Arbitrage possibilities in Bessel processes and their relations to local martingales

Abstract

We show that, if we allow general admissible integrands as trading strategies, the three dimensional Bessel process, Bes3, admits arbitrage possibilities. This is in contrast with the fact that the inverse process is a local martingale and hence is arbitrage free. This leads to some economic interpretation for the analysis of the property of arbitrage in foreign exchange rates. This notion (relative to general admissible integrands) does depend on the fact, which of the two currencies under consideration is chosen as numéraire. The results rely on a general construction of strictly positive local martingales. The construction is related to the Föllmer measure of a positive super-martingale. © 1995 Springer-Verlag.