No Arbitrage and the Growth Optimal Portfolio

Abstract

Recently, several papers have expressed an interest in applying the Growth Optimal Portfolio (GOP) for pricing derivatives. We show that the existence of a GOP is equivalent to the existence of a strictly positive martingale density. Our approach circumvents two assumptions usually set forth in the literature: 1) infinite expected growth rates are permitted and 2) the market does not need to admit an equivalent martingale measure. In particular, our approach shows that models featuring credit constrained arbitrage may still allow a GOP to exist because this type of arbitrage can be removed by a change of numéraire. However, if the GOP exists the market admits an equivalent martingale measure under some numéraire and hence derivatives can be priced. The structure of martingale densities is used to provide a new characterization of the GOP which emphasizes the relation to other methods of pricing in incomplete markets. The case where GOP denominated asset prices are strict supermartingales is analyzed in the case of pure jump driven uncertainty.

Keywords:

• Arbitrage
• Growth optimal portfolio
• Market price of risk
• Sigma martingale density

Mathematics Subject Classification (2000):

• 91B30
• 60H30
• 60G44

JEL Classification:

• G10

ACKNOWLEDGMENTS

The authors would like to thank Tomas Björk, Niels Jøt rgen Nielsen, Eckhard Platen, Martin Schweizer, Gordan Žitković, participants at the PhD workshop in Ebeltoft (2004), participants at the second CCCP-meeting at CMU (2005) and anonymous referees for corrections, comments and fruitful discussions.