How Potential Investments may Change the OptimalPortfolio for the Exponential

We present an example of a nancial market consisting of a deterministic bond and two risky nancial assets (S 1 t ; S 2 t) 0tT such that, for an agent whose preferences are modeled by expected exponential utility at time T , it is optimal to constantly hold one unit of asset S 1. However, if we pass to the market consisting only of the bond and the rst risky asset S 1 , this trading strategy is not optimal any more: now it is optimal for the utility maximizer to invest the initial endowment into the bond. This phenomenon, which at rst glance seems paradoxical from an economic point of view, reveals that the proper deenition of the class of \permitted" trading strategies H is a subtle issue if we consider exponential utility U (x) = ?e ?x or, more generally, utility functions U assuming nite values on all of R. One way of choosing a proper class was introduced in S 00]: those trading strategies H which can be approximated by a sequence (H n) 1 n=1 such that, for each n 2 N, the process H n S is bounded from below, i.e., may be nanced with a nite credit line. In the example it turns out that, for the strategy of holding constantly one unit of the asset S 1 , such an approximation is only possible if the agent also has the second asset S 2 at her disposal. Nevertheless this asset is not used by the optimal strategy and therefore its role might be compared to a catalyst in chemistry. Another way of deening an appropriate class of \permitted" trading strategies H was proposed in DGRSSS 00], namely those predictable processes H such that H S is a Q-martingale, for each equivalent martingale measure Q with nite entropy H (QjP). Under some additional technical condition it was shown in DGRSSS 00] that this class does contain the optimal strategy; in the recent paper KS 00] this result was proved in full generality. In section 2 we also present a proof of this result and extend it to general utility function having reasonable asymptotic elasticity. For this extension to hold true the term Q-martingale has to be replaced by the term Q-supermartingale.