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On Utility-Based Superreplication Prices of Contingent Claims with Unbounded Payoffs

  • Frank Oertel (a1) and Mark Owen (a2)

Abstract

Consider a financial market in which an agent trades with utility-induced restrictions on wealth. For a utility function which satisfies the condition of reasonable asymptotic elasticity at -∞, we prove that the utility-based superreplication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite loss-entropy. For an agent whose utility function is unbounded from above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, the former set is the closure of the latter under a suitable weak topology. Central to our proof is a proof of the duality between the cone of utility-based superreplicable contingent claims and the cone generated by pricing measures with finite loss-entropy.

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Copyright

Corresponding author

Postal address: School of Mathematical Sciences, Aras Na Laoi, University College Cork, Cork, Ireland.
∗∗ Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: mowen@ma.hw.ac.uk

Footnotes

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∗∗∗

The authors gratefully acknowledge support from EPSRC grant number GR/S80202/01.

Footnotes

References

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[1] Biagini, S. and Frittelli, M. (2004). On the super replication price of unbounded claims. Ann. Appl. Prob. 14, 19701970.
[2] Biagini, S. and Frittelli, M. (2005). Utility maximization in incomplete markets for unbounded processes. Finance Stoch. 9, 493517.
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[6] Elliott, R. J. and Kopp, P. E. (2005). Mathematics of Financial Markets, 2nd edn. Springer, New York.
[7] Oertel, F. and Owen, M. P. (2005). Geometry of polar wedges and super-replication prices in incomplete financial markets. Preprint, Department of Actuarial Mathematics and Statistics, Heriot-Watt University.
[8] Owen, M. P. (2003). On utility based super replication prices. Preprint, Department of Actuarial Mathematics and Statistics, Heriot-Watt University.
[9] Owen, M. P. and Žitković, G. (2008). Optimal investment with an unbounded random endowment and utility-based pricing. To appear in Math. Finance.
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[12] Wong, Y.-C. (1993). Some Topics in Functional Analyis and Operator Theory. Science Press, Beijing.

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