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Coherent hedging in incomplete markets

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  • Birgit Rudloff

Abstract

In incomplete financial markets, not every contingent claim can be perfectly replicated by a self-financing strategy. In this paper, we minimize the risk that the value of the hedging portfolio falls below the payoff of the claim at time T. We use a coherent risk measure, introduced by Artzner et al., to measure the risk of the shortfall. The dynamic optimization problem of finding a self-financing strategy that minimizes the coherent risk of the shortfall can be split into a static optimization problem and a representation problem. We will deduce necessary and sufficient optimality conditions 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. Our results improve those obtained by Nakano. The optimal hedging strategy consists of superhedging a modified claim that is the product of the original payoff and the solution to the static problem.

Suggested Citation

  • Birgit Rudloff, 2009. "Coherent hedging in incomplete markets," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 197-206.
    • Handle: RePEc:taf:quantf:v:9:y:2009:i:2:p:197-206
    DOI: 10.1080/14697680802169787
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    Citations

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    Cited by:

    1. Wayne King Ming Chan, 2015. "RAROC-Based Contingent Claim Valuation," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 3-2015.
    2. Wayne King Ming Chan, 2015. "RAROC-Based Contingent Claim Valuation," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 21, July-Dece.
    3. 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.
    4. Grechuk, Bogdan & Zabarankin, Michael, 2016. "Inverse portfolio problem with coherent risk measures," European Journal of Operational Research, Elsevier, vol. 249(2), pages 740-750.
    5. Hirbod Assa & Nikolay Gospodinov, 2017. "A Robust Approach to Hedging and Pricing in Imperfect Markets," Risks, MDPI, vol. 5(3), pages 1-20, July.
    6. 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.
    7. 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.

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