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Generalizations of Functionally Generated Portfolios with Applications to Statistical Arbitrage

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  • Winslow Strong

Abstract

The theory of functionally generated portfolios (FGPs) is an aspect of the continuous-time, continuous-path Stochastic Portfolio Theory of Robert Fernholz. FGPs have been formulated to yield a master equation - a description of their return relative to a passive (buy-and-hold) benchmark portfolio serving as the num\'eraire. This description has proven to be analytically very useful, as it is both pathwise and free of stochastic integrals. Here we generalize the class of FGPs in several ways: (1) the num\'eraire may be any strictly positive wealth process, not necessarily the market portfolio or even a passive portfolio; (2) generating functions may be stochastically dynamic, adjusting to changing market conditions through an auxiliary continuous-path stochastic argument of finite variation. These generalizations do not forfeit the important tractability properties of the associated master equation. We show how these generalizations can be usefully applied to scenario analysis, statistical arbitrage, portfolio risk immunization, and the theory of mirror portfolios.

Suggested Citation

  • Winslow Strong, 2012. "Generalizations of Functionally Generated Portfolios with Applications to Statistical Arbitrage," Papers 1212.1877, arXiv.org, revised Oct 2013.
    • Handle: RePEc:arx:papers:1212.1877
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    References listed on IDEAS

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    1. Adrian Banner & Daniel Fernholz, 2008. "Short-term relative arbitrage in volatility-stabilized markets," Annals of Finance, Springer, vol. 4(4), pages 445-454, October.
    2. Winslow Strong & Jean-Pierre Fouque, 2011. "Diversity and arbitrage in a regulatory breakup model," Annals of Finance, Springer, vol. 7(3), pages 349-374, August.
    3. 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.
    4. Fama, Eugene F & French, Kenneth R, 1992. "The Cross-Section of Expected Stock Returns," Journal of Finance, American Finance Association, vol. 47(2), pages 427-465, June.
    5. Fernholz, Robert, 1999. "On the diversity of equity markets," Journal of Mathematical Economics, Elsevier, vol. 31(3), pages 393-417, April.
    6. Robert Fernholz, 1999. "Portfolio Generating Functions," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 15, pages 344-367, World Scientific Publishing Co. Pte. Ltd..
    7. Robert Fernholz & Ioannis Karatzas, 2005. "Relative arbitrage in volatility-stabilized markets," Annals of Finance, Springer, vol. 1(2), pages 149-177, November.
    8. Fama, Eugene F. & French, Kenneth R., 1993. "Common risk factors in the returns on stocks and bonds," Journal of Financial Economics, Elsevier, vol. 33(1), pages 3-56, February.
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    Cited by:

    1. Soumik Pal & Ting-Kam Leonard Wong, 2016. "Exponentially concave functions and a new information geometry," Papers 1605.05819, arXiv.org, revised May 2017.
    2. Ting-Kam Leonard Wong, 2014. "Optimization of relative arbitrage," Papers 1407.8300, arXiv.org, revised Nov 2014.
    3. Kangjianan Xie, 2020. "Leakage of rank-dependent functionally generated trading strategies," Annals of Finance, Springer, vol. 16(4), pages 573-591, December.
    4. Soumik Pal & Ting-Kam Leonard Wong, 2014. "The geometry of relative arbitrage," Papers 1402.3720, arXiv.org, revised Jul 2015.
    5. Alexander Schied & Leo Speiser & Iryna Voloshchenko, 2016. "Model-free portfolio theory and its functional master formula," Papers 1606.03325, arXiv.org, revised May 2018.
    6. Alexander Vervuurt, 2015. "Topics in Stochastic Portfolio Theory," Papers 1504.02988, arXiv.org.
    7. Andrew L. Allan & Christa Cuchiero & Chong Liu & David J. Promel, 2021. "Model-free Portfolio Theory: A Rough Path Approach," Papers 2109.01843, arXiv.org, revised Oct 2022.
    8. Patrick Mijatovic, 2021. "Beating the Market with Generalized Generating Portfolios," Papers 2101.07084, arXiv.org.
    9. Ioannis Karatzas & Donghan Kim, 2018. "Trading Strategies Generated Pathwise by Functions of Market Weights," Papers 1809.10123, arXiv.org, revised Mar 2019.
    10. Ting-Kam Wong, 2015. "Optimization of relative arbitrage," Annals of Finance, Springer, vol. 11(3), pages 345-382, November.
    11. Winslow Strong, 2014. "Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension," Finance and Stochastics, Springer, vol. 18(3), pages 487-514, July.
    12. repec:ehl:lserod:102424 is not listed on IDEAS
    13. Qingyin Ge & Yunuo Ma & Yuezhi Liao & Rongyu Li & Tianle Zhu, 2020. "Risk Management and Return Prediction," Papers 2007.01194, arXiv.org.
    14. Yves-Laurent Kom Samo & Alexander Vervuurt, 2016. "Stochastic Portfolio Theory: A Machine Learning Perspective," Papers 1605.02654, arXiv.org.
    15. Robert Fernholz, 2016. "A new decomposition of portfolio return," Papers 1606.05877, arXiv.org.
    16. Johannes Ruf & Kangjianan Xie, 2018. "Generalised Lyapunov Functions and Functionally Generated Trading Strategies," Papers 1801.07817, arXiv.org.
    17. Ioannis Karatzas & Donghan Kim, 2020. "Trading strategies generated pathwise by functions of market weights," Finance and Stochastics, Springer, vol. 24(2), pages 423-463, April.

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