Cost-efficient payoffs under model ambiguity

Bernard, Carole; Junike, Gero; Lux, Thibaut; Vanduffel, Steven Article — Published Version Cost-efficient payoffs under model ambiguity Finance and Stochastics Provided in Cooperation with: Springer Nature Suggested Citation: Bernard, Carole; Junike, Gero; Lux, Thibaut; Vanduffel, Steven (2024) : Costefficient payoffs under model ambiguity, Finance and Stochastics, ISSN 1432-1122, Springer, Berlin, Heidelberg, Vol. 28, Iss. 4, pp. 965-997, https://doi.org/10.1007/s00780-024-00547-z This Version is available at: https://hdl.handle.net/10419/315066 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Finance and Stochastics (2024) 28:965–997 https://doi.org/10.1007/s00780-024-00547-z Cost-efficient payoffs under model ambiguity Carole Bernard1,2 ·Gero Junike3·Thibaut Lux4·Steven Vanduffel2 Received: 6 July 2022 / Accepted: 26 December 2023 / Published online: 12 September 2024 © The Author(s) 2024 Abstract Dybvig (1988a,1988b) solves in a complete market setting the problem of finding a payoff that is cheapest possible in reaching a given target distribution (“cost-efficient payoff”). In the presence of ambiguity, the distribution of a payoff is, however, no longer known with certainty. We study the problem of finding the cheapest possible payoff whose worst-case distribution stochastically dominates a given target distribution (“robust cost-efficient payoff”) and determine solutions under certain conditions. We study the link between “robust cost-efficiency” and the maxmin expected utility setting of Gilboa and Schmeidler (1989), as well as more generally in a possibly nonexpected robust utility setting. Specifically, we show that solutions to maxmin robust expected utility are necessarily robust cost-efficient. We illustrate our study with examples involving uncertainty both on the drift and on the volatility of the risky asset. Keywords Cost-efficient payoffs ·Model ambiguity ·Maxmin utility ·Robust preferences ·Drift and volatility uncertainty Mathematics Subject Classification 91B30 ·62E17 JEL Classification C02 ·C63 ·D80 G. Junike [email protected] C. Bernard [email protected] T. Lux [email protected] S. Vanduffel [email protected] 1Department of Accounting, Law and Finance, Grenoble Ecole de Management (GEM), 38000, Grenoble, France 2Department of Economics and Political Sciences, Vrije Universiteit Brussel (VUB), 1050, Brussels, Belgium 3Institute of Mathematics, Carl von Ossietzky Universität Oldenburg, 26111, Oldenburg, Germany 4Baloise Insurance Group, 4002, Basel, Switzerland 966 C. Bernard et al. 1Introduction In a (complete) market without ambiguity, Dybvig [23,22] characterises optimal payoffs for agents having law-invariant increasing preferences (e.g. expected utility maximisers). His result is based on the observation that any optimal payoff Xmust be cost-efficient in the sense that there cannot exist another payoff with the same probability distribution that is strictly cheaper than X. He then derives, for a given target distribution of terminal wealth, the payoff that achieves this target distribution at the lowest possible cost (cost-efficient payoff). Optimal payoffs are thus driven by distributional constraints rather than appearing as a solution to some optimal expected utility problem. In this regard, Brennan and Solanki [11] note that “from a practical point of view, it may well prove easier for the investor to choose directly his optimal payoff function than it would be for him to communicate his utility function to a portfolio manager.” Sharpe et al. [58] and Goldstein et al. [28] introduce a tool called the distribution builder, which makes it possible for investors to analyse distributions of terminal wealth and to choose their preferred one among alternatives with equal cost; see also Sharpe [57, Sect. 7.9] and Monin [44]. Moreover, the authors of [28] argue that such a tool makes it possible to better elicit investor’s preferences. Our main objective is to extend Dybvig’s results when there is uncertainty on the real-world probability measure. Uncertainty has become a prime issue in many academic domains, from economics to environmental science and psychology. Model ambiguity refers to random phenomena or outcomes whose probabilities are themselves unknown. For instance, the random outcome of a coin toss is subject to model uncertainty when the probability of the coin showing either a head or a tail is not or is at most partially known. This notion of model ambiguity goes back to Knight [40, Chap. VII] and is therefore commonly referred to as Knightian uncertainty. In the presence of ambiguity, the probability distribution of a payoff is not anymore determined. Thus looking for a minimum cost payoff with a given probability distribution is no longer possible. However, investors may still determine a desired distribution function that they would like to achieve “at least”. In this paper, we look for a minimum cost payoff that dominates a target distribution for a chosen integral stochastic order under any plausible real-world probability distribution. Our