On the pricing of contingent claims under constraints

Abstract

We discuss the problem of pricing contingent claims, such as European call options, based on the fundamental principle of "absence of arbitrage" and in the presence of constraints on portfolio choice, for example, incomplete markets and markets with short-selling constraints. Under such constraints, we show that there exists an arbitrage-free interval which contains the celebrated Black-Scholes price (corresponding to the unconstrained case); no price in the interior of this interval permits arbitrage, but every price outside the interval does. In the case of convex constraints, the endpoints of this interval are characterized in terms of auxiliary stochastic control problems, in the manner of Cvitanić and Karatzas. These characterizations lead to explicit computations, or bounds, in several interesting cases. Furthermore, a unique fair price $\hat{p}$ is selected inside this interval, based on utility maximization and "marginal rate of substitution" principles. Again, characterizations are provided for $\hat{p}$, and these lead to very explicit computations. All these results are also extended to treat the problem of pricing contingent claims in the presence of a higher interest rate for borrowing. In the special case of a European call option in a market with constant coefficients, the endpoints of the arbitrage-free interval are the Black-Scholes prices corresponding to the two different interest rates, and the fair price coincides with that of Barron and Jensen.