In which financial markets do mutual fund theorems hold true?

The mutual fund theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T , where agents maximize expected utility of terminal wealth. The main results are: (i) Let N be the wealth process of the numéraire portfolio (i.e., the optimal portfolio for the log utility). If any path-independent option with maturity T written on the numéraire portfolio can be replicated by trading only in N and the risk-free asset, then the MFT holds true for general utility functions, and the numéraire portfolio may serve as mutual fund. This generalizes Merton’s classical result on Black– Merton–Scholes markets as well as the work of Chamberlain in the framework of Brownian filtrations (Chamberlain in Econometrica 56:1283–1300, 1988). Conversely, under a supplementary weak completeness assumption, we show that the validity of the MFT for general utility functions implies the replicability property for options on the numéraire portfolio described above. (ii) If for a given class of utility functions (i.e., investors) the MFT holds true in all complete Brownian financial markets S, then all investors use the same utility Financial support from the Austrian Science Fund (FWF) under the grant P19456, from Vienna Science and Technology Fund (WWTF) under Grant MA13 and by the Christian Doppler Research Association (CDG) is gratefully acknowledged by the first author. The research of the second author was partially supported by the National Science Foundation under Grant DMS-0604643. W. Schachermayer ( ) Vienna University of Technology, Wiedner Hauptstrasse 8-10/105, 1040 Wien, Austria e-mail: [email protected] M. Sîrbu University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA e-mail: [email protected] E. Taflin Chair in Mathematical Finance, EISTI, Ecole International des Sciences du Traitement de l’Information, Avenue du Parc, 95011 Cergy, France e-mail: [email protected] 50 W. Schachermayer et al. function U , which must be of HARA type. This is a result in the spirit of the classical work by Cass and Stiglitz.