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Optimal static-dynamic hedges for exotic options under convex risk measures

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  • Ilhan, Aytaç
  • Jonsson, Mattias
  • Sircar, Ronnie
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    Abstract

    We study the problem of optimally hedging exotic derivatives positions using a combination of dynamic trading strategies in underlying stocks and static positions in vanilla options when the performance is quantified by a convex risk measure. We establish conditions for the existence of an optimal static position for general convex risk measures, and then analyze in detail the case of shortfall risk with a power loss function. Here we find conditions for uniqueness of the static hedge. We illustrate the computational challenge of computing the market-adjusted risk measure in a simple diffusion model for an option on a non-traded asset.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 119 (2009)
    Issue (Month): 10 (October)
    Pages: 3608-3632

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    Handle: RePEc:eee:spapps:v:119:y:2009:i:10:p:3608-3632

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    Keywords: Risk measures Hedging Exotic options;

    References

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    1. Zhegal, Amina & Touzi, Nizar & Bouchard, Bruno, 2004. "Dual Formulation of the Utility Maximization Problem : the case of Nonsmooth Utility," Economics Papers from University Paris Dauphine 123456789/1531, Paris Dauphine University.
    2. 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.
    3. David Hobson, 2004. "STOCHASTIC VOLATILITY MODELS, CORRELATION, AND THE "q"-OPTIMAL MEASURE," Mathematical Finance, Wiley Blackwell, vol. 14(4), pages 537-556.
    4. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    5. Birgit Rudloff, 2007. "Convex Hedging in Incomplete Markets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(5), pages 437-452.
    6. B. Bouchard & N. Touzi & A. Zeghal, 2004. "Dual formulation of the utility maximization problem: the case of nonsmooth utility," Papers math/0405290, arXiv.org.
    7. Aytaç İlhan & Ronnie Sircar, 2006. "Optimal Static-Dynamic Hedges For Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 359-385.
    8. Hans FÃllmer & Peter Leukert, 2000. "Efficient hedging: Cost versus shortfall risk," Finance and Stochastics, Springer, vol. 4(2), pages 117-146.
    9. Ioannis Karatzas & Jaksa Cvitanic, 1999. "On dynamic measures of risk," Finance and Stochastics, Springer, vol. 3(4), pages 451-482.
    10. Aytaç Ílhan & Mattias Jonsson & Ronnie Sircar, 2005. "Optimal investment with derivative securities," Finance and Stochastics, Springer, vol. 9(4), pages 585-595, October.
    11. Mattias Jonsson & K. Ronnie Sircar, 2002. "Partial Hedging In A Stochastic Volatility Environment," Mathematical Finance, Wiley Blackwell, vol. 12(4), pages 375-409.
    12. Marek Musiela & Thaleia Zariphopoulou, 2004. "An example of indifference prices under exponential preferences," Finance and Stochastics, Springer, vol. 8(2), pages 229-239, 05.
    13. Freddy Delbaen & Peter Grandits & Thorsten Rheinländer & Dominick Samperi & Martin Schweizer & Christophe Stricker, 2002. "Exponential Hedging and Entropic Penalties," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 99-123.
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