contributions are three-fold. First, we solve this problem explicitly for a general stochastic ordering under certain assumptions. Solutions to this problem are called “robust costefficient.” Second, we draw connections between such a minimum cost payoff and the problem of finding an optimal payoff under ambiguity for general sets of robust preferences. Third, we present a number of examples, including one on the robust payoff choice in the presence of volatility uncertainty. Our results generalise the results on cost-efficiency given in Dybvig [23,22], Cox and Leland [20] and Bernard et al. [5,6]. When there is no ambiguity on the realworld probability, the robust cost efficient payoffs coincide with the cost-efficient payoffs studied in the literature. To derive our results, we build on the so-called quantile approach to solve the optimisation of a law-invariant increasing functional; see e.g. Schied [53], Carlier and Dana [12,13,14], Jin and Zhou [38], He and Zhou [36,37], Bernard et al. [5], Xu [63] and Rüschendorf and Vanduffel [50]. Cost-efficient payoffs under model ambiguity 967 Specifically, we consider a static setting and are able to address uncertainty about volatility. We show that under certain conditions, the solution to a general robust payoff maximisation problem is equal to the solution of a classical payoff maximisation problem under a least favourable measure P∗with respect to some stochastic ordering. This was already shown by Schied [54] for the case of robust expected utility theory and using first order stochastic dominance. Here, however, we show that these results extend to the case of more general preferences, focusing on the case of first order and second order stochastic dominance. To reach this conclusion, we make certain assumptions. Most notably, we assume the existence of a least favourable measure. This assumption was also made in [54] for the case of first order stochastic dominance, but in this paper, we deal also with the case of second order stochastic dominance (weaker assumption). We also assume that the pricing kernel is continuously distributed under this least favourable measure, as in e.g. [38,36,37,5,63], among many others. These assumptions are important to solve the robust cost-efficient problem. Furthermore, we show that there is a natural correspondence between optimal payoffs in the maxmin utility setting of Gilboa and Schmeidler [27] with a concave increasing utility, and robust cost-efficient payoffs: for any robust cost-efficient payoff X∗, there is a utility function such that X∗solves the maxmin expected utility maximisation problem. We further show that the solution to a robust maximisation problem with respect to a general family of preferences is cost-efficient. This result implies that instead of solving a robust maximisation problem with respect to a general family of preferences, one could solve an expected utility maximisation problem under the single measure P∗for a suitable concave utility function. The literature on optimal payoff choice under ambiguity includes the seminal setting of [27], that is, the so-called “maxmin expected utility,” which was later referred to as robust utility functional by Schied et al. [55]. Specifically, these authors characterise preferences that have a robust utility numerical representation minP∈PEP[u(X)]for some set of probabilities P. Gundel [34] provides a dual characterisation of the solution for robust utility maximisation in both a complete and an incomplete market model. Klibanoff et al. [39] distinguish between subjective beliefs, i.e., the definition of the set of possible or plausible subjective probability measures, and ambiguity attitude, i.e., a characterisation of the agent’s behaviour toward ambiguity. Based on [39], Gollier [30] analyses the effect of ambiguity aversion on the demand for the uncertain asset in a payoff choice problem. Schied [54] solves the maximisation problem of maxmin expected utility of [27] in a general complete market model with dynamic trading, provided there is a least favourable measure with respect to first order stochastic dominance. Specifically, he finds that the optimum for the maxmin utility setting of [27] can be derived in the standard expected utility setting under the least favourable measure. Schied [54] works with a complete market model and mainly in a static setting; dynamics only come into play when the martingale method is applied to the static solutions. A survey on robust preferences and robust payoff choice can be found in Schied et al. [55]. The paper is organised as follows. The robust cost-efficiency problem is described in Sect. 2. In Sect. 3, we solve the robust cost-efficiency problem, and we include two examples in a lognormal market with uncertainty on the drift and the volatility along with another example in a Lévy market in which the physical measure is obtained 968 C. Bernard et al. by the Esscher transform. In Sect. 4, we develop the correspondence between robust cost-efficient payoffs and strategies that solve a robust optimal payoff problem, including the maxmin utility setting of [27] as a special case. In Sect. 5, we show that the solution to a general robust optimal payoff problem can also be obtained as the solution to the maximisation of the maxmin utility setting of [27] for a well-chosen concave utility function. Section 6concludes. 