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Worst-Case Value at Risk of Nonlinear Portfolios


  • Steve Zymler

    () (Department of Computing, Imperial College of Science, Technology and Medicine, London SW7 2AZ, United Kingdom)

  • Daniel Kuhn

    () (Department of Computing, Imperial College of Science, Technology and Medicine, London SW7 2AZ, United Kingdom)

  • Berç Rustem

    () (Department of Computing, Imperial College of Science, Technology and Medicine, London SW7 2AZ, United Kingdom)


Portfolio optimization problems involving value at risk (VaR) are often computationally intractable and require complete information about the return distribution of the portfolio constituents, which is rarely available in practice. These difficulties are compounded when the portfolio contains derivatives. We develop two tractable conservative approximations for the VaR of a derivative portfolio by evaluating the worst-case VaR over all return distributions of the derivative underliers with given first- and second-order moments. The derivative returns are modelled as convex piecewise linear or--by using a delta-gamma approximation--as (possibly nonconvex) quadratic functions of the returns of the derivative underliers. These models lead to new worst-case polyhedral VaR (WPVaR) and worst-case quadratic VaR (WQVaR) approximations, respectively. WPVaR serves as a VaR approximation for portfolios containing long positions in European options expiring at the end of the investment horizon, whereas WQVaR is suitable for portfolios containing long and/or short positions in European and/or exotic options expiring beyond the investment horizon. We prove that--unlike VaR that may discourage diversification--WPVaR and WQVaR are in fact coherent risk measures. We also reveal connections to robust portfolio optimization. This paper was accepted by Dimitris Bertsimas, optimization.

Suggested Citation

  • Steve Zymler & Daniel Kuhn & Berç Rustem, 2013. "Worst-Case Value at Risk of Nonlinear Portfolios," Management Science, INFORMS, vol. 59(1), pages 172-188, July.
  • Handle: RePEc:inm:ormnsc:v:59:y:2013:i:1:p:172-188
    DOI: 10.1287/mnsc.1120.1615

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    References listed on IDEAS

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    3. Gabrel, Virginie & Murat, Cécile & Thiele, Aurélie, 2014. "Recent advances in robust optimization: An overview," European Journal of Operational Research, Elsevier, vol. 235(3), pages 471-483.
    4. Barrieu, Pauline & Scandolo, Giacomo, 2014. "Assessing financial model risk," LSE Research Online Documents on Economics 60084, London School of Economics and Political Science, LSE Library.
    5. Lotfi, Somayyeh & Zenios, Stavros A., 2018. "Robust VaR and CVaR optimization under joint ambiguity in distributions, means, and covariances," European Journal of Operational Research, Elsevier, vol. 269(2), pages 556-576.
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    10. Amir Ahmadi-Javid & Malihe Fallah-Tafti, 2017. "Portfolio Optimization with Entropic Value-at-Risk," Papers 1708.05713,
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