arXiv math/05035502005-03-24math.PRq-fin.PR

On the super replication price of unbounded claims

Sara Biagini, Marco Frittelli

备注：: 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)
发表信息：: Annals of Applied Probability 2004, Vol. 14, No. 4, 1970-1991
DOI：: 10.1214/105051604000000459

英文摘要：

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_Φ}E_Q[f], where M_Φ is the class of pricing measures with finite generalized entropy (i.e., E[Φ(\frac{dQ}{dP})]<\infty) and where Φis the convex conjugate of the utility function of the investor.

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