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Capital distribution and portfolio performance in the mean-field Atlas model

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  • Benjamin Jourdain

    (CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTech, MATHRISK - Mathematical Risk handling - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en Automatique - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École des Ponts ParisTech)

  • Julien Reygner

    (CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTech, LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

Abstract

We study a mean-field version of rank-based models of equity markets such as the Atlas model introduced by Fernholz in the framework of Stochastic Portfolio Theory. We obtain an asymptotic description of the market when the number of companies grows to infinity. Then, we discuss the long-term capital distribution. We recover the Pareto-like shape of capital distribution curves usually derived from empirical studies, and provide a new description of the phase transition phenomenon observed by Chatterjee and Pal. Finally, we address the performance of simple portfolio rules and highlight the influence of the volatility structure on the growth of portfolios.

Suggested Citation

  • Benjamin Jourdain & Julien Reygner, 2015. "Capital distribution and portfolio performance in the mean-field Atlas model," Post-Print hal-00921151, HAL.
    • Handle: RePEc:hal:journl:hal-00921151
    DOI: 10.1007/s10436-014-0258-5
    Note: View the original document on HAL open archive server: https://enpc.hal.science/hal-00921151v3
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    References listed on IDEAS

    as
    1. Shkolnikov, Mykhaylo, 2013. "Large volatility-stabilized markets," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 212-228.
    2. Fernholz, E. Robert & Ichiba, Tomoyuki & Karatzas, Ioannis, 2013. "Two Brownian particles with rank-based characteristics and skew-elastic collisions," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2999-3026.
    3. Robert Fernholz & Tomoyuki Ichiba & Ioannis Karatzas, 2013. "A second-order stock market model," Annals of Finance, Springer, vol. 9(3), pages 439-454, August.
    4. Shkolnikov, Mykhaylo, 2012. "Large systems of diffusions interacting through their ranks," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1730-1747.
    5. Tomoyuki Ichiba & Vassilios Papathanakos & Adrian Banner & Ioannis Karatzas & Robert Fernholz, 2009. "Hybrid Atlas models," Papers 0909.0065, arXiv.org, revised Apr 2011.
    6. Robert Fernholz & Tomoyuki Ichiba & Ioannis Karatzas, 2013. "A second-order stock market model," Papers 1302.3870, arXiv.org.
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    Cited by:

    1. Andrey Sarantsev, 2019. "Comparison Techniques for Competing Brownian Particles," Journal of Theoretical Probability, Springer, vol. 32(2), pages 545-585, June.
    2. Sergio A. Almada Monter & Mykhaylo Shkolnikov & Jiacheng Zhang, 2018. "Dynamics of observables in rank-based models and performance of functionally generated portfolios," Papers 1802.03593, arXiv.org.
    3. Brandon Flores & Blessing Ofori-Atta & Andrey Sarantsev, 2021. "A stock market model based on CAPM and market size," Annals of Finance, Springer, vol. 17(3), pages 405-424, September.

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