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Asymptotic power utility-based pricing and hedging

Kallsen, Jan ; Muhle-Karbe, Johannes ; Vierthauer, Richard

In: Mathematics and Financial Economics, 2014, vol. 8, no. 1, p. 1-28

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    Title
    • Asymptotic power utility-based pricing and hedging
    Author
    • Kallsen, Jan. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Westring 383, 24118, Kiel, Germany
    • Muhle-Karbe, Johannes. Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland
    • Vierthauer, Richard. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Westring 383, 24118, Kiel, Germany
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Mathematics and Financial Economics, 2014, vol. 8, no. 1, p. 1-28. Springer Berlin Heidelberg
    Other Version
    OAI-PMH Identifier
    • oai:doc.rero.ch:326241
    Summary
    Kramkov and Sîrbu (Ann. Appl. Probab., 16:2140-2194, 2006; Stoch. Proc. Appl., 117:1606-1620, 2017) have shown that first-order approximations of power utility-based prices and hedging strategies for a small number of claims can be computed by solving a mean-variance hedging problem under a specific equivalent martingale measure and relative to a suitable numeraire. For power utilities, we propose an alternative representation that avoids the change of numeraire. More specifically, we characterize the relevant quantities using semimartingale characteristics similarly as in Černý and Kallsen (Ann. Probab., 35:1479-1531, 2007) for mean-variance hedging. These results are illustrated by applying them to exponential Lévy processes and stochastic volatility models of Barndorff-Nielsen and Shephard type (J. R. Stat. Soc. B, 63:167-241, 2001). We find that asymptotic utility-based hedges are virtually independent of the investor's risk aversion. Moreover, the price adjustments compared to the Black-Scholes model turn out to be almost linear in the investor's risk aversion, and surprisingly small unless very high levels of risk aversion are considered.