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.