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Asymptotic Power Utility-Based Pricing and Hedging

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  • Jan Kallsen
  • Johannes Muhle-Karbe
  • Richard Vierthauer
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    Abstract

    Kramkov and Sirbu (2006, 2007) have shown that first-order approximations of power utility-based prices and hedging strategies can be computed by solving a mean-variance hedging problem under a specific equivalent martingale measure and relative to a suitable numeraire. In order to avoid the introduction of an additional state variable necessitated by the change of numeraire, we propose an alternative representation in terms of the original numeraire. More specifically, we characterize the relevant quantities using semimartingale characteristics similarly as in Cerny and Kallsen (2007) for mean-variance hedging. These results are illustrated by applying them to exponential L\'evy processes and stochastic volatility models of Barndorff-Nielsen and Shephard type.

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    File URL: http://arxiv.org/pdf/0912.3362
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0912.3362.

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    Date of creation: Dec 2009
    Date of revision: Jan 2013
    Handle: RePEc:arx:papers:0912.3362

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    1. Kramkov, D. & Sîrbu, M., 2007. "Asymptotic analysis of utility-based hedging strategies for small number of contingent claims," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1606-1620, November.
    2. Ales Čern�, 2007. "Optimal Continuous-Time Hedging With Leptokurtic Returns," Mathematical Finance, Wiley Blackwell, vol. 17(2), pages 175-203.
    3. Friedrich Hubalek & Jan Kallsen & Leszek Krawczyk, 2006. "Variance-optimal hedging for processes with stationary independent increments," Papers math/0607112, arXiv.org.
    4. Michael Mania & Martin Schweizer, 2005. "Dynamic exponential utility indifference valuation," Papers math/0508489, arXiv.org.
    5. Marcel Nutz, 2009. "The Bellman equation for power utility maximization with semimartingales," Papers 0912.1883, arXiv.org, revised Mar 2012.
    6. Marcel Nutz, 2009. "The Opportunity Process for Optimal Consumption and Investment with Power Utility," Papers 0912.1879, arXiv.org, revised Jun 2010.
    7. Ale\v{s} \v{C}ern\'y & Jan Kallsen, 2007. "On the structure of general mean-variance hedging strategies," Papers 0708.1715, arXiv.org.
    8. Julien Hugonnier & Dmitry Kramkov, 2004. "Optimal investment with random endowments in incomplete markets," Papers math/0405293, arXiv.org.
    9. Goll, Thomas & Kallsen, Jan, 2000. "Optimal portfolios for logarithmic utility," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 31-48, September.
    10. Jan Kallsen & Richard Vierthauer, 2009. "Quadratic hedging in affine stochastic volatility models," Review of Derivatives Research, Springer, vol. 12(1), pages 3-27, April.
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