The Fundamental Theorem of Utility Maximization and Num\'eraire Portfolio
AbstractThe fundamental theorem of utility maximization (called FTUM hereafter) says that the utility maximization admits solution if and only if there exists an equivalent martingale measure. This theorem is true for discrete market models (where the number of scenarios is finite), and remains valid for general discrete-time market models when the utility is smooth enough. However, this theorem --in this current formulation-- fails in continuous-time framework even with nice utility function, where there might exist arbitrage opportunities and optimal portfolio. This paper addresses the question how far we can weaken the non-arbitrage condition as well as the utility maximization problem to preserve their complete and strong relationship described by the FTUM. As application of our version of the FTUM, we establish equivalence between the No-Unbounded-Profit-with-Bounded-Risk condition, the existence of num\'eraire portfolio, and the existence of solution to the utility maximization under equivalent probability measure. The latter fact can be interpreted as a sort of weak form of market's viability, while this equivalence is established with a much less technical approach. Furthermore, the obtained equivalent probability can be chosen as close to the real-world probability measure as we want (but might not be equal).
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Bibliographic InfoPaper provided by arXiv.org in its series Papers with number 1211.4598.
Date of creation: Nov 2012
Date of revision: Nov 2012
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