Geometry of polar wedges and super-replication prices in incomplete financial markets

Consider a financial market in which an agent trades with utility-induced restrictions on wealth. By introducing a general convex-analytic framework which includes the class of umbrella wedges in certain Riesz spaces (cf. Definition 2.2 and Remark 2.3) and faces of convex sets, consisting of probability measures (cf. Lemma 4.5 and Proposition 5.4), together with a duality theory for polar wedges, we provide a representation of the super-replication price of an unbounded (but sufficiently integrable) contingent claim that can be dominated approximately by a zero-financed terminal wealth as the the supremum of its discounted expectation under pricing measures which appear as faces of a given set of separating probability measures (cf. Proposition 5.3 and Corollary 5.5). As a first direct application of our general framework, we see that the utility-based super-replication price of an unbounded contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite entropy (cf. Corollary 6.5). Central to our proof is the representation of a wedge CΦ of utility-based super-replicable contingent claims as the polar wedge of the set of finite entropy separating measures. CΦ is shown to be the closure, under a relevant weak topology, of the wedge of all (sufficiently integrable) contingent claims that can be dominated by a zero-financed terminal wealth. Our general approach shows, that those terminal wealths need not necessarily stem from admissible trading strategies only. We investigate also the natural dual of this result, and show in particular that the polar wedge of CΦ is the wedge 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 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 (cf. Theorem 6.4).