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Exponentially concave functions and high dimensional stochastic portfolio theory

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  • Pal, Soumik

Abstract

We construct an explicit example of asymptotic short term relative arbitrage. Specifically, for every n we assume an n dimensional semimartingale market model that starts from a heavy-tailed initial position in the unit simplex and impose weak assumptions on its volatility. We then construct a sequence of portfolios, one for each dimension, that outperform the market portfolio in dimension n by an amount Mn by time δn with a probability at least 1−qn. Here Mn→∞ exponentially fast in n and δn,qn decrease to zero. Moreover, these portfolios never underperform below a pre-specified lower bound. The key fact is that it is possible to construct a sequence of exponentially concave functions on the unit simplex of increasing concavity because the typical diameter of the unit simplex in dimension n is O(1∕n).

Suggested Citation

  • Pal, Soumik, 2019. "Exponentially concave functions and high dimensional stochastic portfolio theory," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3116-3128.
  • Handle: RePEc:eee:spapps:v:129:y:2019:i:9:p:3116-3128
    DOI: 10.1016/j.spa.2018.09.004
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    References listed on IDEAS

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    1. Y.M. Kabanov & D.O. Kramkov, 1998. "Asymptotic arbitrage in large financial markets," Finance and Stochastics, Springer, vol. 2(2), pages 143-172.
    2. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    3. Soumik Pal & Ting-Kam Leonard Wong, 2014. "The geometry of relative arbitrage," Papers 1402.3720, arXiv.org, revised Jul 2015.
    4. Irene Klein, 2000. "A Fundamental Theorem of Asset Pricing for Large Financial Markets," Mathematical Finance, Wiley Blackwell, vol. 10(4), pages 443-458, October.
    5. Robert Fernholz, 2015. "An example of short-term relative arbitrage," Papers 1510.02292, arXiv.org.
    6. Robert Fernholz & Ioannis Karatzas & Constantinos Kardaras, 2005. "Diversity and relative arbitrage in equity markets," Finance and Stochastics, Springer, vol. 9(1), pages 1-27, January.
    7. Fernholz, Robert, 1999. "On the diversity of equity markets," Journal of Mathematical Economics, Elsevier, vol. 31(3), pages 393-417, April.
    8. Fernholz, E. Robert & Karatzas, Ioannis & Ruf, Johannes, 2018. "Volatility and arbitrage," LSE Research Online Documents on Economics 75234, London School of Economics and Political Science, LSE Library.
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    Cited by:

    1. Martin Larsson & Johannes Ruf, 2020. "Relative Arbitrage: Sharp Time Horizons and Motion by Curvature," Papers 2003.13601, arXiv.org, revised Feb 2021.

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