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The geometry of relative arbitrage

Abstract

Consider an equity market with n stocks. The vector of proportions of the total market capitalizations that belong to each stock is called the market weight. The market weight defines the market portfolio which is a buy-and-hold portfolio representing the performance of the entire stock market. Consider a function that assigns a portfolio vector to each possible value of the market weight, and we perform self-financing trading using this portfolio function. We study the problem of characterizing functions such that the resulting portfolio will outperform the market portfolio in the long run under the conditions of diversity and sufficient volatility. No other assumption on the future behavior of stock prices is made. We prove that the only solutions are functionally generated portfolios in the sense of Fernholz. A second characterization is given as the optimal maps of a remarkable optimal transport problem. Both characterizations follow from a novel property of portfolios called multiplicative cyclical monotonicity.

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Acknowledgments

We are grateful to Prof. Walter Schachermayer for a thorough reading of the manuscript and suggesting numerous comments for improvement. We also thank the anonymous reviewers for detailed comments about the presentation. This research is partially supported by NSF grant DMS-1308340.

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Correspondence to Soumik Pal.

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Pal, S., Wong, TK.L. The geometry of relative arbitrage. Math Finan Econ 10, 263–293 (2016). https://doi.org/10.1007/s11579-015-0159-z

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Keywords

  • Stochastic portfolio theory
  • Rebalancing
  • Functionally generated portfolios
  • Optimal transport
  • Model-free finance

JEL Classification

  • G11
  • G14