Incomplete markets and short-sales constraints : An equilibrium approach

We consider a general discrete-time dynamic nancial market with three assets: a riskless bond, a security and a derivative. The market is incomplete (apriori) and at equilibrium. We assume also that the agents of the economy have short-sales constraints on the stock and that the payo at the expiry of the derivative asset is a monotone function of the underlying security price. The derivative price process is not identi ed ex ante. This leads the agents to act as if there were no market for this asset at the intermediary dates. Using some nice properties of the pricing probabilities, which are admissible at the equilibrium, we prove that it suffices to consider the subset of the risk-neutral probabilities that overestimate the low values of the security and underestimate its high values with respect to the true probability. This approach greatly reduces the interval of admissible prices for the derivative asset with respect to no-arbitrage, as showed numerically.