Utility-based super-replication prices of unbounded contingent claims and duality of cones

Consider a financial market in which an agent trades with utility-induced restrictions on wealth. We prove that the utility-based super-replication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite entropy. Central to our proof is the representation of a cone CV of utilitybased super-replicable contingent claims as the polar cone of the set of finite entropy separating measures. CV is shown to be the closure, under a relevant weak topology, of the cone of all (sufficiently integrable) contingent claims that can be dominated by a zero-financed terminal wealth. As our approach shows, those terminal wealths need not necessarily stem from admissible trading strategies only. We investigate also the natural dual of this result, and show that the polar cone of CV is the cone generated by separating measures with finite loss-entropy. For an agent whose utility function is unbounded from above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, we prove that the former set is the closure of the latter under a suitable weak topology. The full two-sided polarity we achieve between measures and contingent claims yields an economic justification for the use of the cone CV : the utilitybased restrictions which this cone imposes on terminal wealth arise only from the investor’s preferences to asymptotically large negative wealth. An application of our results to the special case of admissible trading reveals that under a suitable weak topology CV is the closure of the cone of all a. s. bounded contingent claims that can be dominated by a terminal wealth originating from admissible trading strategies. Finally, we show how our framework can be applied to another field of mathematical economics and how it sheds a different light on Farkas’ Lemma and its infinite dimensional version there.