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May 1996 On the pricing of contingent claims under constraints
I. Karatzas, S. G. Kou
Ann. Appl. Probab. 6(2): 321-369 (May 1996). DOI: 10.1214/aoap/1034968135


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.


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I. Karatzas. S. G. Kou. "On the pricing of contingent claims under constraints." Ann. Appl. Probab. 6 (2) 321 - 369, May 1996.


Published: May 1996
First available in Project Euclid: 18 October 2002

zbMATH: 0856.90012
MathSciNet: MR1398049
Digital Object Identifier: 10.1214/aoap/1034968135

Primary: 60H30 , 90A09 , 93E20
Secondary: 49N15 , 60G44 , 90A10 , 90A16

Keywords: Black-Scholes formula , Constrained portfolios , equivalent martingale measures , incomplete markets , martingale representations , minimization of relative entropy , Pricing of contingent claims , Stochastic control , two different interest rates , utility maximization

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.6 • No. 2 • May 1996
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