Diversity and No Arbitrage

Abstract

A stock market is called diverse if no stock can dominate the market in terms of relative capitalization. On one hand, this natural property leads to arbitrage in diffusion models under mild assumptions. On the other hand, it is also easy to construct diffusion models which are both diverse and free of arbitrage. Can one tell whether an observed diverse market admits arbitrage? In this article, we argue that this may well be impossible by proving that the known examples of diverse markets in the literature (which do admit arbitrage) can be approximated uniformly (on the logarithmic scale) by models that are both diverse and arbitrage-free.

Keywords:

• Diverse market
• Consistent price system
• Conditonal full support
• Multidimensional continuous semimartingale

Mathematics Subject Classification (2000):

• Primary: 91B25
• Secondary: 60H05

Additional information

Funding

The European Union and the European Social Fund have provided financial support to the project under the grant agreement no. TÁMOP 4.2.1./B-09/1/KMR-2010-0003.