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Relative Arbitrage Opportunities in $N$ Investors and Mean-Field Regimes

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  • Tomoyuki Ichiba
  • Tianjiao Yang

Abstract

The relative arbitrage portfolio, formulated in Stochastic Portfolio Theory (SPT), outperforms a benchmark portfolio over a given time-horizon with probability one. This paper analyzes the market behavior and optimal investment strategies to attain relative arbitrage both in the $N$ investors and mean field regimes under some market conditions. An investor competes with a benchmark of market and peer investors, expecting to outperform the benchmark and minimizing the initial capital. With market price of risk processes depending on the market portfolio and investors, we develop a systematic way to solve multi-agent optimization problem within SPT's framework. The objective can be characterized by the smallest nonnegative continuous solution of a Cauchy problem. By a modification in the structure of the extended mean field game with common noise and its notion of the uniqueness of Nash equilibrium, we show a unique equilibrium in $N$-player games and mean field games with mild conditions on the equity market.

Suggested Citation

  • Tomoyuki Ichiba & Tianjiao Yang, 2020. "Relative Arbitrage Opportunities in $N$ Investors and Mean-Field Regimes," Papers 2006.15158, arXiv.org, revised Jan 2021.
    • 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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