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

Abstract

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 and faces of convex sets (consisting of probability measures), 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. Central to our investigation is the representation of a wedge $C_Φ$ of utility-based super-replicable contingent claims as the polar wedge of the set of finite entropy separating measures. Our general approach shows, that those terminal wealths need {\it not} necessarily stem from {\it admissible} trading strategies only. The full two-sided polarity we achieve between measures and contingent claims yields an economic justification for the use of the wedge $C_Φ$: the utility-based restrictions which this wedge imposes on terminal wealth arise only from the investor's preferences to asymptotically large negative wealth.