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Asymptotic Power Utility-Based Pricing and Hedging

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  • Jan Kallsen
  • Johannes Muhle-Karbe
  • Richard Vierthauer

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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  • Jan Kallsen & Johannes Muhle-Karbe & Richard Vierthauer, 2009. "Asymptotic Power Utility-Based Pricing and Hedging," Papers 0912.3362, arXiv.org, revised Jan 2013.
    • Handle: RePEc:arx:papers:0912.3362
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    References listed on IDEAS

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    1. Marcel Nutz, 2009. "The Opportunity Process for Optimal Consumption and Investment with Power Utility," Papers 0912.1879, arXiv.org, revised Jun 2010.
    2. Alev{s} v{C}ern'y & Jan Kallsen, 2007. "On the Structure of General Mean-Variance Hedging Strategies," Papers 0708.1715, arXiv.org, revised Jul 2017.
    3. Julien Hugonnier & Dmitry Kramkov, 2004. "Optimal investment with random endowments in incomplete markets," Papers math/0405293, arXiv.org.
    4. Aleš Černý, 2007. "Optimal Continuous‐Time Hedging With Leptokurtic Returns," Mathematical Finance, Wiley Blackwell, vol. 17(2), pages 175-203, April.
    5. Goll, Thomas & Kallsen, Jan, 2000. "Optimal portfolios for logarithmic utility," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 31-48, September.
    6. Friedrich Hubalek & Jan Kallsen & Leszek Krawczyk, 2006. "Variance-optimal hedging for processes with stationary independent increments," Papers math/0607112, arXiv.org.
    7. Marcel Nutz, 2009. "The Bellman equation for power utility maximization with semimartingales," Papers 0912.1883, arXiv.org, revised Mar 2012.
    8. Jan Kallsen & Richard Vierthauer, 2009. "Quadratic hedging in affine stochastic volatility models," Review of Derivatives Research, Springer, vol. 12(1), pages 3-27, April.
    9. 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.
    10. Michael Mania & Martin Schweizer, 2005. "Dynamic exponential utility indifference valuation," Papers math/0508489, arXiv.org.
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