2 Problem statement We assume a static market setting in which trading only takes place today and at the end of the planning horizon T>0. There is a bank account earning the continuously compounded risk-free interest rate r∈R.LetR+=[0,∞).LetST:→R+represent the random value of a risky asset at maturity. We denote by S0>0 its current value and by Fthe σ-algebra generated by ST.LetPbe a set of equivalent realworld probability measures on (, F).ThesetPcan be thought of as a collection of probability measures that the investor deems plausible for the market. We define the set of payoffs X={g(ST):g:R+→R+measurable, EQ[g(ST)]<∞}, where Qis a fixed pricing measure equivalent to all P∈P. A payoff Xis also called acontingent claim. Furthermore, for any X∈X, its price is given by e−rT EQ[X]. By FP X, we denote the cumulative distribution function of X∈Xunder P∈P. Remark 2.1 As in Dybvig [22], we could use the discount factor (1+r)−Tfor some r∈(−1,∞), based on compounded interest rates. However, working with exponential stock price models, it is more convenient and consistent with more recent literature to use the discount factor e−rT for r∈R. Remark2.2 Under the assumption that all call options (ST−K)+,K≥0, are traded, the market can be completed. This is shown by Ross [48] for discrete and by Nachman [46, Corollary 6] for general static markets. Market completion by spanning call options as in [46] has been further developed by Madan and Milne [41], Bakshi and Madan [2], Carr and Wu [16] and many others. Rogge [47, Theorem 2] shows that n-period models are complete if a call-completeness condition is satisfied. Carr and Madan [15, Eq. (1)] provide an explicit replication strategy of a payoff g(ST) if gis twice continuously differentiable. See also Breeden and Litzenberger [10]for earlier results. Consider an investor with a finite budget and planning horizon T>0 who wishes to invest in the market while having ambiguous views on the real-world probability measure. How can she find her optimal investment strategy? As in Schied [54], she could maximise some robust expected utility à la Gilboa and Schmeidler [27]. The basic idea is then to look for a payoff that maximises the worst case expected utility, reflecting the idea that the investor aims to protect against the worst while hoping for the best. However, it seems easier for investors to specify the desired probability distribution of the terminal wealth rather than a utility function (see Brennan and Solanki [11], Sharpe et al. [58] and Goldstein et al. [28]). Cost-efficient payoffs under model ambiguity 969 As in [23,58], Vrecko and Langer [61], Bernard et al. [5], we thus assume in this paper that the investor specifies a desired (cumulative) distribution function F0of future terminal wealth. Once the investor understands which distribution function F0 is acceptable to her, the natural question arises as to how to find under ambiguity the cheapest payoff with a distribution function at maturity that is “at least as good” as F0. This is the robust cost-efficiency problem formalised hereafter. In this regard, we need to recall the concept of integral stochastic ordering; see e.g. Denuit et al. [21, Sect. 3]. In this paper, we denote by Fa set of measurable functions from R+to R. Definition2.3 Let Gand Fbe two distribution functions with support on R+. Then G dominates Fin integral stochastic ordering with respect to F, written as FFG,if R+ f(x)dF ≤R+ f(x)dG for all f∈Fsuch that the expectations are finite. Let FFSD denote the set of all nondecreasing functions from R+to R. The corresponding integral stochastic ordering is called first order stochastic dominance (FSD). Furthermore, let FSSD denote the set of all nondecreasing and concave functions from R+to R. The corresponding integral stochastic ordering is called second order stochastic dominance (SSD). It is well known that FSD reflects the common agreement of all investors with law-invariant increasing preferences, see Bernard et al. [6, Theorem 1], whereas SSD reflects the common agreement of those who have law-invariant increasing and diversification-loving preferences (risk-averse investors); see Bernard and Sturm [7, Corollary 2.6]. Problem 2.4 The F-robust cost-efficiency problem for a distribution function F0is defined as inf X∈BF F0 e−rT EQ[X],(2.1) where BF F0denotes the class of admissible payoffs defined as BF F0={X∈X:F0FFP X,∀P∈P}. A solution to (2.1)iscalledanF-robust cost-efficient payoff. As discussed above, the target distribution function of the investor is F0. That is, we are interested in all payoffs that have a distribution function at maturity that is at least as good as F0under all plausible scenarios P∈P. For example, when F=FFSD, we care about payoffs having distribution functions FP X,P∈P, that dominate F0in FSD. In order not to “throw away investors’ money”, see Dybvig [23], we then aim to determine the cheapest among the payoffs in the admissible set BF F0. In Theorem 3.1, we provide solutions to the F-robust cost-efficiency problem (2.1) under regularity conditions on the set Fand F0. 