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Tight Bounds for a Class of Data-Driven Distributionally Robust Risk Measures

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  • Derek Singh
  • Shuzhong Zhang

Abstract

This paper expands the notion of robust moment problems to incorporate distributional ambiguity using Wasserstein distance as the ambiguity measure. The classical Chebyshev-Cantelli (zeroth partial moment) inequalities, Scarf and Lo (first partial moment) bounds, and semideviation (second partial moment) in one dimension are investigated. The infinite dimensional primal problems are formulated and the simpler finite dimensional dual problems are derived. A principal motivating question is how does data-driven distributional ambiguity affect the moment bounds. Towards answering this question, some theory is developed and computational experiments are conducted for specific problem instances in inventory control and portfolio management. Finally some open questions and suggestions for future research are discussed.

Suggested Citation

  • Derek Singh & Shuzhong Zhang, 2020. "Tight Bounds for a Class of Data-Driven Distributionally Robust Risk Measures," Papers 2010.05398, arXiv.org, revised Oct 2020.
    • Handle: RePEc:arx:papers:2010.05398
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    References listed on IDEAS

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    1. Ioana Popescu, 2005. "A Semidefinite Programming Approach to Optimal-Moment Bounds for Convex Classes of Distributions," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 632-657, August.
    2. James E. Smith, 1995. "Generalized Chebychev Inequalities: Theory and Applications in Decision Analysis," Operations Research, INFORMS, vol. 43(5), pages 807-825, October.
    3. Derek Singh & Shuzhong Zhang, 2019. "Distributionally Robust XVA via Wasserstein Distance Part 2: Wrong Way Funding Risk," Papers 1910.03993, arXiv.org.
    4. Dimitris Bertsimas & Ioana Popescu, 2002. "On the Relation Between Option and Stock Prices: A Convex Optimization Approach," Operations Research, INFORMS, vol. 50(2), pages 358-374, April.
    5. Li Chen & Simai He & Shuzhong Zhang, 2011. "Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection," Operations Research, INFORMS, vol. 59(4), pages 847-865, August.
    6. Keiiti Isii, 1960. "The extrema of probability determined by generalized moments (I) bounded random variables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 12(2), pages 119-134, June.
    7. Derek Singh & Shuzhong Zhang, 2020. "Distributionally Robust Profit Opportunities," Papers 2006.11279, arXiv.org.
    8. Lo, Andrew W., 1987. "Semi-parametric upper bounds for option prices and expected payoffs," Journal of Financial Economics, Elsevier, vol. 19(2), pages 373-387, December.
    9. Simai He & Jiawei Zhang & Shuzhong Zhang, 2010. "Bounding Probability of Small Deviation: A Fourth Moment Approach," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 208-232, February.
    10. Luis F. Zuluaga & Javier F. Peña, 2005. "A Conic Programming Approach to Generalized Tchebycheff Inequalities," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 369-388, May.
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    Cited by:

    1. Derek Singh & Shuzhong Zhang, 2020. "Distributionally Robust Newsvendor with Moment Constraints," Papers 2010.16369, arXiv.org.

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