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Dynamic exponential utility indifference valuation

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  • Michael Mania
  • Martin Schweizer
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    Abstract

    We study the dynamics of the exponential utility indifference value process C(B;\alpha) for a contingent claim B in a semimartingale model with a general continuous filtration. We prove that C(B;\alpha) is (the first component of) the unique solution of a backward stochastic differential equation with a quadratic generator and obtain BMO estimates for the components of this solution. This allows us to prove several new results about C_t(B;\alpha). We obtain continuity in B and local Lipschitz-continuity in the risk aversion \alpha, uniformly in t, and we extend earlier results on the asymptotic behavior as \alpha\searrow0 or \alpha\nearrow\infty to our general setting. Moreover, we also prove convergence of the corresponding hedging strategies.

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    File URL: http://arxiv.org/pdf/math/0508489
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    Paper provided by arXiv.org in its series Papers with number math/0508489.

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    Date of creation: Aug 2005
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    Publication status: Published in Annals of Applied Probability 2005, Vol. 15, No. 3, 2113-2143
    Handle: RePEc:arx:papers:math/0508489

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    1. Pauline Barrieu & Nicole El Karoui, 2005. "Inf-convolution of risk measures and optimal risk transfer," Finance and Stochastics, Springer, vol. 9(2), pages 269-298, 04.
    2. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
    3. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52.
    4. Becherer, Dirk, 2003. "Rational hedging and valuation of integrated risks under constant absolute risk aversion," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 1-28, August.
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    Cited by:
    1. Frei, Christoph & Mocha, Markus & Westray, Nicholas, 2012. "BSDEs in utility maximization with BMO market price of risk," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2486-2519.
    2. Michail Anthropelos & Nikolaos E. Frangos & Stylianos Z. Xanthopoulos & Athanasios N. Yannacopoulos, 2008. "On contingent claims pricing in incomplete markets: A risk sharing approach," Papers 0809.4781, arXiv.org, revised Feb 2012.
    3. Keita Owari, 2013. "A Robust Version of Convex Integral Functionals," CARF F-Series CARF-F-319, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    4. Tevzadze, Revaz, 2008. "Solvability of backward stochastic differential equations with quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 118(3), pages 503-515, March.
    5. Mingxin Xu, 2004. "Risk Measure Pricing and Hedging in Incomplete Markets," Finance 0406004, EconWPA, revised 06 Apr 2005.
    6. Christoph Frei & Markus Mocha & Nicholas Westray, 2011. "BSDEs in Utility Maximization with BMO Market Price of Risk," Papers 1107.0183, arXiv.org, revised Feb 2012.
    7. Vicky Henderson & Gechun Liang, 2014. "Pseudo Linear Pricing Rule for Utility Indifference Valuation," Papers 1403.7830, arXiv.org.
    8. Claudia Ceci & Anna Gerardi, 2011. "Utility indifference valuation for jump risky assets," Decisions in Economics and Finance, Springer, vol. 34(2), pages 85-120, November.
    9. Beatrice Acciaio & Irina Penner, 2010. "Dynamic risk measures," Papers 1002.3794, arXiv.org.
    10. Ale\v{s} \v{C}ern\'y & Stephan Denkl & Jan Kallsen, 2013. "Hedging in L\'evy models and the time step equivalent of jumps," Papers 1309.7833, arXiv.org, revised Nov 2013.
    11. Arnaud Porchet & Nizar Touzi & Xavier Warin, 2009. "Valuation of power plants by utility indifference and numerical computation," Computational Statistics, Springer, vol. 70(1), pages 47-75, August.
    12. Jan Kallsen & Johannes Muhle-Karbe & Richard Vierthauer, 2009. "Asymptotic Power Utility-Based Pricing and Hedging," Papers 0912.3362, arXiv.org, revised Jan 2013.
    13. Marie-Amélie Morlais, 2009. "Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem," Finance and Stochastics, Springer, vol. 13(1), pages 121-150, January.
    14. Morlais, Marie-Amelie, 2010. "A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1966-1995, September.
    15. Michail Anthropelos & Gordan Žitković, 2010. "Partial equilibria with convex capital requirements: existence, uniqueness and stability," Annals of Finance, Springer, vol. 6(1), pages 107-135, January.

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