970 C. Bernard et al. Dybvig [23,22] introduced the standard cost-efficiency problem without ambiguity on the set of physical measures, that is, when P={P}. Specifically, for some fixed P∈P, the problem he considered reads as inf X∈AP F0 e−rT EQ[X],(2.2) where AP F0={X∈X:F0=FP X}. We refer to this problem as the standard cost-efficiency problem. Furthermore, we say that a payoff Xthat is distributed with FP Xis P-cost-efficient if Xsolves the standard cost-efficiency problem (2.2) under Pwith respect to F0=FP X. By Bernard et al. [5], a payoff Xis P-cost-efficient if and only if Xis nonincreasing in the state price ξP=e−rT dQ dPP-a.s.; see also Schied [53, Proposition 2.5]. The standard cost-efficiency problem (2.2) has been solved in [23,22]; see also Lemma B.1 in the Appendix. In Corollary 3.5, we show that if Pis a singleton, the solution to the F-robust cost-efficiency problem is unique and coincides with the solution to the standard cost-efficiency problem. Remark 2.5 For a fixed X∈X, in general, FP Xcannot be equal to F0for all P∈P. Therefore, we replace the condition F0=FP Xin the standard cost-efficiency problem with F0FFP Xin the robust setting. The next example anticipates Sects. 3.2.1 and 3.2.2 and is designed to help distinguish between the standard and the robust cost-efficiency problems. As in Embrechts and Hofert [24], we define for a nondecreasing function T:R→Rthe generalised inverse T−1by T−1(y) =inf{x∈R:T(x)≥y},y∈R. Example 2.6 Assume the real-world distribution of STis lognormal. There are three investors: one investor assumes that the drift of STunder the physical measure, denoted by Pμ1, is equal to μ1>r. Another investor assumes that the drift is given by μ2>μ 1under the physical measure, denoted by Pμ2. A third investor has ambiguity and assumes that the drift lies in the interval [μ1,μ 2], and thus considers the set P={Pμ:μ∈[μ1,μ 2]} as the set of all plausible probability measures on (, F). The cheapest payoffs to obtain a fixed target distribution function F0are well known for investors one and two and are given by X∗ 1:= F−1 0FPμ1 ST(ST),X ∗ 2:= F−1 0FPμ2 ST(ST), respectively; see Bernard et al. [5, Proposition 3]. Within the set P,Pμ1corresponds to a pessimistic view of the stock price behaviour, and we shall see in Sect. 3.2.1 that X∗ 1is a solution to the cost-efficiency problem of the third investor if F=FFSD. In the case in which F=FSSD,X∗ 1also solves the cost-efficiency problem of the Cost-efficient payoffs under model ambiguity 971 third investor if additionally F−1 0◦FPμ1 STis concave; see Sect. 3.2.2. This example illustrates that the solution to the standard cost-efficiency problem for arbitrary P∈P and the solution to the robust cost-efficiency problem do not coincide in general. Note that the case μ1<μ 2<ris economically less relevant. However, it can be shown that in this case, X∗ 1:= F−1 0(1−FPμ1 ST(ST)),X∗ 2:= F−1 0(1−FPμ2 ST(ST)) and that the latter payoff solves the cost-efficiency problem of the third investor if F=FFSD;see also Remark 3.6. Remark 2.7 As in Rüschendorf and Wolf [51], we could also consider uncertainty on the target distribution function F0. Specifically, it is assumed in [51] that the investor specifies finitely many acceptable distribution functions F1 0,...,FN 0.Asall Ndistribution functions are acceptable to the investor, she could solve the robust cost-efficiency problem Ntimes and buy the cheapest among the Nsolutions. 2.1 Assumptions In order to solve the robust cost efficiency problem (2.1), we need some regularity conditions on the set Fand on the target distribution F0. In this regard, we define some concepts. Recall first the concept of a least favourable measure introduced by Schied [54] for the case F=FFSD.ForP∈P, we define the corresponding likelihood ratio by P=dP dQ. We remark that the random variable e−rT Pis also called state price because the price of a payoff X∈Xcan be expressed by e−rT EQ[X]=EPe−rT PX,P∈P. Definition 2.8 A measure P∗∈Pwith corresponding likelihood ratio ∗:= dP∗ dQis called a least favourable measure with respect to Fif FP∗ ∗FFP ∗for all P∈P. Definition 2.8 generalises [54, Definition 2.1] which assumed the existence of a least favourable measure with respect to FFSD to determine payoffs that solve the robust expected utility problem of Gilboa–Schmeidler [27]. We also need the following definition. Definition 2.9 The set Fis said to be composition-consistent if for f, g ∈F,also f◦g∈F. Note that the sets FFSD and FSSD are composition-consistent. This follows from the fact that the composition of nondecreasing (resp. nondecreasing and concave) functions is again nondecreasing (resp. nondecreasing and concave). The following result provides conditions that guarantee the existence of a least favourable measure and turns out to be very useful for applications. Proposition 2.10 Assume that Fis composition-consistent.If FP STFFP STfor some P∈Pand all P∈P,and if P=f(S T)for some f∈F,then Pis a least 972 C. Bernard et al. favourable measure with respect to F.If,additionally,STis continuously distributed under Pand fis strictly increasing,then Pis continuously distributed under P. Proof Let P,P∈P.LetXbe a payoff and f∈F. Recall that FP XFFP Xif and only if EP[g(X)]≤EP[g(X)]for all g∈Fsuch that the expectations are finite. Because Fis composition-consistent, it follows that FP XFFP X=⇒ FP f(X) FFP f(X).