Game-Theoretic Foundations for Probability and Finance

17Game‐Theoretic Portfolio Theory

In an idealized market with a finite number of securities, the capital‐weighted market portfolio can be outperformed by a long‐only portfolio that gives greater weights to securities with smaller capitalization, provided that the securities' price paths are positive and trading never dies out, in some sense. This mathematical fact has been emphasized by E. Robert Fernholz, who has made it a centerpiece of his measure‐theoretic stochastic portfolio theory [138,139,141]. In this chapter we show that Fernholz's picture can be reshaped into a relatively simple and transparent game‐theoretic picture in the martingale space that uses the total market capitalization as numeraire. In particular, our way of formalizing trading never dying out is that the sum of the quadratic variations of the securities is unbounded.

In order to apply our game‐theoretic portfolio theory to an actual market, we need to assume that transaction costs and complications arising from dividends can be neglected. We also need to assume that the martingale space we construct is valid in the trading‐in‐the‐small sense; a trader cannot become very rich very suddenly. But application of the theory does not require the long‐run efficiency assumption used in the preceding chapter: we need not assume that the total market capitalization is an efficient numeraire.

Whereas Fernholz has contended that his strategies can outperform the market, our exposition takes a more neutral stance. ...