Asset valuation and optimal portfolio choice in incomplete markets

In this research we develop methods for valuing private assets in economies where uncertainty affects any asset's payoff. The economy is assumed to be incomplete, that is, the marketed assets in the economy do not span the complete payoff space. A private asset is an asset that cannot be replicated by marketed assets. We show that in general the perception of a private asset's value depends on an investor's preferences and wealth.;We analyze models where the marketed prices are characterized by either discrete or continuous time stochastic processes. Our analysis uses the theory of vector space optimization more explicitly than much of the theory developed in finance.;When the market is complete optimal portfolio theory, asset pricing theory, and equilibrium theory are integrated into a common theory by virtue of a unique state-price vector. When the market is incomplete there exist an infinite number of state-price vectors and thus asset pricing is not uniquely defined. Furthermore, a Pareto-optimal competitive equilibrium generally does not exist.;However, although the value of a private asset is not uniquely defined in an incomplete market but depends on the specific assumptions of the economic model, it is still necessary to determine an investor-specific state-price vector to find the investor's perception of a private asset's value. The investor-specific state-price vector can be decomposed into a marketed component and a non-marketed component. The marketed component correctly prices any marketed asset, whereas the non-marketed component can be used to develop the appropriate valuation formula that corresponds to the model assumptions.;We show how to determine the state-price vector in an incomplete market that corresponds to an investor's preferences and opportunities. Furthermore, we show how to use this state-price vector to find the private asset's value that reflects the same investor's economic environment. Finally, because the valuation for an incomplete market formulation can be very calculation intensive, we present an approximate valuation algorithm that simplifies the calculations. This algorithm determines a lower bound on the asset's value.