A company with $n$ geographically widely dispersed sites seeks insurance that pays off if $m$ out of the $n$ sites experience rarely occurring catastrophes (e.g., earthquakes) during a year. This study describes an adaptive dynamic strategy that enables an insurance company to offer the policy with smaller loss probability than more conventional static policies induce, but at a comparable or lower premium. The strategy accomplishes this by periodically purchasing reinsurance on individual sites. Exploiting rarity, the policy induces zero loss with probability one if no more than one quake occurs during any review interval. The policy also may induce a profit if $m$ or more quakes occur in an interval if no quakes have occurred in previous intervals. The study also examines the benefit of more than one reinsurance policy per active site. The study relies on expected utility to determine indifference premiums and derives an upper bound on loss probability independent of premium.

Motivated by the observationthat the gain-loss criterion, while offering economically meaningful prices of contingent claims,is sensitive to the reference measure governing the underlying stock price process (a situationreferred to as ambiguity of measure), we propose a gain-loss pricing model robust to shifts in the reference measure. Using a dual representation property of polyhedral risk measures we obtain a one-step, gain-loss criterion based theorem ofasset pricing under ambiguity of measure, and illustrate its use.

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 -∞, we prove that the utility-based superreplication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing 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, the former set is the closure of the latter under a suitable weak topology. Central to our proof is a proof of the duality between the cone of utility-based superreplicable contingent claims and the cone generated by pricing measures with finite loss-entropy.