Shadow Probability Theory for Asset Pricing under Ambiguity

Assuming that probabilities (capacities) of events are random, this paper introduces a novel model of decision making under ambiguity, called Shadow probability theory, a generalization of the Choquet expected utility. In this model, probabilities of observable events in a subordinated outcome-space are dominated by second-order unobservable events in a directing probability space. The level of ambiguity, and the decision maker's attitude toward it, are measured with respect to the directing space. Risk and risk attitude, on the other hand, apply to the subordinated space, as in classical expected utility theory. The desired distinction between preferences and beliefs and between risk and ambiguity is then obtained. A measure of ambiguity is naturally carried over on these settings. The present paper proves that in most cases ambiguity cannot be diversified. Counterintuitively, adding an ambiguous lottery to a portfolio of lotteries usually increases its ambiguity. This result, which implies that full diversification is not always optimal, challenges the common notion in the financial literature that investors should hold a fully diversified portfolio. Using this model of choice, the current paper generalizes the classical asset pricing theory by incorporating ambiguous probabilities. It proposes a well defined ambiguity premium that can be measured empirically