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 .
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 .
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capitalization
capitalizations
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weights. Consider two portfolios, one is a passive
weight. The market weight defines a
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representing the entire market , and the other
called the market portfolio whose value represents the performance of the entire stock market. Consider a function that
R
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
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
R
. This class of `relative arbitrage' portfolios were discovered by Fernholz and are called functionally generated portfolios.
under the conditions of diversity and sufficient volatility. No other assumption on the future behavior of stock prices is made.
R
, 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
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
D
. Our primary tool is a property of multidimensional functions that we call multiplicative cyclical monotonicity