On the Super Replication Price of Unbounded Claims

• Published in: The Annals of applied probability 2004-11, Vol.14 (4), p.1970-1991
• Main Authors: Biagini, Sara, Frittelli, Marco
• Format: Article
• Language: English
• Subjects: 60G42; 60G44; duality; Entropy; generalized entropy; incomplete markets; Investors; Market prices; Martingales; Mathematics; preferences; reasonable asymptotic elasticity; Super replication price; Topological theorems; Topological vector spaces; Topology; Utility functions; utility maximization; Weighting functions
• ISSN: 1050-5164; 2168-8737
• DOI: 10.1214/105051604000000459

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$sup_Q E_Q[f]$, 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.,$E[\Phi({dQ\over dP})] < \infty$) and where Φ is the convex conjugate of the utility function of the investor.