(2.3) The result FP PFFP Pthen follows by (2.3). Let fbe strictly increasing. By [24], the generalised inverse f−1of fis continuous on the range of f. Thus we have P[P≤x]=P[ST≤f−1(x)]=FP STf−1(x),x∈R, and so Pis continuously distributed under Psince FP STis continuous.  We also need a definition that is, to the best of our knowledge, new to the literature. Definition 2.11 The set Fis called cost-consistent if for all X, Y ∈Xand all P∈Psuch that X, Y are P-cost-efficient, FP XFFP Yimplies EQ[X]≤EQ[Y] and additionally, FP X=FP Yimplies EQ[X]<E Q[Y]. Proposition 2.12 The set FSSD is cost-consistent.Moreover,if FSSD ⊆F,then Fis cost-consistent. Proof The cost-consistency of FSSD can be proved along the lines of Bernard et al. [8, proof of Lemma 2]. Furthermore, FP XFFP Yimplies FP XFSSD FP Y, which finishes the proof.  As the set FSSD is contained in FFSD, Proposition 2.12 implies that FFSD and FSSD are cost-consistent. We provide examples of sets Fof functions that are not costor composition-consistent. Example2.13 Third order stochastic dominance is the integral stochastic ordering that arises from the set FTSD consisting of all functions R→f:R+such that f>0, f ≤0 and f ≥0. The set FTSD is composition-consistent, but in general not cost-consistent; see Appendix A. Example 2.14 Müller et al. [45] introduced the (1+γ)-stochastic dominance order for γ∈(0,1), which lies between FSD and SSD ordering. The set induced by (1+γ)-stochastic dominance order is in general not composition-consistent, but is cost-consistent in light of Proposition 2.12. Example 2.15 Rothschild and Stiglitz [49] introduced the concave stochastic order which is defined via the set of all concave (but not necessarily nondecreasing) functions. The concave stochastic order coincides with SSD if we compare two payoffs with the same mean; see Föllmer and Schied [25, Remark 2.63]. The set of all concave functions is cost-consistent, but not composition-consistent. Cost-efficient payoffs under model ambiguity 979 3.3 Robust cost-efficient payoffs in general markets using Esscher transform Inspired by Corcuera et al. [19], fix S0>0 and let s>0 and Zbe a payoff with mean zero and variance one. Under Q, assume that Zhas density fQ Z(x) > 0, x∈R, and model the future stock price at date Tby ST=S0e(r+ω)T +s√TZ, where ω∈Ris a mean-correcting term, i.e., ωis chosen such that e−rT EQ[ST]=S0. The density of X=logSTunder Qis fQ X(x) =1 s√TfQ Zx−logS0−(r +ω)T s√T,x>0. The corresponding density of STunder Qis denoted by fQ ST, and it holds that fQ ST(x) =1 xfQ X(logx), x > 0. Let h∗>0 and H⊆[h∗,∞)be a set containing h∗such that EQ[(ST)h]exists for all h∈H. Define a family of probability measures P=(Ph)h∈Has follows: Phis a measure such that Xhas density fPh Xunder Ph, where fPh Xis obtained from fQ Xby applying the Esscher transform. The use of the Esscher transform can be supported by a utility maximising argument; see Gerber and Shiu [26]. More precisely, we define Phsuch that fPh X(x) =ehxfQ X(x) Rehy fQ X(y)dy =ehx fQ X(x) EQ[(ST)h],x>0. It follows that fPh ST(x) =xh x fQ X(logx) EQ[(ST)h],x>0. The density fPh∗ STcrosses fPh STonly once from above for h∗<h; hence by Denuit et al. [21, Property 3.3.32], it follows that FPh∗ STFFSD FPh ST=⇒ FPh∗ STFSSD FPh ST,h∈H. For the likelihood ratio, it holds that Ph∗=fPh∗ ST(ST) fQ ST(ST)=(ST)h∗ EQ[(ST)h∗], 980 C. Bernard et al. which is strictly increasing in STas h∗>0, and concave if h∗∈(0,1]. We can apply Proposition 2.10 to show that Assumptions 2.18 and 2.19 are satisfied for the sets FFSD and FSSD with least favourable measure P∗=Ph∗and corresponding likelihood ratio ∗=Ph∗. We can use Theorem 3.1 to compute the cost-efficient payoff of a distribution function F0. 4 Robust payoff selection Gilboa and Schmeidler [27] provide axioms that justify a maxmin expected utility framework to make robust decisions when there is ambiguity on the probability measure P, i.e., when Pcontains more than one element. In this framework, Schied [54] shows that when a least favourable measure P∗∈Pwith respect to the FSD ordering (i.e., the integral stochastic ordering induced by the set FFSD defined in Sect. 2)exists, an optimal payoff can be derived. In this section, we extend the work of [54]in two different ways. First, we account for preferences beyond expected utility. Specifically, we derive optimal payoffs for robust preferences that are in accord with expected utility theory, rank-dependent utility theory and Yaari’s dual theory. Second, assuming the existence of a least favourable measure P∗with respect to a general integral stochastic ordering induced by some set F, not necessarily identical to FFSD, we derive the optimal payoff. Specifically, we derive optimal payoffs when a least favourable measure P∗∈Pwith respect to the SSD ordering exists (see Proposition 2.10 for a sufficient condition) and the target distribution F0is sufficiently light-tailed (see Remark 3.4). 