Finance and Stochastics

, Volume 13, Issue 1, pp 49–77 | Cite as

In which financial markets do mutual fund theorems hold true?

Article

Abstract

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:
  1. (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.

     
  2. (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 function U, which must be of HARA type. This is a result in the spirit of the classical work by Cass and Stiglitz.

     

Keywords

Mutual fund Numéraire portfolio European option Replication Completeness 

Mathematics Subject Classification (2000)

91B16 91B28 91B70 

JEL Classification

G11 C61 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Walter Schachermayer
    • 1
  • Mihai Sîrbu
    • 2
  • Erik Taflin
    • 3
  1. 1.Vienna University of TechnologyWienAustria
  2. 2.University of Texas at AustinAustinUSA
  3. 3.Chair in Mathematical Finance, EISTIEcole International des Sciences du Traitement de l’InformationCergyFrance

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