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On the Super Replication Price of Unbounded Claims

Sara Biagini and Marco Frittelli
The Annals of Applied Probability
Vol. 14, No. 4 (Nov., 2004), pp. 1970-1991
Stable URL: http://www.jstor.org/stable/4140455
Page Count: 22
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On the Super Replication Price of Unbounded Claims
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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 $sup_Q E_Q[f]$, 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.

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