IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2010.05398.html
   My bibliography  Save this paper

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2010.05398
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. 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.
    4. 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.
    5. 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.
    6. Derek Singh & Shuzhong Zhang, 2020. "Distributionally Robust Profit Opportunities," Papers 2006.11279, arXiv.org.
    7. 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.
    8. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

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

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Georgia Perakis & Guillaume Roels, 2008. "Regret in the Newsvendor Model with Partial Information," Operations Research, INFORMS, vol. 56(1), pages 188-203, February.
    2. 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.
    3. 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.
    4. 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.
    5. J. A. Primbs, 2010. "SDP Relaxation of Arbitrage Pricing Bounds Based on Option Prices and Moments," Journal of Optimization Theory and Applications, Springer, vol. 144(1), pages 137-155, January.
    6. 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.
    7. 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.
    8. Zhi Chen & Melvyn Sim & Huan Xu, 2019. "Distributionally Robust Optimization with Infinitely Constrained Ambiguity Sets," Operations Research, INFORMS, vol. 67(5), pages 1328-1344, September.
    9. András Prékopa & Anh Ninh & Gabriela Alexe, 2016. "On the relationship between the discrete and continuous bounding moment problems and their numerical solutions," Annals of Operations Research, Springer, vol. 238(1), pages 521-575, March.
    10. Shao, Hui, 2017. "Decomposing aggregate risk into marginal risks under partial information: A top-down method," Statistics & Probability Letters, Elsevier, vol. 124(C), pages 97-100.
    11. Roy H. Kwon & Jonathan Y. Li, 2016. "A stochastic semidefinite programming approach for bounds on option pricing under regime switching," Annals of Operations Research, Springer, vol. 237(1), pages 41-75, February.
    12. Karthik Natarajan & Melvyn Sim & Joline Uichanco, 2018. "Asymmetry and Ambiguity in Newsvendor Models," Management Science, INFORMS, vol. 64(7), pages 3146-3167, July.
    13. Adrian Gepp & Geoff Harris & Bruce Vanstone, 2020. "Financial applications of semidefinite programming: a review and call for interdisciplinary research," Accounting and Finance, Accounting and Finance Association of Australia and New Zealand, vol. 60(4), pages 3527-3555, December.
    14. Xi Chen & Simai He & Bo Jiang & Christopher Thomas Ryan & Teng Zhang, 2021. "The Discrete Moment Problem with Nonconvex Shape Constraints," Operations Research, INFORMS, vol. 69(1), pages 279-296, January.
    15. Henry Lam & Clementine Mottet, 2017. "Tail Analysis Without Parametric Models: A Worst-Case Perspective," Operations Research, INFORMS, vol. 65(6), pages 1696-1711, December.
    16. Qiaoming Han & Donglei Du & Luis F. Zuluaga, 2014. "Technical Note---A Risk- and Ambiguity-Averse Extension of the Max-Min Newsvendor Order Formula," Operations Research, INFORMS, vol. 62(3), pages 535-542, June.
    17. Jun-ya Gotoh & Yoshitsugu Yamamoto & Weifeng Yao, 2011. "Bounding Contingent Claim Prices via Hedging Strategy with Coherent Risk Measures," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 613-632, December.
    18. Roy Kwon & Jonathan Li, 2016. "A stochastic semidefinite programming approach for bounds on option pricing under regime switching," Annals of Operations Research, Springer, vol. 237(1), pages 41-75, February.
    19. Dimitris Bertsimas & Aurélie Thiele, 2006. "A Robust Optimization Approach to Inventory Theory," Operations Research, INFORMS, vol. 54(1), pages 150-168, February.
    20. Wentao Hu, 2019. "calculation worst-case Value-at-Risk prediction using empirical data under model uncertainty," Papers 1908.00982, arXiv.org.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2010.05398. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: http://arxiv.org/ .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.