Arbitrage and Optimal Portfolio Choice with Financial Constraints

We analyze the pricing of risky income streams in a world with competitive security markets where investors are constrained by restrictions on possible portfolio holdings. We investigate how we can transfer concepts and pricing techniques from a world without frictions to such a more realistic situation. We show that basically two new aspects arise: First, portfolio constraints can lead to situations where not all arbitrage opportunities are necessarily eliminated in equilibrium. For a world with portfolio constraints the concept of no arbitrage has to be replaced by a weaker concept which we call no unlimited arbitrage. Second, though we can characterize prices which allow no unlimited arbitrage by the existence of certain state prices as in the unconstrained case, additional computational work is needed for deriving from this fact a pricing theory for contingent claims. We propose a technique which can achieve this task and which renders itself for a computationally simple implementation for many constraint situations which are of practical interest. The power of no arbitrage techniques is preserved in the sense that no specific assumptions about utility functions of investors have to be made. We relate our analysis to the optimal decision problem of an investor and show the various relations between the properties of an optimal solution to this problem and the arbitrage-free values of risky income streams. This opens a unified view on the different approaches to asset pricing under portfolio constraints used in the literature and conveys their common underlying logic.