Variations on an example of Karatzas and Ruf

TL;DR

This paper explores various market models with positive continuous semimartingales, examining conditions for arbitrage and covariance matrix properties, thereby extending classical results with new examples and insights.

Contribution

It introduces new market examples with different covariance structures and arbitrage properties, expanding understanding of market dynamics under bounded excess growth rates.

Findings

Markets with singular covariance matrices can exhibit arbitrage or no arbitrage.

Markets with nonsingular covariance matrices can also have arbitrage or be arbitrage-free.

Relative arbitrage opportunities can exist over arbitrary time horizons.

Abstract

Markets composed of stocks with capitalization processes represented by positive continuous semimartingales are studied under the condition that the market excess growth rate is bounded away from zero. The following examples of these markets are given: i) a market with a singular covariance matrix and instantaneous relative arbitrage; ii) a market with a singular covariance matrix and no arbitrage; iii) a market with a nonsingular covariance matrix and no arbitrage; iv) a market with a nonsingular covariance matrix and relative arbitrage over an arbitrary time horizon.