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Convex Hedging in Incomplete Markets


  • Birgit Rudloff


In incomplete financial markets not every contingent claim can be replicated by a self-financing strategy. The risk of the resulting shortfall can be measured by convex risk measures, recently introduced by Follmer and Schied (2002). The dynamic optimization problem of finding a self-financing strategy that minimizes the convex risk of the shortfall can be split into a static optimization problem and a representation problem. It follows that the optimal strategy consists in superhedging the modified claim [image omitted] , where H is the payoff of the claim and [image omitted] is a solution of the static optimization problem, an optimal randomized test. In this paper, necessary and sufficient optimality conditions are deduced for the static problem using convex duality methods. The solution of the static optimization problem turns out to be a randomized test with a typical 0-1-structure.

Suggested Citation

  • Birgit Rudloff, 2007. "Convex Hedging in Incomplete Markets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(5), pages 437-452.
  • Handle: RePEc:taf:apmtfi:v:14:y:2007:i:5:p:437-452
    DOI: 10.1080/13504860701352206

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    Cited by:

    1. Hirbod Assa & Nikolay Gospodinov, 2017. "A Robust Approach to Hedging and Pricing in Imperfect Markets," Risks, MDPI, Open Access Journal, vol. 5(3), pages 1-20, July.
    2. Li, Jing & Xu, Mingxin, 2009. "Minimizing Conditional Value-at-Risk under Constraint on Expected Value," MPRA Paper 26342, University Library of Munich, Germany, revised 25 Oct 2010.
    3. Raimund M. Kovacevic, 2019. "Valuation and pricing of electricity delivery contracts: the producer’s view," Annals of Operations Research, Springer, vol. 275(2), pages 421-460, April.
    4. Ilhan, Aytaç & Jonsson, Mattias & Sircar, Ronnie, 2009. "Optimal static-dynamic hedges for exotic options under convex risk measures," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3608-3632, October.
    5. Zhenyu Cui & Jun Deng, 2018. "Shortfall risk through Fenchel duality," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(02), pages 1-14, June.
    6. Tim Leung & Qingshuo Song & Jie Yang, 2013. "Outperformance portfolio optimization via the equivalence of pure and randomized hypothesis testing," Finance and Stochastics, Springer, vol. 17(4), pages 839-870, October.
    7. Micha{l} Barski, 2014. "On the shortfall risk control -- a refinement of the quantile hedging method," Papers 1402.3725,, revised Dec 2015.
    8. Ludovic Tangpi, 2018. "Efficient hedging under ambiguity in continuous time," Papers 1812.10876,, revised Mar 2019.
    9. L. Rüschendorf & Steven Vanduffel, 2020. "On the construction of optimal payoffs," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(1), pages 129-153, June.
    10. Mustafa Ç. Pinar, 2010. "Buyer's quantile hedge portfolios in discrete-time trading," Quantitative Finance, Taylor & Francis Journals, vol. 13(5), pages 729-738, October.
    11. Tomasz Tkalinski, 2014. "Convex hedging of non-superreplicable claims in discrete-time market models," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 79(2), pages 239-252, April.


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