Utility maximization under increasing risk aversion in one-period models

It has been shown at different levels of generality that under increasing risk aversion utility indifference sell prices of a contingent claim converge to the super-replication price and the shortfalls of utility maximizing hedging portfolios starting from the super-replication price tend to zero in L-1. In this paper we give an example of a one-period financial model with bounded prices where utility optimal strategies and terminal wealths stay bounded but do not converge when the risk aversion is going to infinity. Then we give general results on the behavior of utility maximizing strategies and terminal wealths under increasing risk aversion in one-period models. The concept of a balanced strategy turns out to play a crucial role.