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

Part of: Miscellaneous applications of functional analysis Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Frank Oertel  and
Mark Owen
Frank Oertel*
Affiliation:
University College Cork
Mark Owen*
Affiliation:
Heriot-Watt University
*
∗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
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Abstract

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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.

Keywords

Superreplicationincomplete marketcontingent claimduality theoryweak topology

MSC classification

Secondary: 46N10: Applications in optimization, convex analysis, mathematical programming, economics 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 880 - 888
Copyright
Copyright © Applied Probability Trust 2007 

Footnotes

∗∗∗

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

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