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Relative Arbitrage Opportunities with Interactions among $N$ Investors

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  • Tomoyuki Ichiba
  • Nicole Tianjiao Yang

Abstract

The relative arbitrage portfolio outperforms a benchmark portfolio over a given time-horizon with probability one. With market price of risk processes depending on the market portfolio and investors, this paper analyzes the multi-agent optimization of relative arbitrage opportunities in the coupled system of market and wealth dynamics. We construct a well-posed market dynamical system of McKean-Vlasov type under an empirical measure of investors, where each investor seeks for relative arbitrage with respect to a benchmark dependent on market and all the agents. We show the conditions to guarantee relative arbitrage opportunities among competitive investors through the Fichera drift. Under mild conditions, we derive the optimal strategies for investors and the unique Nash equilibrium that depends on the smallest nonnegative solution of a Cauchy problem.

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  • Tomoyuki Ichiba & Nicole Tianjiao Yang, 2020. "Relative Arbitrage Opportunities with Interactions among $N$ Investors," Papers 2006.15158, arXiv.org, revised Jul 2024.
    • Handle: RePEc:arx:papers:2006.15158
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Daniel Lacker & Thaleia Zariphopoulou, 2019. "Mean field and n‐agent games for optimal investment under relative performance criteria," Mathematical Finance, Wiley Blackwell, vol. 29(4), pages 1003-1038, October.
    4. Erhan Bayraktar & Yu-Jui Huang & Qingshuo Song, 2010. "Outperforming the market portfolio with a given probability," Papers 1006.3224, arXiv.org, revised Aug 2012.
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