Diversity and relative arbitrage in equity markets

An equity market is called diverse if no single stock is ever allowed to dominate the entire market in terms of relative capitalization. In the context of the standard Itô-process model initiated by Samuelson (1965) we formulate this property (and the allied, successively weaker notions of weak diversity and asymptotic weak diversity) in precise terms. We show that diversity is possible to achieve, but delicate. Several examples are provided which illustrate these notions and show that weakly-diverse markets contain relative arbitrage opportunities: it is possible to outperform or underperform such markets over any given time-horizon. The existence of this type of relative arbitrage does not interfere with the development of contingent claim valuation, and has consequences for the pricing of long-term warrants and for put-call parity. Several open questions are suggested for further study.Received: January 2004, Mathematics Subject Classification (2000): 60H10, 91B28; 60J55JEL Classification: G10We are grateful for the helpful remarks offered by seminar audiences at Columbia, Yale, Princeton, the Sloan School of MIT, Boston University, the Mathematical Institute in Oberwolfach, and the Universities of Athens, Connecticut/Storrs and Texas/Austin. Special thanks go to Professors Jérôme Detemple, Julien Hugonnier, Ralf Korn, Andrew Lo, Mark Lowenstein and Steven Shreve. We are also indebted to Dr. Adrian Banner for a number of discussions that helped sharpen our thinking about these problems, and to the referees and editors for suggestions that improved the exposition. A significant part of this work was completed in the spring semester of 2002, while the second author was on sabbatical leave at the Cowles Foundation for Research in Economics, Yale University. He is grateful to the Foundation for its hospitality.