4.1 Family-consistent preferences Apreference is understood as a binary relation on the set Xof payoffs. The interpretation is that YXif Yis preferred to X. A functional W:X→Ris called arepresentation of if YXif and only if W(Y) ≥W(X). The functional W is also called a utility functional; see Schied et al. [55], He et al. [35] and Assa and Zimper [1]. In general, Wmay depend on the different measures P∈Pin a complicated way. In what follows, we denote a utility functional that depends solely on some P∈Pby WP. Definition 4.1 Let (WP)P∈Pbe a family of utility functionals. The utility functional WP,P∈P, is called P-law-invariant if FP X=FP Yimplies that WP(X) =WP(Y ). The family (WP)P∈Pis called law-invariant if each individual utility functional WP is P-law-invariant. Example 4.2 A standard example of a P-law-invariant utility functional is given by WP(X) =EP[u(X)]for some increasing utility function u. In this case, the functional W(X) =infP∈PWP(X) amounts to the worst-case expected utility, commonly called robust expected utility, which was introduced in [27]. It is also referred to as a robust utility functional in [55]. Cost-efficient payoffs under model ambiguity 981 To the best of our knowledge, the next definition is new to the literature. It will be helpful in solving robust payoff choice problems. Definition 4.3 Let (WP)P∈Pbe a family of utility functionals. Let Y⊆X.Thefamily of utility functionals (WP)P∈Pis called F-family-consistent on Ywith respect to P∈Pif for all Y∈Y, the inequality FP YFFP Y,P∈P implies that WP(Y) ≤WP(Y), P∈P. F-family-consistency of (WP)P∈Pon Ywith respect to some P∈Phas the following interpretation: If a measure Pyields the most pessimistic view of any payoff Ywith respect to the stochastic ordering induced by some set F, then the preference under that measure is the lowest as well. Next, we discuss some examples. Let Y⊆Xbe a set of payoffs and let Dbe the set of cumulative distribution functions induced by Y, i.e., D={FP Y:Y∈Y,P∈P}. Let us consider an agent taking into account a family of law-invariant utility functionals (WP)P∈P, i.e., WP(Y) =w(FP Y), P∈P,(4.1) for some well-defined w:D→R.Ifwrespects the integral stochastic ordering, i.e., FFG=⇒ w(F) ≤w(G), F, G ∈D,(4.2) then (WP)P∈Pis F-family-consistent on Ywith respect to all P∈P. We provide some specific examples in the contexts of expected utility theory, Yaari’s dual theory of choice and rank-dependent expected utility theory. Example 4.4 Let u:R+→R.Letφ:[0,1]→[0,1]with φ(0)=0 and φ(1)=1. For a given distribution function F, define wEUT(F ) =R+ u(x)dF (x), wYaari(F ) =R+ φ1−F(x) dx, wRDEU(F ) =R+ u(x)d1−φ1−F(x) , where we tacitly assume that all integrals exist. It is straightforward to show that when uand φare nondecreasing, the family of utility functionals induced by wEUT, 982 C. Bernard et al. wYaari or wRDEU as in (4.1)isFFSD-family-consistent on Ywith respect to all P∈P, where Yis restricted to contain random variables such that all relevant integrals exist. Furthermore, if uis strictly increasing and concave and φis strictly increasing, continuously differentiable and convex, we obtain that such a family is FSSD-familyconsistent on Ywith respect to all P∈P; see Yaari [64], Wang and Young [62], He et al. [35] and Ryan [52]. Remark 4.5 One could allow the function win (4.1) to depend on P, i.e., define WP(Y) =wP(F P Y)for some wP:D→R,P∈P. The family of utility functionals (WP)P∈Pis then F-family-consistent on Ywith respect to some P∈Pif both wP(F ) ≤wP(F ), F ∈D,P∈P and (4.2) hold. To see this, let Y∈Ywith FP Y≤FP Yfor all P∈P. It follows that WP(Y) =wP(F P Y)≤wP(F P Y)≤wP(F P Y)=WP(Y). 4.2 Optimal payoffs under robust preferences Inspired by Gilboa and Schmeidler [27] and Schied [54], we consider the following problem. Problem 4.6 Let x0>0 be the initial wealth. Let (WP)P∈Pbe a family of utility functionals. We consider the robust maximisation problem max X∈Yx0 (WP)P∈P inf P∈PWP(X), (4.3) where Yx0 (WP)P∈P=P∈PYx0 WPand Yx0 WP:={X∈X:WP[X]∈R,e −rT EQ[X]≤x0},P∈P. It turns out that under certain conditions, a solution to the robust optimisation problem (4.3) can be found as a solution to a maximisation problem under a single measure P∈P. Problem 4.7 Let x0>0 be the initial wealth. Let WPfor some P∈Pbe a utility functional. We consider the maximisation problem max X∈Yx0 WP WP(X). (4.4) Under the assumption of the existence of a least favourable measure with respect to the FSD ordering, [54] showed that in order to solve the robust maximisation problem (4.3) for utility functionals (WP)P∈P,WP(x) =EP[u(X)], it actually suffices to solve the single measure maximisation problem (4.4). The following result generalises this beyond the expected utility setting to a general law-invariant family of utility functionals (WP)P∈P. The result is illustrated in Sect. 4.3, where we consider a robust rank-dependent expected utility maximisation problem for an investor with ambiguity on the trend and/or volatility of the risky asset. Cost-efficient payoffs under model ambiguity 983 Theorem 4.8 Let F=FFSD.Under Assumptions 2.18 and 2.19,suppose (WP)P∈P is law-invariant and FFSD-family-consistent on Yx0 (WP)P∈Pwith respect to P∗∈P. Assume that the maximisation problem (4.4)under P∗has a solution ˜ X∈Yx0 (WP)P∈P. Then it holds that max X∈Yx0 (WP)P∈P inf P∈PWP(X) =max X∈Yx0 WP∗ WP∗(X). Proof Let h∈FFSD be such that h(∗)∈Yx0 (WP)P∈P. Then by Assumption 2.18 and (2.3) and since (WP)P∈Pis FFSD-family-consistent on Yx0 (WP)P∈Pwith respect to P∗, we have WP∗h(∗)≤inf P∈PWPh(∗).