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The joint distribution of stock returns is not elliptical

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  • R'emy Chicheportiche
  • Jean-Philippe Bouchaud

Abstract

Using a large set of daily US and Japanese stock returns, we test in detail the relevance of Student models, and of more general elliptical models, for describing the joint distribution of returns. We find that while Student copulas provide a good approximation for strongly correlated pairs of stocks, systematic discrepancies appear as the linear correlation between stocks decreases, that rule out all elliptical models. Intuitively, the failure of elliptical models can be traced to the inadequacy of the assumption of a single volatility mode for all stocks. We suggest several ideas of methodological interest to efficiently visualise and compare different copulas. We identify the rescaled difference with the Gaussian copula and the central value of the copula as strongly discriminating observables. We insist on the need to shun away from formal choices of copulas with no financial interpretation.

Suggested Citation

  • R'emy Chicheportiche & Jean-Philippe Bouchaud, 2010. "The joint distribution of stock returns is not elliptical," Papers 1009.1100, arXiv.org, revised Jun 2012.
    • Handle: RePEc:arx:papers:1009.1100
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    Cited by:

    1. Petr Koldanov, 2019. "Testing new property of elliptical model for stock returns distribution," Papers 1907.10306, arXiv.org.
    2. Lasko Basnarkov & Viktor Stojkoski & Zoran Utkovski & Ljupco Kocarev, 2019. "Option Pricing With Heavy-Tailed Distributions Of Logarithmic Returns," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(07), pages 1-35, November.
    3. Fung, Thomas & Seneta, Eugene, 2016. "Tail asymptotics for the bivariate skew normal," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 129-138.
    4. Broda, Simon A. & Krause, Jochen & Paolella, Marc S., 2018. "Approximating expected shortfall for heavy-tailed distributions," Econometrics and Statistics, Elsevier, vol. 8(C), pages 184-203.
    5. Simon Hediger & Jeffrey Näf & Marc S. Paolella & Paweł Polak, 2023. "Heterogeneous tail generalized common factor modeling," Digital Finance, Springer, vol. 5(2), pages 389-420, June.
    6. Donald Geman & H'elyette Geman & Nassim Nicholas Taleb, 2014. "Tail Risk Constraints and Maximum Entropy," Papers 1412.7647, arXiv.org.
    7. Bücher Axel & Jaser Miriam & Min Aleksey, 2021. "Detecting departures from meta-ellipticity for multivariate stationary time series," Dependence Modeling, De Gruyter, vol. 9(1), pages 121-140, January.
    8. Jonathan Raimana Chan & Thomas Huckle & Antoine Jacquier & Aitor Muguruza, 2021. "Portfolio optimisation with options," Papers 2111.12658, arXiv.org, revised Sep 2024.
    9. Gambacciani, Marco & Paolella, Marc S., 2017. "Robust normal mixtures for financial portfolio allocation," Econometrics and Statistics, Elsevier, vol. 3(C), pages 91-111.
    10. Petr Koldanov & Nina Lozgacheva, 2016. "Multiple Testing Of Sign Symmetry For Stock Return Distributions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(08), pages 1-14, December.
    11. Koldanov, A. & Koldanov, P. & Semenov, D., 2021. "Confidence set for connected stocks of stock market," Journal of the New Economic Association, New Economic Association, vol. 50(2), pages 12-34.
    12. R'emy Chicheportiche & Jean-Philippe Bouchaud, 2013. "A nested factor model for non-linear dependences in stock returns," Papers 1309.3102, arXiv.org.
    13. Fung, Thomas & Seneta, Eugene, 2014. "Convergence rate to a lower tail dependence coefficient of a skew-t distribution," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 62-72.

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