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Num\'{e}raire-invariant preferences in financial modeling

  • Constantinos Kardaras
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    We provide an axiomatic foundation for the representation of num\'{e}raire-invariant preferences of economic agents acting in a financial market. In a static environment, the simple axioms turn out to be equivalent to the following choice rule: the agent prefers one outcome over another if and only if the expected (under the agent's subjective probability) relative rate of return of the latter outcome with respect to the former is nonpositive. With the addition of a transitivity requirement, this last preference relation has an extension that can be numerically represented by expected logarithmic utility. We also treat the case of a dynamic environment where consumption streams are the objects of choice. There, a novel result concerning a canonical representation of unit-mass optional measures enables us to explicitly solve the investment--consumption problem by separating the two aspects of investment and consumption. Finally, we give an application to the problem of optimal num\'{e}raire investment with a random time-horizon.

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    File URL: http://arxiv.org/pdf/0903.3736
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    Paper provided by arXiv.org in its series Papers with number 0903.3736.

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    Date of creation: Mar 2009
    Date of revision: Nov 2010
    Publication status: Published in Annals of Applied Probability 2010, Vol. 20, No. 5, 1697-1728
    Handle: RePEc:arx:papers:0903.3736
    Contact details of provider: Web page: http://arxiv.org/

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    1. Eckhard Platen, 2004. "A Benchmark Approach to Finance," Research Paper Series 138, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Stefan Ankirchner & Steffen Dereich & Peter Imkeller, 2005. "The Shannon information of filtrations and the additional logarithmic utility of insiders," Papers math/0503013, arXiv.org, revised May 2006.
    3. Blanchet-Scalliet, Christophette & El Karoui, Nicole & Jeanblanc, Monique & Martellini, Lionel, 2008. "Optimal investment decisions when time-horizon is uncertain," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1100-1113, December.
    4. Bruno Bouchard & Huyên Pham, 2004. "Wealth-path dependent utility maximization in incomplete markets," Finance and Stochastics, Springer, vol. 8(4), pages 579-603, November.
    5. Gordan Zitkovic, 2005. "Utility Maximization with a Stochastic Clock and an Unbounded Random Endowment," Papers math/0503516, arXiv.org.
    6. Samuelson, Paul A., 1979. "Why we should not make mean log of wealth big though years to act are long," Journal of Banking & Finance, Elsevier, vol. 3(4), pages 305-307, December.
    7. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, vol. 5(3), pages 327-341.
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