Consider an equity market with n stocks. The vector of proportions of the total market capitalization that belongs to each stock is called the market weights. Consider two portfolios, one is a passive buy-and-hold portfolio representing the entire market, and the other assigns a portfolio vector for each possible value of the market weights and requires trading to maintain this assignment. The evolution of stocks is taken to be any jump process in discrete time, or a path of any continuous semimartingale in continuous time. We do not make any stochastic modeling assumptions on the evolutions. We provide necessary and sufficient conditions on the assignment on portfolios to guarantee that, for all such evolutions in a  sufficiently volatile market, the actively traded portfolio outperforms the buy-and-hold portfolio in the long run. This class of `relative arbitrage' portfolios were discovered by Fernholz and are called functionally generated portfolios. We show that, in an appropriate sense, a slight generalization of these are the only possible ones. Remarkably, such portfolios can be constructed using solutions of Monge-Kantorovich optimal transport problems on the unit simplex with a special choice of the cost function. We provide conditions under which these portfolios lead to statistical arbitrage in high-frequency trading. Our primary tool is a property of multidimensional functions that we call multiplicative cyclical monotonicity.
Consider an equity market with n stocks. The vector of proportions of the total market capitalization that belongs to each stock is called the market weights. Consider two portfolios, one is a passive buy-and-hold portfolio representing the entire market, and the other assigns a portfolio vector for each possible value of the market weights and requires trading to maintain this assignment. The evolution of stocks is taken to be any jump process in discrete time, or a path of any continuous semimartingale in continuous time. We do not make any stochastic modeling assumptions on the evolutions. We provide necessary and sufficient conditions on the assignment on portfolios to guarantee that, for all such evolutions in a diverse and sufficiently volatile market, the actively traded portfolio outperforms the buy-and-hold portfolio in the long run. This class of `relative arbitrage' portfolios were discovered by Fernholz and are called functionally generated portfolios. We prove that, in an appropriate sense, a slight generalization of these are the only possible ones. Remarkably, such portfolios correspond to solutions of Monge-Kantorovich optimal transport problems on the unit simplex with a cost function that can be described as log of the partition function. We also show how the presence of microstructure noise in stock price data leads to statistical arbitrage in high-frequency trading. Our primary tool is a property of multidimensional functions that we call multiplicative cyclical monotonicity.
A
diverse and
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show
prove
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can be constructed using
correspond to
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special choice of the cost
cost function that can be described as log of the partition
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provide conditions under which these portfolios lead
also show how the presence of microstructure noise in stock price data leads