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Maximum Spectral Measures of Risk with Given Risk Factor Marginal Distributions

Author

Listed:
  • Mario Ghossoub

    (Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

  • Jesse Hall

    (Credit Risk Technology, Scotiabank, Toronto, Ontario M5H 1H1, Canada)

  • David Saunders

    (Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

Abstract

We consider the problem of determining an upper bound for the value of a spectral risk measure of a loss that is a general nonlinear function of two factors whose marginal distributions are known but whose joint distribution is unknown. The factors may take values in complete separable metric spaces. We introduce the notion of Maximum Spectral Measure (MSM), as a worst-case spectral risk measure of the loss with respect to the dependence between the factors. The MSM admits a formulation as a solution to an optimization problem that has the same constraint set as the optimal transport problem but with a more general objective function. We present results analogous to the Kantorovich duality, and we investigate the continuity properties of the optimal value function and optimal solution set with respect to perturbation of the marginal distributions. Additionally, we provide an asymptotic result characterizing the limiting distribution of the optimal value function when the factor distributions are simulated from finite sample spaces. The special case of Expected Shortfall and the resulting Maximum Expected Shortfall is also examined.

Suggested Citation

  • Mario Ghossoub & Jesse Hall & David Saunders, 2023. "Maximum Spectral Measures of Risk with Given Risk Factor Marginal Distributions," Mathematics of Operations Research, INFORMS, vol. 48(2), pages 1158-1182, May.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:2:p:1158-1182
    DOI: 10.1287/moor.2022.1299
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    References listed on IDEAS

    as
    1. Alfred Galichon, 2016. "Optimal Transport Methods in Economics," Economics Books, Princeton University Press, edition 1, number 10870, December.
    2. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," SciencePo Working papers hal-03460952, HAL.
    3. Cheung, Ka Chun & Yam, Sheung Chi Phillip & Zhang, Yiying, 2019. "Risk-adjusted Bowley reinsurance under distorted probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 64-72.
    4. Alfred Galichon, 2016. "Optimal transport methods in economics," SciencePo Working papers hal-03256830, HAL.
    5. Mario Ghossoub & David Saunders, 2021. "On the continuity of the feasible set mapping in optimal transport," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(1), pages 113-117, April.
    6. Alexander Shapiro, 2013. "On Kusuoka Representation of Law Invariant Risk Measures," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 142-152, February.
    7. Efe A. Ok, 2007. "Preliminaries of Real Analysis, from Real Analysis with Economic Applications," Introductory Chapters, in: Real Analysis with Economic Applications, Princeton University Press.
    8. Alfred Galichon, 2017. "A Survey of Some Recent Applications of Optimal Transport Methods to Econometrics," Sciences Po Economics Publications (main) hal-03948107, HAL.
    9. Paul Embrechts & Bin Wang & Ruodu Wang, 2015. "Aggregation-robustness and model uncertainty of regulatory risk measures," Finance and Stochastics, Springer, vol. 19(4), pages 763-790, October.
    10. Hardy,Mary R. & Saunders,David, 2022. "Quantitative Enterprise Risk Management," Cambridge Books, Cambridge University Press, number 9781009098465.
    11. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    12. Gretsky, Neil E & Ostroy, Joseph M & Zame, William R, 1992. "The Nonatomic Assignment Model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(1), pages 103-127, January.
    13. Alfred Galichon, 2016. "Optimal transport methods in economics," Post-Print hal-03256830, HAL.
    14. Mathias Beiglbock & Pierre Henry-Labord`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929, arXiv.org, revised Feb 2013.
    15. Volker Krätschmer & Alexander Schied & Henryk Zähle, 2014. "Comparative and qualitative robustness for law-invariant risk measures," Finance and Stochastics, Springer, vol. 18(2), pages 271-295, April.
    16. Alfred Galichon, 2017. "A survey of some recent applications of optimal transport methods to econometrics," Econometrics Journal, Royal Economic Society, vol. 20(2), pages 1-11.
    17. Krätschmer, Volker & Schied, Alexander & Zähle, Henryk, 2012. "Qualitative and infinitesimal robustness of tail-dependent statistical functionals," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 35-47, January.
    18. Alfred Galichon, 2016. "Optimal transport methods in economics," Sciences Po Economics Publications (main) hal-03256830, HAL.
    19. Alfred Galichon, 2017. "A Survey of Some Recent Applications of Optimal Transport Methods to Econometrics," Post-Print hal-03948107, HAL.
    20. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Sciences Po Economics Publications (main) hal-03460952, HAL.
    21. Chan, Fung-Yee & Gerber, Hans U., 1985. "The Reinsurer's Monopoly and the Bowley Solution," ASTIN Bulletin, Cambridge University Press, vol. 15(2), pages 141-148, November.
    22. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Post-Print hal-03460952, HAL.
    23. Paul Glasserman & Linan Yang, 2018. "Bounding Wrong†Way Risk In Cva Calculation," Mathematical Finance, Wiley Blackwell, vol. 28(1), pages 268-305, January.
    24. A. Galichon & P. Henry-Labord`ere & N. Touzi, 2014. "A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options," Papers 1401.3921, arXiv.org.
    25. Memartoluie, Amir & Saunders, David & Wirjanto, Tony, 2017. "Wrong-way risk bounds in counterparty credit risk management," Journal of Risk Management in Financial Institutions, Henry Stewart Publications, vol. 10(2), pages 150-163, April.
    26. Volker Kratschmer & Alexander Schied & Henryk Zahle, 2012. "Comparative and qualitative robustness for law-invariant risk measures," Papers 1204.2458, arXiv.org, revised Jan 2014.
    27. Pichler, Alois, 2013. "The natural Banach space for version independent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 405-415.
    28. James Bergin, 1999. "On the continuity of correspondences on sets of measures with restricted marginals," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 13(2), pages 471-481.
    29. Mathias Beiglböck & Pierre Henry-Labordère & Friedrich Penkner, 2013. "Model-independent bounds for option prices—a mass transport approach," Finance and Stochastics, Springer, vol. 17(3), pages 477-501, July.
    30. Alfred Galichon, 2017. "A survey of some recent applications of optimal transport methods to econometrics," Econometrics Journal, Royal Economic Society, vol. 20(2), pages 1-11, June.
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    Cited by:

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    2. Max Nendel, 2026. "Asymptotic Behaviour of Unexpected Losses and Risk Ratios for Co-Monotonic Alternatives," Papers 2605.18049, arXiv.org.
    3. De Vecchi, Corrado & Nendel, Max & Streicher, Jan, 2025. "Upper Comonotonicity and Risk Aggregation under Dependence Uncertainty," Center for Mathematical Economics Working Papers 739, Center for Mathematical Economics, Bielefeld University.

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