On utility-based super-replication prices of contingent claims with unbounded payoffs

Abstract

Consider a financial market in which an agent trades with utility-induced restrictions on wealth. For a utility function which satisfies the condition of reasonable asymptotic elasticity at $-\infty$ 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 {\it 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, the former set is the closure of the latter under a suitable weak topology. Central to our proof is the representation of a cone $C_U$ of utility-based super-replicable contingent claims as the polar cone to the set of finite loss-entropy pricing measures. The cone $C_U$ is defined as 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. We investigate also the natural dual of this result and show that the polar cone to $C_U$ is generated by those separating measures with finite loss-entropy. The full two-sided polarity we achieve between measures and contingent claims yields an economic justification for the use of the cone $C_U$, and an open question.