Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T15:56:42.104Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

[1] Biagini, S. and Frittelli, M. (2004). On the super replication price of unbounded claims. Ann. Appl. Prob. 14, 1970–1970.Google Scholar
[2] Biagini, S. and Frittelli, M. (2005). Utility maximization in incomplete markets for unbounded processes. Finance Stoch. 9, 493–517.Google Scholar
[3] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520.Google Scholar
[4] Delbaen, F. and Schachermayer, W. (1997). The Banach space of workable contingent claims in arbitrage theory. Ann. Inst. H. Poincaré Prob. Statist. 33, 113–144.CrossRefGoogle Scholar
[5] Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250.Google Scholar
[6] Elliott, R. J. and Kopp, P. E. (2005). Mathematics of Financial Markets, 2nd edn. Springer, New York.Google Scholar
[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.Google Scholar
[8] Owen, M. P. (2003). On utility based super replication prices. Preprint, Department of Actuarial Mathematics and Statistics, Heriot-Watt University.Google Scholar
[9] Owen, M. P. and Žitković, G. (2008). Optimal investment with an unbounded random endowment and utility-based pricing. To appear in Math. Finance.Google Scholar
[10] Rockafellar, R. T. (1972). Convex Analysis. Princeton University Press.Google Scholar
[11] Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. Ann. App. Prob. 11, 694–734.Google Scholar
[12] Wong, Y.-C. (1993). Some Topics in Functional Analyis and Operator Theory. Science Press, Beijing.Google Scholar