(4.5) Let X∗=(F P∗ ˜ X)−1FP∗ ∗(∗). Then X∗solves the standard cost-efficiency problem for FP∗ ˜ X;soEQ[X∗]≤EQ[˜ X] and FP∗ ˜ X=FP∗ X∗, and thus the law-invariance of (WP)P∈Pyields X∗∈Yx0 (WP)P∈P. Moreover, X∗is a nondecreasing function of ∗. It follows by (4.5) that max X∈Yx0 WP∗ WP∗(X) =WP∗(˜ X) =WP∗(X∗) ≤inf P∈PWP(X∗) ≤max X∈Yx0 (WP)P∈P inf P∈PWP(X) ≤max X∈Yx0 (WP)P∈P WP∗(X) ≤max X∈Yx0 WP∗ WP∗(X), where the last inequality follows because Yx0 (WP)P∈P⊆Yx0 WP∗. From Theorem 4.8, it follows immediately that solving robust preference maximisation problems may reduce to solving an optimisation problem under a single probability measure. The following example illustrates this consequence. Example4.9 Assume F=FFSD and that Assumptions 2.18 and 2.19 are satisfied. Let WP(F ) =w(FP Y)as in (4.1), where w∈{wEUT,wYaari,wRDEU}as in Example 4.4. Assumingasolutionto(4.4) under P∗∈Pexists, it follows that max X∈Yx0 (WP)P∈P inf P∈PWP(X) =max X∈Yx0 WP∗ WP∗(X). 984 C. Bernard et al. The main assumption in Theorem 4.8 that is needed to solve the robust maximisation problem (4.3) in the case of a family of law-invariant utility functionals (WP)P∈Pis the existence of a least favourable measure P∗with respect to FFSD.In the following result, we show that it is possible to weaken this assumption in that we only require existence of a least favourable measure P∗with respect to some F⊆FFSD,e.g.F=FSSD. The theorem is illustrated in Sect. 4.3, where we consider a robust rank-dependent expected utility maximisation problem for an investor who faces ambiguity on expected return and volatility of the risky asset. Theorem 4.10 Consider a given set F.Under Assumptions 2.18 and 2.19,suppose that the maximisation problem (4.4)under P∗has a solution ˜ X∈Yx0 (WP)P∈P,which can P∗-a.s.be expressed as f( ∗)for some f∈F.Further,assume that (WP)P∈Pis F-family-consistent on Yx0 (WP)P∈Pwith respect to P∗.Then it holds that max X∈Yx0 (WP)P∈P inf P∈PWP(X) =max X∈Yx0 WP∗ WP∗(X). Proof Let h∈Fbe such that h(∗)∈Yx0 (WP)P∈P. Then (4.5) holds by the F-familyconsistency and (2.3). By assumption, ˜ X=f( ∗)P∗-a.s. for some f∈F. Hence we obtain max X∈Yx0 WP∗ WP∗(X) =WP∗(˜ X) ≤inf P∈PWP(˜ X) ≤max X∈Yx0 (WP)P∈P inf P∈PWP(X) ≤max X∈Yx0 (WP)P∈P WP∗(X) ≤max X∈Yx0 WP∗ WP∗(X), where the last inequality follows because Yx0 (WP)P∈P⊆Yx0 WP∗. Note that in comparison to the statements in Theorem 4.8,theWPin Theorem 4.10 need not be law-invariant. Moreover, as long as the solution can be expressed as a certain function of the likelihood ratio ∗, the utility functionals need not be increasing, i.e., X≤YP-a.s. need not imply WP(X) ≤WP(Y ). As pointed out, Theorem 4.10 is applicable in particular for the case F=FSSD. However, the requirement that ˜ X∈Yx0 (WP)P∈Pcan be P∗-a.s. expressed as h(∗)for some h∈FSSD is equivalent to hbeing increasing and concave. This property is difficult to verify ex ante. Hereafter, we show that if WPis an expected utility, this condition translates into an easy-to-verify condition on the utility function. We formulate the following result. Cost-efficient payoffs under model ambiguity 985 Theorem 4.11 Fix P∈Pwith likelihood ratio P.Let u:R+→Rbe a differentiable,concave and strictly increasing utility function such that uis strictly decreasing.If the maximisation problem (4.4)under Phas a solution,then the solution is a nondecreasing and concave function of Pif and only if 1 uis convex.If uis three times differentiable,1 uis convex if and only if a(x) ≥p(x) 2,(4.6) where a(x) := −u(x) u(x) is the absolute risk aversion and p(x) :=−u(x) u(x) the absolute prudence. Proof By Bernard et al. [6, Lemma 2], the solution to (4.4) is unique and given by (u)−1(c0 P)for some c0>0. See also Merton [43] for a proof in a context in which the Inada conditions are satisfied. Note that u>0 and that 1 uis strictly increasing. Observe that the inverse of 1 uis x→ (u)−1(1 x), which is hence also strictly increasing. The inverse of a convex (concave) and strictly increasing function is concave (convex). For the second assertion, observe that u <0 and that a function is convex on an open interval if and only if its second derivative is nonnegative. Then (4.6) follows immediately.  