DIVERSITY AND RELATIVE ARBITRAGE IN FINANCIAL MARKETS

Ioannis Karatzas
(joint work with R. Fernholz and C. Kardaras)

Abstract

A financial 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 Ito-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 indeed possible, though rather delicate, to achieve. Several illustrative examples are provided, including two which demonstrate that weakly-diverse financial markets contain relative arbitrage opportunities: it is possible to outperform (or underperform) such markets significantly, over arbitrary time-horizons. The existence of such relative arbitrage does not interfere with the development of option pricing, and has interesting consequences for the pricing of long-term warrants and for put-call parity. Several open questions are suggested for further study.