Diversity and no arbitrage

TL;DR

This paper investigates the relationship between market diversity and arbitrage opportunities, showing that diverse markets can be approximated by models that are both arbitrage-free and diverse, raising questions about detectability.

Contribution

It demonstrates that known diverse markets with arbitrage can be approximated by arbitrage-free models, challenging the ability to identify arbitrage in observed diverse markets.

Findings

Diverse markets can be approximated by arbitrage-free models.

Known diverse markets with arbitrage are not distinguishable from arbitrage-free models.

The approximation is uniform on the logarithmic scale.

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 the present paper 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 which are both diverse and arbitrage-free.