On the super replication price of unbounded claims

Abstract

In an incomplete market the price of a claim f in general cannot be uniquely identified by no arbitrage arguments. However, the ``classical'' super replication price is a sensible indicator of the (maximum selling) value of the claim. When f satisfies certain pointwise conditions (e.g., f is bounded from below), the super replication price is equal to sup_QE_Q[f], where Q varies on the whole set of pricing measures. Unfortunately, this price is often too high: a typical situation is here discussed in the examples. We thus define the less expensive weak super replication price and we relax the requirements on f by asking just for ``enough'' integrability conditions. By building up a proper duality theory, we show its economic meaning and its relation with the investor's preferences. Indeed, it turns out that the weak super replication price of f coincides with sup_{Q\in M_{\Phi}}E_Q[f], where M_{\Phi} is the class of pricing measures with finite generalized entropy (i.e., E[\Phi (\frac{dQ}{dP})]<\infty) and where \Phi is the convex conjugate of the utility function of the investor.Comment: Published at http://dx.doi.org/10.1214/105051604000000459 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org