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Optimization of relative arbitrage

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  • Ting-Kam Wong

Abstract

In stochastic portfolio theory, a relative arbitrage is an equity portfolio which is guaranteed to outperform a benchmark portfolio over a finite horizon. When the market is diverse and sufficiently volatile, and the benchmark is the market or a buy-and-hold portfolio, functionally generated portfolios introduced by Fernholz provide a systematic way of constructing relative arbitrages. In this paper we show that if the market portfolio is replaced by the equal or entropy weighted portfolio among many others, no relative arbitrages can be constructed under the same conditions using functionally generated portfolios. We also introduce and study a shaped-constrained optimization problem for functionally generated portfolios in the spirit of maximum likelihood estimation of a log-concave density. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Ting-Kam Wong, 2015. "Optimization of relative arbitrage," Annals of Finance, Springer, vol. 11(3), pages 345-382, November.
    • Handle: RePEc:kap:annfin:v:11:y:2015:i:3:p:345-382
    DOI: 10.1007/s10436-015-0261-5
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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. Victor DeMiguel & Lorenzo Garlappi & Raman Uppal, 2009. "Optimal Versus Naive Diversification: How Inefficient is the 1-N Portfolio Strategy?," The Review of Financial Studies, Society for Financial Studies, vol. 22(5), pages 1915-1953, May.
    3. Winslow Strong, 2012. "Generalizations of Functionally Generated Portfolios with Applications to Statistical Arbitrage," Papers 1212.1877, arXiv.org, revised Oct 2013.
    4. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987, arXiv.org.
    5. Madeleine Cule & Richard Samworth & Michael Stewart, 2010. "Maximum likelihood estimation of a multi‐dimensional log‐concave density," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(5), pages 545-607, November.
    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..
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    Citations

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    Cited by:

    1. Alexander Schied & Leo Speiser & Iryna Voloshchenko, 2016. "Model-free portfolio theory and its functional master formula," Papers 1606.03325, arXiv.org, revised May 2018.
    2. Peter Baxendale & Ting-Kam Leonard Wong, 2019. "Random concave functions," Papers 1910.13668, arXiv.org, revised May 2021.
    3. Ali Al-Aradi & Sebastian Jaimungal, 2018. "Outperformance and Tracking: Dynamic Asset Allocation for Active and Passive Portfolio Management," Papers 1803.05819, arXiv.org, revised Jul 2018.
    4. Ali Al-Aradi & Sebastian Jaimungal, 2019. "Active and Passive Portfolio Management with Latent Factors," Papers 1903.06928, arXiv.org.
    5. Yves-Laurent Kom Samo & Alexander Vervuurt, 2016. "Stochastic Portfolio Theory: A Machine Learning Perspective," Papers 1605.02654, arXiv.org.
    6. Erhan Bayraktar & Donghan Kim & Abhishek Tilva, 2024. "Quantifying dimensional change in stochastic portfolio theory," Mathematical Finance, Wiley Blackwell, vol. 34(3), pages 977-1021, July.
    7. Alexander Vervuurt & Ioannis Karatzas, 2015. "Diversity-Weighted Portfolios with Negative Parameter," Papers 1504.01026, arXiv.org, revised Jul 2015.
    8. Ting-Kam Leonard Wong, 2017. "On portfolios generated by optimal transport," Papers 1709.03169, arXiv.org, revised Sep 2017.
    9. Zihao Zhang & Stefan Zohren & Stephen Roberts, 2020. "Deep Learning for Portfolio Optimization," Papers 2005.13665, arXiv.org, revised Jan 2021.
    10. Ali Al-Aradi & Sebastian Jaimungal, 2018. "Outperformance and Tracking: Dynamic Asset Allocation for Active and Passive Portfolio Management," Applied Mathematical Finance, Taylor & Francis Journals, vol. 25(3), pages 268-294, May.
    11. Nicole Tianjiao Yang & Tomoyuki Ichiba, 2023. "Relative Arbitrage Opportunities in an Extended Mean Field System," Papers 2311.02690, arXiv.org, revised Oct 2024.
    12. Soumik Pal & Ting-Kam Leonard Wong, 2016. "Exponentially concave functions and a new information geometry," Papers 1605.05819, arXiv.org, revised May 2017.
    13. Steven Campbell & Qien Song & Ting-Kam Leonard Wong, 2024. "Macroscopic properties of equity markets: stylized facts and portfolio performance," Papers 2409.10859, arXiv.org, revised Oct 2024.
    14. Alexander Vervuurt & Ioannis Karatzas, 2015. "Diversity-weighted portfolios with negative parameter," Annals of Finance, Springer, vol. 11(3), pages 411-432, November.

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    More about this item

    Keywords

    Stochastic portfolio theory; Relative arbitrage; Functionally generated portfolio; Shape-constrained optimization ; Portfolio management; G11; C61;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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