03078, arXiv.org, revised Nov 2023.
    5. Raul Merino & Jan Posp'iv{s}il & Tom'av{s} Sobotka & Tommi Sottinen & Josep Vives, 2019. "Decomposition formula for rough Volterra stochastic volatility models," Papers 1906.07101, arXiv.org, revised Aug 2019.
    6. Ofelia Bonesini & Antoine Jacquier & Alexandre Pannier, 2023. "Rough volatility, path-dependent PDEs and weak rates of convergence," Papers 2304.03042, arXiv.org.
    7. Wu, Peng & Muzy, Jean-François & Bacry, Emmanuel, 2022. "From rough to multifractal volatility: The log S-fBM model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    8. Huy N. Chau & Duy Nguyen & Thai Nguyen, 2024. "On short-time behavior of implied volatility in a market model with indexes," Papers 2402.16509, arXiv.org, revised Apr 2024.
    9. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Papers 2210.12393, arXiv.org.
    10. Etienne Chevalier & Sergio Pulido & Elizabeth Zúñiga, 2021. "American options in the Volterra Heston model," Working Papers hal-03178306, HAL.
    11. Alexandre Pannier, 2023. "Path-dependent PDEs for volatility derivatives," Papers 2311.08289, arXiv.org, revised Jan 2024.
    12. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Working Papers hal-03827332, HAL.
    13. Etienne Chevalier & Sergio Pulido & Elizabeth Zúñiga, 2022. "American options in the Volterra Heston model," Post-Print hal-03178306, HAL.
    14. Etienne Chevalier & Sergio Pulido & Elizabeth Z'u~niga, 2021. "American options in the Volterra Heston model," Papers 2103.11734, arXiv.org, revised May 2022.
    15. Bolko, Anine E. & Christensen, Kim & Pakkanen, Mikko S. & Veliyev, Bezirgen, 2023. "A GMM approach to estimate the roughness of stochastic volatility," Journal of Econometrics, Elsevier, vol. 235(2), pages 745-778.
    16. Jonathan Haynes & Daniel Schmitt & Lukas Grimm, 2019. "Estimating stochastic volatility: the rough side to equity returns," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 449-469, December.
    17. Blanka Horvath & Josef Teichmann & Žan Žurič, 2021. "Deep Hedging under Rough Volatility," Risks, MDPI, vol. 9(7), pages 1-20, July.
    18. Alexandre Pannier & Cristopher Salvi, 2024. "A path-dependent PDE solver based on signature kernels," Papers 2403.11738, arXiv.org.
    19. Paul Gassiat, 2022. "Weak error rates of numerical schemes for rough volatility," Papers 2203.09298, arXiv.org, revised Feb 2023.
    20. Todorov, Viktor & Tauchen, George & Grynkiv, Iaryna, 2014. "Volatility activity: Specification and estimation," Journal of Econometrics, Elsevier, vol. 178(P1), pages 180-193.

    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:2401.02819. 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.