Pricing Risk Under Risk Measures : An Introduction to Stochastic-Endogenous Equilibria

We investigate risk averse agents who manage risk by trading financial securities in a market that we call a risk market. We assume this market is perfectly competitive and complete. When risk aversion is expressed using risk measures, the (bundle of) prices for financial securities turns out to be a probability density function, PDF. This endogeneity of probability distributions defines a novel template for equilibria under uncertainty and, more specifically, equilibria under risk. It is particularly striking that when agents use coherent risk measures to assess uncertain outcomes, a most risk neutral agent emerges from the risk market: its risk set is precisely the intersection of all agents' risk sets, and the endogenous price of risk is simultaneously a worst case PDF for the most risk neutral agent and for each agent individually. We also show that risk markets can be conveniently adapted to decision making models like Nash games under risk, where agents are risk averse and optimize their actions under uncertainty