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Shortfall risk through Fenchel duality

Author

Listed:
  • Zhenyu Cui

    (School of Business, Stevens Institute of Technology, Hoboken, USA)

  • Jun Deng

    (School of Banking and Finance, University of International Business and Economics, Beijing, P. R. China)

Abstract

In this paper, we propose a Fenchel duality approach to study the minimization problem of the shortfall risk. We consider a general increasing and strictly convex loss function, which may be more general than the situation of convex risk measures usually assumed in the literature. We first translate the associated stochastic optimization problem to an equivalent static optimization problem, and then obtain the explicit structure of the optimal randomized test for both complete and incomplete markets. For the incomplete market case, to the best of our knowledge, we obtain for the first time the explicit randomized test, while previous literature only established the existence through the supermartingale optional decomposition approach. We also solve the shortfall risk minimization problem for an insider through the enlargement of filtrations approach.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:ijfexx:v:05:y:2018:i:02:n:s2424786318500196
    DOI: 10.1142/S2424786318500196
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    References listed on IDEAS

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    1. Amendinger, Jürgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 263-286, July.
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    3. Pierre Collin-Dufresne & Vyacheslav Fos, 2015. "Do Prices Reveal the Presence of Informed Trading?," Journal of Finance, American Finance Association, vol. 70(4), pages 1555-1582, August.
    4. Kyle, Albert S, 1985. "Continuous Auctions and Insider Trading," Econometrica, Econometric Society, vol. 53(6), pages 1315-1335, November.
    5. Birgit Rudloff, 2009. "Coherent hedging in incomplete markets," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 197-206.
    6. Birgit Rudloff, 2007. "Convex Hedging in Incomplete Markets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(5), pages 437-452.
    7. Amendinger, Jürgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," SFB 373 Discussion Papers 1998,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    8. Pierre Collin-Dufresne & Vyacheslav Fos, 2013. "Do Prices Reveal the Presence of Informed Trading?," Swiss Finance Institute Research Paper Series 13-69, Swiss Finance Institute, revised Sep 2015.
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    Cited by:

    1. Małgorzata Przybyła-Kasperek, 2019. "Three Conflict Methods in Multiple Classifiers that Use Dispersed Knowledge," International Journal of Information Technology & Decision Making (IJITDM), World Scientific Publishing Co. Pte. Ltd., vol. 18(02), pages 555-599, March.

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