In which Financial Markets do Mutual Fund Theorems hold true?

TL;DR

This paper investigates the conditions under which the Mutual Fund Theorem holds in general semimartingale markets, linking it to replicability of options on the numéraire portfolio and utility function characteristics.

Contribution

It generalizes Merton's classical results by establishing conditions for the Mutual Fund Theorem in broad market settings and characterizes utility functions for which MFT holds across markets.

Findings

MFT holds if options on the numéraire portfolio are replicable using only the numéraire portfolio

In complete Brownian markets, MFT validity implies investors share a HARA utility function

The numéraire portfolio can serve as a mutual fund under certain replicability conditions

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.