IDEAS home Printed from https://ideas.repec.org/a/taf/quantf/v13y2013i8p1185-1197.html
   My bibliography  Save this article

Using relative returns to accommodate fat-tailed innovations in processes and option pricing

Author

Listed:
  • Cathy O'Neil
  • Gilles Zumbach

Abstract

The canonical process used to describe financial time series is based on a logarithmic random walk. Adding a fat-tail distribution for the innovations in this framework creates fundamental inconsistencies, essentially related to a diverging integral. The problems are related to (1) the (infinite) values of statistical quantities for processes with fat-tail innovations, (2) the robustness of computations when dealing with large events (genuine or noise), and (3) the pricing of options with heteroskedasticity and fat tails. Instead of logarithmic processes, we advocate using geometric processes and relative returns so that these problems do not occur. The mathematical properties of geometric processes are explored for setups of increasing complexity. For empirical time series and for processes, the skews are evaluated using a robust estimator and are compared for both return definitions. Finally, the European option pricing framework is modified to use geometric processes for the underlying, allowing the incorporation of natural fat-tail innovations.

Suggested Citation

  • Cathy O'Neil & Gilles Zumbach, 2013. "Using relative returns to accommodate fat-tailed innovations in processes and option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 13(8), pages 1185-1197, July.
    • Handle: RePEc:taf:quantf:v:13:y:2013:i:8:p:1185-1197
    DOI: 10.1080/14697688.2012.727462
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/14697688.2012.727462
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/14697688.2012.727462?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Peter Christoffersen & Redouane Elkamhi & Bruno Feunou & Kris Jacobs, 2010. "Option Valuation with Conditional Heteroskedasticity and Nonnormality," The Review of Financial Studies, Society for Financial Studies, vol. 23(5), pages 2139-2183.
    2. Gilles Zumbach, 2004. "Volatility processes and volatility forecast with long memory," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 70-86.
    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. Gilles Zumbach & Luis Fernández, 2012. "Fast and realistic European ARCH option pricing and hedging," Quantitative Finance, Taylor & Francis Journals, vol. 13(5), pages 713-728, November.
    2. Zumbach, Gilles, 2012. "Option pricing and ARCH processes," Finance Research Letters, Elsevier, vol. 9(3), pages 144-156.
    3. Salman Huseynov, 2021. "Long and short memory in dynamic term structure models," CREATES Research Papers 2021-15, Department of Economics and Business Economics, Aarhus University.
    4. Monfort, Alain & Pegoraro, Fulvio, 2012. "Asset pricing with Second-Order Esscher Transforms," Journal of Banking & Finance, Elsevier, vol. 36(6), pages 1678-1687.
    5. Sigurd Emil Rømer & Rolf Poulsen, 2020. "How Does the Volatility of Volatility Depend on Volatility?," Risks, MDPI, vol. 8(2), pages 1-18, June.
    6. Alexandre Carbonneau & Fr'ed'eric Godin, 2021. "Deep equal risk pricing of financial derivatives with non-translation invariant risk measures," Papers 2107.11340, arXiv.org.
    7. Lars Stentoft, 2008. "Option Pricing using Realized Volatility," CREATES Research Papers 2008-13, Department of Economics and Business Economics, Aarhus University.
    8. Matthias R. Fengler & Helmut Herwartz & Christian Werner, 2012. "A Dynamic Copula Approach to Recovering the Index Implied Volatility Skew," Journal of Financial Econometrics, Oxford University Press, vol. 10(3), pages 457-493, June.
    9. Wenting Liu & Zhaozhong Gui & Guilin Jiang & Lihua Tang & Lichun Zhou & Wan Leng & Xulong Zhang & Yujiang Liu, 2023. "Stock Volatility Prediction Based on Transformer Model Using Mixed-Frequency Data," Papers 2309.16196, arXiv.org.
    10. Torben G. Andersen & Luca Benzoni, 2008. "Realized volatility," Working Paper Series WP-08-14, Federal Reserve Bank of Chicago.
    11. Yan Liu & Xiong Zhang, 2023. "Option Pricing Using LSTM: A Perspective of Realized Skewness," Mathematics, MDPI, vol. 11(2), pages 1-21, January.
    12. Christoffersen, Peter & Feunou, Bruno & Jacobs, Kris & Meddahi, Nour, 2014. "The Economic Value of Realized Volatility: Using High-Frequency Returns for Option Valuation," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 49(3), pages 663-697, June.
    13. Boris David & Gilles Zumbach, 2022. "Multivariate backtests and copulas for risk evaluation," Papers 2206.03896, arXiv.org, revised Nov 2023.
    14. Omar El Euch & Jim Gatheral & Radov{s} Radoiv{c}i'c & Mathieu Rosenbaum, 2018. "The Zumbach effect under rough Heston," Papers 1809.02098, arXiv.org.
    15. Liu, Ruipeng & Lux, Thomas, 2010. "Flexible and robust modelling of volatility comovements: a comparison of two multifractal models," Kiel Working Papers 1594, Kiel Institute for the World Economy (IfW Kiel).
    16. Alexandru Badescu & Robert J. Elliott & Juan-Pablo Ortega, 2012. "Quadratic hedging schemes for non-Gaussian GARCH models," Papers 1209.5976, arXiv.org, revised Dec 2013.
    17. Jean-Guy Simonato & Lars Stentoft, 2015. "Which pricing approach for options under GARCH with non-normal innovations?," CREATES Research Papers 2015-32, Department of Economics and Business Economics, Aarhus University.
    18. Paolella, Marc S. & Polak, Paweł, 2015. "COMFORT: A common market factor non-Gaussian returns model," Journal of Econometrics, Elsevier, vol. 187(2), pages 593-605.
    19. Challet, Damien & Peirano, Pier Paolo, 2008. "The ups and downs of the renormalization group applied to financial time series," MPRA Paper 9770, University Library of Munich, Germany.
    20. Yu-Hua Zeng & Shou-Lei Wang & Yu-Fei Yang, 2014. "Calibration of the Volatility in Option Pricing Using the Total Variation Regularization," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-9, March.

    More about this item

    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:taf:quantf:v:13:y:2013:i:8:p:1185-1197. 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: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/RQUF20 .

    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.