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Convex Hedging in Incomplete Markets

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  • Birgit Rudloff

Abstract

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. repec:gam:jrisks:v:5:y:2017:i:3:p:36-:d:105112 is not listed on IDEAS
    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. 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.
    4. 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.
    5. Micha{l} Barski, 2014. "On the shortfall risk control -- a refinement of the quantile hedging method," Papers 1402.3725, arXiv.org, revised Dec 2015.
    6. 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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