Uncertainty and Risk in Financial Markets

Abstract

This paper considers a general equilibrium model in which the distinction between uncertainty and risk is formalized by assuming agents have incomplete preferences over state-contingent consumption bundles, as in Bewley (1986). Without completeness, individual decision making depends on a set of probability distributions over the state space. A bundle is preferred to another if and only if it has larger expected utility for all probabilities in this set. When preferences are complete this set is a singleton, and the model reduces to standard expected utility. In this setting, we characterize Pareto optima and equilibria, and show that the presence of uncertainty generates robust indeterminacies in equilibrium prices and allocations for any specification of initial endowments. We derive comparative statics results linking the degree of uncertainty with changes in equilibria. Despite the presence of robust indeterminacies, we show that equilibrium prices and allocations vary continuously with underlying fundamentals. Equilibria in a standard risk economy are thus robust to adding small degrees of uncertainty. Finally, we give conditions under which some assets are not traded due to uncertainty aversion. Copyright The Econometric Society 2005.

Ambiguity and the Bayesian Paradigm

“…Other (published) applications of ambiguity averse preferences include Epstein andWang (1994, 1995), who explain …nancial crashes and booms, Mukerji (1998), who explains incompleteness of contracts, Chateauneuf, Dana, and Tallon (2000), who study optimal risk-sharing rules with ambiguity averse agents, Greenberg (2000), who …nds that in a strategic set-up a player may …nd it bene…cial to generate ambiguity about her strategy choice, Mukerji and Tallon (2001), who show how incompleteness of …nancial markets my arise because of ambiguity aversion, Rigotti and Shannon (2005), who characterize equilibria and optima and study how they depend on the degree of ambiguity, Bose, Ozdenoren and Pape (2006), who study auctions under ambiguity, Nishimura and Ozaki (2007), who…”

Risk, ambiguity, and state-preference theory

“…Suppose that B-events are unambiguous in the sense that they are stochastically independent of the credal state, while A-events are potentially ambiguous in the sense that they are dependent on the credal state but independent of the B-events. Then p jkji in equation (36) has the factorization p jkji = p jji q k : This is a model-speci…c de…nition of ambiguity that refers to unobservable states of mind, rather than a behavioral de…nition, but it describes a scenario in which the second-order uncertainty model should be expected to produce ambiguity-averse behavior. In particular, it might be expected that a decision maker with these beliefs and aversion to credal uncertainty would be more averse to betting on A-events than B-events in the sense of assigning higher ambiguity premia to A:B-bets than to B:A-bets, analogous to the property of the two-source utility function that was established in Corollary 2.…”

“…A model of preferences that are incomplete due to indeterminacy of probabilities was introduced into microeconomics by Bewley (1986), who referred to it as "Knightian" uncertainty 1 . Its implications for phenomena such as general equilibrium have been elaborated by Rigotti and Shannon (2005) and others. A decision maker with incomplete preferences does not have …xed indi¤erence curves, and she may exhibit indecision or inertia when called upon to make a choice under ambiguous conditions.…”

Efficient allocations and equilibria with short-selling and incomplete preferences

“…Such incomplete preferences include Bewley's [2] and Rigotti and Shannon [15] incomplete preferences. Under this representation assumption, it is easy to define and characterize the concepts of no-arbitrage prices and no unbounded utility arbitrage.…”

“…Agents are assumed not to have enough information to quantify uncertainty by a single probability, hence each agent has a set of priors. Agents are further assumed to have risk averse Bewley's [2] (or Rigotti and Shannon [15] ) incomplete preferences. Under standard conditions on utility indices (strict concavity and increasingness) and sets of priors (convexity and compactness), it is shown that a necessary and su cient for the existence of an individually rational e cient allocation or equilibrium is that the relative interiors of the agents' sets of risk adjusted probabilities intersect or that agents do not engage in mutually compatible trades that have non negative expectations with respect to their risk adjusted probabilities.…”