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In which Financial Markets do Mutual Fund Theorems hold true?

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  • Walter Schachermayer
  • Mihai Sirbu
  • Erik Taflin

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. It is established that: 1) Let N be the wealth process of the num\'eraire portfolio (i.e. the optimal portfolio for the log utility). If any path-independent option with maturity T written on the num\'eraire portfolio can be replicated by trading \emph{only} in N, then the (MFT) holds true for general utility functions, and the num\'eraire portfolio may serve as mutual fund. This generalizes Merton's classical result on Black-Scholes markets. Conversely, under a supplementary weak completeness assumption, we show that the validity of the (MFT) for general utility functions implies the same replicability property for options on the num\'eraire portfolio described above. 2) 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.

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  • Walter Schachermayer & Mihai Sirbu & Erik Taflin, 2007. "In which Financial Markets do Mutual Fund Theorems hold true?," Papers 0710.1909, arXiv.org.
    • Handle: RePEc:arx:papers:0710.1909
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    References listed on IDEAS

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    1. (**), Hui Wang & Jaksa Cvitanic & (*), Walter Schachermayer, 2001. "Utility maximization in incomplete markets with random endowment," Finance and Stochastics, Springer, vol. 5(2), pages 259-272.
    2. Mark Rubinstein, 1976. "The Valuation of Uncertain Income Streams and the Pricing of Options," Bell Journal of Economics, The RAND Corporation, vol. 7(2), pages 407-425, Autumn.
    3. Hicks, J. R., 1986. "A Revision of Demand Theory," OUP Catalogue, Oxford University Press, number 9780198285502.
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