Consider an equity market with $n$ stocks. The vector of proportions of the total market capitalizations that belongs to each stock is called the market weight. The market weight defines a buy-and-hold portfolio called the market portfolio whose value represents the performance of the entire stock market. Consider a function that assigns a portfolio vector for each possible value of the market weight. Suppose 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 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.
