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On the construction of optimal payoffs

Author

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  • L. Rüschendorf

    (University of Freiburg)

  • Steven Vanduffel

    (Vrije Universiteit Brussel)

Abstract

In the framework of continuous-time market models with specified pricing density, optimal payoffs under increasing law-invariant preferences are known to be anti-monotonic with the pricing density. Consequently, optimal portfolio selection problems can be reformulated as optimization problems on real functions under monotonicity conditions. We solve two basic types of these optimization problems, which makes it possible to obtain in a fairly unified way the optimal payoff for several portfolio selection problems of interest. In particular, we completely solve the optimal portfolio selection problem for an investor with preferences as in cumulative prospect theory or as in Yaari’s dual theory. Extending previous work, we also characterize optimal payoffs when the payoff is required to have a fixed copula with some benchmark (state-dependent constraint). Specifically, we show that if one can determine the optimal payoff under a concave law-invariant objective, then one can also determine the optimal payoff when adding the state-dependent constraint. In the final part of the paper, we consider an extension to (incomplete) market models in which the pricing density is not completely specified. When a sufficient number of payoffs have a known market price, we show that optimal payoffs are anti-monotonic to some pricing density that we explicitly derive from these market prices. As examples, we deal with some exponential Lévy market models and some market models involving Itô processes.

Suggested Citation

  • L. Rüschendorf & Steven Vanduffel, 2020. "On the construction of optimal payoffs," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(1), pages 129-153, June.
    • Handle: RePEc:spr:decfin:v:43:y:2020:i:1:d:10.1007_s10203-019-00272-9
    DOI: 10.1007/s10203-019-00272-9
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    Cited by:

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    2. Fangyuan Zhang, 2023. "Non-concave portfolio optimization with average value-at-risk," Mathematics and Financial Economics, Springer, volume 17, number 3, February.
    3. Chen, An & Vanduffel, Steven & Wilke, Morten, 2025. "Optimal payoffs under smooth ambiguity," European Journal of Operational Research, Elsevier, vol. 320(3), pages 754-764.
    4. Xue Dong He & Zhaoli Jiang, 2020. "Optimal Payoff under the Generalized Dual Theory of Choice," Papers 2012.00345, arXiv.org.
    5. Carole Bernard & Gero Junike & Thibaut Lux & Steven Vanduffel, 2024. "Cost-efficient payoffs under model ambiguity," Finance and Stochastics, Springer, vol. 28(4), pages 965-997, October.
    6. Cai, Jun & Liu, Fangda & Yin, Mingren, 2024. "Worst-case risk measures of stop-loss and limited loss random variables under distribution uncertainty with applications to robust reinsurance," European Journal of Operational Research, Elsevier, vol. 318(1), pages 310-326.
    7. Bernard, C. & De Gennaro Aquino, L. & Vanduffel, S., 2023. "Optimal multivariate financial decision making," European Journal of Operational Research, Elsevier, vol. 307(1), pages 468-483.
    8. Yichun Chi & Zuo Quan Xu & Sheng Chao Zhuang, 2022. "Distributionally Robust Goal-Reaching Optimization in the Presence of Background Risk," North American Actuarial Journal, Taylor & Francis Journals, vol. 26(3), pages 351-382, August.
    9. Felix-Benedikt Liebrich & Cosimo Munari, 2021. "Law-invariant functionals that collapse to the mean: Beyond convexity," Papers 2106.01281, arXiv.org, revised Jul 2021.
    10. Bi, Xiuchun & Cui, Zhenyu & Fan, Jiacheng & Yuan, Lvning & Zhang, Shuguang, 2023. "Optimal investment problem under behavioral setting: A Lagrange duality perspective," Journal of Economic Dynamics and Control, Elsevier, vol. 156(C).

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