Before revision:  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 .
After revision:  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 these characterizations follow from a novel property of portfolios called multiplicative cyclical monotonicity. Using this framework we show how the presence of microstructure noise in stock price data leads to statistical arbitrage in high-frequency trading .