  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
possible to achieve, but delicate. Several illustrative examples are provided,
which demonstrate that weakly-diverse financial markets contain relative
arbitrage opportunities: it is possible to outperform (or underperform) such
markets over sufficiently long time-horizons, and to underperform them
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.
