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Worst-Case Value at Risk of Nonlinear Portfolios

Author

Listed:
  • 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)

Abstract

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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    File URL: http://dx.doi.org/10.1287/mnsc.1120.1615
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    References listed on IDEAS

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    Cited by:

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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.
    6. Yu, Jing-Rung & Chiou, Wan-Jiun Paul & Mu, Da-Ren, 2015. "A linearized value-at-risk model with transaction costs and short selling," European Journal of Operational Research, Elsevier, vol. 247(3), pages 872-878.
    7. Zhu, Shushang & Zhu, Wei & Pei, Xi & Cui, Xueting, 2020. "Hedging crash risk in optimal portfolio selection," Journal of Banking & Finance, Elsevier, vol. 119(C).
    8. Ling, Aifan & Sun, Jie & Wang, Meihua, 2020. "Robust multi-period portfolio selection based on downside risk with asymmetrically distributed uncertainty set," European Journal of Operational Research, Elsevier, vol. 285(1), pages 81-95.
    9. Ling, Aifan & Sun, Jie & Yang, Xiaoguang, 2014. "Robust tracking error portfolio selection with worst-case downside risk measures," Journal of Economic Dynamics and Control, Elsevier, vol. 39(C), pages 178-207.
    10. Amir Ahmadi-Javid & Malihe Fallah-Tafti, 2017. "Portfolio Optimization with Entropic Value-at-Risk," Papers 1708.05713, arXiv.org.
    11. Chen, Jingnan, 2020. "Optimal liquidation of financial derivatives," Finance Research Letters, Elsevier, vol. 34(C).
    12. Asimit, Alexandru V. & Hu, Junlei & Xie, Yuantao, 2019. "Optimal robust insurance with a finite uncertainty set," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 67-81.
    13. Erick Delage & Daniel Kuhn & Wolfram Wiesemann, 2019. "“Dice”-sion–Making Under Uncertainty: When Can a Random Decision Reduce Risk?," Management Science, INFORMS, vol. 65(7), pages 3282-3301, July.
    14. Ahmadi-Javid, Amir & Fallah-Tafti, Malihe, 2019. "Portfolio optimization with entropic value-at-risk," European Journal of Operational Research, Elsevier, vol. 279(1), pages 225-241.
    15. Jin, Xiu & Chen, Na & Yuan, Ying, 2019. "Multi-period and tri-objective uncertain portfolio selection model: A behavioral approach," The North American Journal of Economics and Finance, Elsevier, vol. 47(C), pages 492-504.
    16. Asimit, Alexandru V. & Bignozzi, Valeria & Cheung, Ka Chun & Hu, Junlei & Kim, Eun-Seok, 2017. "Robust and Pareto optimality of insurance contracts," European Journal of Operational Research, Elsevier, vol. 262(2), pages 720-732.
    17. Sio Chong U & Jacky So & Deng Ding & Lihong Liu, 2016. "An efficient Fourier expansion method for the calculation of value-at-risk: Contributions of extra-ordinary risks," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(01), pages 1-27, March.
    18. Ramponi, Federico Alessandro & Campi, Marco C., 2018. "Expected shortfall: Heuristics and certificates," European Journal of Operational Research, Elsevier, vol. 267(3), pages 1003-1013.
    19. Zhu, Shushang & Fan, Minjie & Li, Duan, 2014. "Portfolio management with robustness in both prediction and decision: A mixture model based learning approach," Journal of Economic Dynamics and Control, Elsevier, vol. 48(C), pages 1-25.
    20. Zhilin Kang & Zhongfei Li, 2018. "An exact solution to a robust portfolio choice problem with multiple risk measures under ambiguous distribution," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(2), pages 169-195, April.
    21. Xuan Vinh Doan & Xiaobo Li & Karthik Natarajan, 2015. "Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals," Operations Research, INFORMS, vol. 63(6), pages 1468-1488, December.

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