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On the construction of optimal payoffs

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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.

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Notes

  1. Note that in the mathematical finance literature one often narrows the scope of these market models to so-called complete market models in which the pricing density is unique, and all payoffs are attainable by a self-financing strategy. However, all results on optimal payoffs that are available in the literature for complete market models are also valid in incomplete markets, under the assumption that all participants agree on using a specific pricing density. In this case, however, optimal payoffs are no longer guaranteed to be attainable.

  2. The quantile formulation of the optimal portfolio selection and related optimization problems has a long history. It is intimately connected with the cost minimizing distributional analysis of Dybvig (1988b) and with Hoeffding–Fréchet bounds. For portfolio optimization, it has, for instance, been used in Föllmer and Schied (2004), Carlier and Dana (2006), Burgert and Rüschendorf (2006), Bernard et al. (2014a, b, 2015b), Bernard and Vanduffel (2014), and Kassberger and Liebmann (2012); see also Bernard et al. (2015a) and Bernard et al. (2019) for applications to the ranking of portfolios and explaining their demand. This technique has recently also been put forward and studied in a series of papers, including He and Zhou (2011); Zhang et al. (2011); Jin and Zhou (2008); Xu and Zhou (2013), and Xu (2016); see also the survey paper of Zhou (2011) and the references herein. With respect to risk optimization, it traces its pedigree back to the early eighties (see Rüschendorf 1983).

  3. Xu (2016) develops a Lagrangian relaxation technique that allows derivation of an optimal payoff under rank-dependent utility preferences. It is claimed that this approach also allows to derive the optimal payoff under cumulative prospect theory preferences or under preferences as in Yaari’s dual theory. There is however no proof, and moreover, the stated approach requires inversion of a concave utility, which is at odds with the use of linear utility in Yaari’s dual theory and the use of S-shaped utility functions in cumulative prospect theory.

  4. This is always possible up to a suitable enlargement of the probability space \((\Omega ,\mathcal {A},(\mathcal {A} _{t})_{0\le t\le T},P)\).

  5. We call a functional \(\Psi \) on \(L^{0}(\Omega , P)\) (weakly) increasing if \(\Psi (X+a) \ge \Psi (X)\) for all \(a \ge 0\) and \(X\in L^{0}(\Omega ,P)\).

  6. Recall that two random variables X and Y are called anti-monotonic if \(X(\omega _1) < Y(\omega _2)\) implies \(Y(\omega _2) \le X(\omega _1)\). Anti-monotonicity between two real functions is defined in a similar way.

  7. A distortion of the tail probability function of the payment is also used in other theories like Lopes’ SP/A model (Lopes 1987; Lopes and Oden 1999) and Kahnemann and Tversky’s influential prospect theory (Kahneman and Tversky 1979; Tversky and Kahneman 1992).

  8. Note that we do not assume the concavity of w, as is predominantly the case in the CPT literature.

  9. Exponential Lévy market model with Esscher pricing has been introduced in the mathematical finance literature by Gerber and Shiu (1994) and Madan and Milne (1991) and can be considered as one of the classic models; see Eberlein and Keller (1995), Chan (1999), Kallsen and Shiryaev (2002), Esche and Schweizer (2005), Hubalek and Sgarra (2006), Vanduffel et al. (2009), and Benth and Sgarra (2012) for studies of its properties and further motivation.

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Acknowledgements

We thank two reviewers for valuable comments and suggestions that helped improving the paper.

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  1. University of Freiburg, Eckerstraße 1, 79104, Freiburg, Germany

    L. Rüschendorf

  2. Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium

    Steven Vanduffel

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Rüschendorf, L., Vanduffel, S. On the construction of optimal payoffs. Decisions Econ Finan 43, 129–153 (2020). https://doi.org/10.1007/s10203-019-00272-9

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