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Existence, uniqueness, and stability of optimal payoffs of eligible assets

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  • Michel Baes
  • Pablo Koch‐Medina
  • Cosimo Munari

Abstract

In a capital adequacy framework, risk measures are used to determine the minimal amount of capital that a financial institution has to raise and invest in a portfolio of prespecified eligible assets in order to pass a given capital adequacy test. From a capital efficiency perspective, it is important to be able to do so at the lowest possible cost and to identify the corresponding portfolios, or, equivalently, their payoffs. We study the existence and uniqueness of such optimal payoffs as well as their behavior under a perturbation or an approximation of the underlying capital position. This behavior is naturally linked to the continuity properties of the set‐valued map that associates to each capital position the corresponding set of optimal eligible payoffs. Upper continuity can be ensured under fairly natural assumptions. Lower continuity is typically less easy to establish. While it is always satisfied in a polyhedral setting, it generally fails otherwise, even when the reference risk measure is convex. However, lower continuity can often be established for eligible payoffs that are close to being optimal. Besides capital adequacy, our results have a variety of natural applications to pricing, hedging, and capital allocation problems.

Suggested Citation

  • Michel Baes & Pablo Koch‐Medina & Cosimo Munari, 2020. "Existence, uniqueness, and stability of optimal payoffs of eligible assets," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 128-166, January.
    • Handle: RePEc:bla:mathfi:v:30:y:2020:i:1:p:128-166
    DOI: 10.1111/mafi.12205
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    Citations

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    Cited by:

    1. Laudagé, Christian & Sass, Jörn & Wenzel, Jörg, 2022. "Combining multi-asset and intrinsic risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 254-269.
    2. Cosimo Munari, 2020. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Papers 2009.04151, arXiv.org.
    3. Matteo Burzoni & Cosimo Munari & Ruodu Wang, 2020. "Adjusted Expected Shortfall," Papers 2007.08829, arXiv.org, revised Aug 2021.
    4. Erio Castagnoli & Giacomo Cattelan & Fabio Maccheroni & Claudio Tebaldi & Ruodu Wang, 2021. "Star-shaped Risk Measures," Papers 2103.15790, arXiv.org, revised Apr 2022.
    5. Maria Arduca & Cosimo Munari, 2020. "Fundamental theorem of asset pricing with acceptable risk in markets with frictions," Papers 2012.08351, arXiv.org, revised Apr 2022.
    6. Maria Arduca & Cosimo Munari, 2023. "Fundamental theorem of asset pricing with acceptable risk in markets with frictions," Finance and Stochastics, Springer, vol. 27(3), pages 831-862, July.
    7. Maria Arduca & Cosimo Munari, 2021. "Risk measures beyond frictionless markets," Papers 2111.08294, arXiv.org.
    8. Burzoni, Matteo & Munari, Cosimo & Wang, Ruodu, 2022. "Adjusted Expected Shortfall," Journal of Banking & Finance, Elsevier, vol. 134(C).
    9. Cosimo Munari, 2021. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Finance and Stochastics, Springer, vol. 25(1), pages 77-99, January.
    10. Sascha Desmettre & Christian Laudagé & Jörn Sass, 2020. "Good-Deal Bounds for Option Prices under Value-at-Risk and Expected Shortfall Constraints," Risks, MDPI, vol. 8(4), pages 1-22, October.

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