Remark 4.12 Maggi et al. [42] have shown that a(x) > p(x) if and only if the utility has increasing absolute risk aversion, which is somewhat unusual (it is typically assumed that agents have decreasing absolute risk aversion given that they become less risk averse as their wealth increases). But our condition (4.6) is not incompatible with decreasing absolute risk aversion due to the factor 1 2. Condition (4.6) has appeared several times in the literature. It has been found to play a role in the context of insurance models in Bourlès [9], but also appeared as a condition in the opening of a new asset market (see Gollier and Kimball [32]) when there is uncertainty on the size (see Gollier et al. [31]) or the probability of losses (see Gollier [29]) and under contingent auditing (see Sinclair-Desgagné and Gabel [59]). Further interpretation of this condition and in particular of the degree of concavity of the inverse of the marginal utility can be found in [9]. This condition also appears in Varian [60] in the context of payoff selection under ambiguity. Example 4.13 As an illustration of Theorem 4.11, we provide two utility functions which are differentiable, concave and strictly increasing and such that the reciprocal of the marginal utility is convex: •The exponential utility for risk-averse agents: u:R+→R,x → 1−e−λx for λ>0. It holds that 1 u(x) =eλx, which is strictly increasing and convex. •CRRA utility: u:R+→R,x → x1−η 1−ηfor η>1. It holds that 1 u(x) =xη, which is strictly increasing and convex. 4.3 Rank-dependent utility in lognormal markets We now discuss some examples to illustrate Sect. 4.2 in a lognormal market setting with uncertainty on the drift and volatility. In particular, we explicitly solve a robust rank-dependent expected utility problem using Theorems 4.8 and 4.10.Asin 986 C. Bernard et al. Sect. 3.2.2, we assume that the real-world distribution of STis lognormal with parameters log S0+(μ −σ2 2)T and σ√Tand that the investor has uncertainty on the trend and potentially also the volatility. She may expect the true parameters to lie within the cube, for r<μ 1<μ 2and 0 <σ 1≤s, Dμ1,μ2,σ1,s ={(μ, σ ) ⊆R2:μ1≤μ≤μ2,σ 1≤σ≤s}, where sis the volatility under the pricing measure Q. The investor thus considers P=(Pμ,σ )(μ,σ)∈Dμ1,μ2,σ1,s as the set of all plausible probability measures on (, F). Note that if σ1=s, the investor only faces drift ambiguity; otherwise, she considers ambiguity on both trend and volatility. Under Pμ,σ ,STis lognormal with density fμ,σ defined in (3.1). In the next example, is the standard normal distribution function. Example 4.14 Let U(x) =x1−η 1−η,x≥0, for η∈(0,1)be the CRRA utility function. Let γ∈Rand let w(u) =(−1(u) +γ),u∈[0,1], denote the so-called Wang transform, which is increasing concave if γ>0 and increasing convex if γ<0. Consider the payoff choice problem in which the investor maximises her expected rank-dependent utility, i.e., max X∈Yx0 WPμ,σ ∞ 0U(x)d1−w1−FPμ,σ X(x),(4.7) where x0>0 is the initial wealth and (μ, σ ) ∈Dμ1,μ2,σ1,s .Letθ:= √Tμ−r σ.The solution to (4.7)isgivenby X∗ μ,σ :=⎧ ⎨ ⎩ λ−1 ηexp(rT η−1 2γ η(θ +γ ))(P) γ θη+1 η,γ>−θ, λ−1 η,otherwise, (4.8) where λdepends on η,γand θ;see(4.10)–(4.12) below. The solution to the robust rank-dependent utility problem max X∈Yx0 (WP)P∈P inf P∈P∞ 0U(x)d1−w1−FP X(x)dx (4.9) is given by X∗ μ1,s if there is no ambiguity on the volatility, i.e., when σ1=s. If there is ambiguity on the volatility, γ<0 and μ1−r s2∈(0,1], then X∗ μ1,s still solves (4.9). Proof We first prove that X∗ μ,σ solves (4.7). Let P=Pμ,σ . The state price ξP:= e−rT P is lognormally distributed with parameters −rT −1 2θ2and θ>0; see (3.2). Hence as μ>r, it holds that FP ξP(x) =P[ξP≤x]=logx+rT +1 2θ2 θ,x>0, Cost-efficient payoffs under model ambiguity 987 and (F P ξP)−1(p) =exp −1(p)θ −rT −1 2θ2,p∈(0,1). Let H(z) =−w−1(1−z) 0(F P ξP)−1(t)dt, z ∈[0,1]. The solution to the classical rank-utility problem (4.7) is well known (see for instance Xu [63, Theorem 4.1] or Rüschendorf and Vanduffel [50, Sect. 3.2]) and is given by X∗ μ,σ =(U)−1λˆ H1−wFP ξP(ξP), where λis determined by EP[ξPX∗ μ,σ ]=x0and ˆ His the concave envelope of H. Using w(u) =(−1(u)+γ) (−1(u)) , we obtain after some calculations that H(z) =(F P ξP)−1(w−1(1−z)) w(w−1(1−z)) =exp −1(1−z)(γ +θ)−rT −1 2(γ +θ)2. We distinguish two cases to find a more explicit expression for X∗ μ,σ . Case 1:γ+θ>0. Then His nonincreasing and hence His concave and equal to ˆ H.As(U)−1(y) =y−1 η, it is easy to see that X∗ μ,σ =λ−1 ηexp −rTγ θη −1 2 γ η(θ +γ) (ξP)−γ θη−1 η =λ−1 ηexp rT η−1 2 γ η(θ +γ) (P) γ θη+1 η. If 1 −γ θη −1 η=0, then ξPX∗ μ,σ is constant and it holds that λ=x−η 0exp −rTγ θ−1 2γ(θ +γ) .(4.10) Otherwise, ξPX∗ μ,σ is lognormally distributed and it follows that λ=x−η 0exp rT(1−η) +1 2θ2(1−η) +γ2+θη −(γ +θ)2.(4.11) Case 2:γ+θ≤0. Then His convex. Note that H(0)=−1 and H(1)=0. As H is convex, the concave envelope ˆ Hof His given by ˆ H(x) =x−1. Then ˆ H≡1 and X∗ μ,σ =(U)−1(λ)=λ−1 η. 988 C. Bernard et al. Therefore ξPX∗ μ,σ =λ−1 ηξP, and hence λ=x−η 0e−rTη.(4.12) Assume that there is no ambiguity on the volatility, i.e., σ1=s. Section 3.2.1 shows that the least favourable measure with respect to FFSD is given by P∗=Pμ1,s with corresponding likelihood ratio ∗=Pμ1,s . By Example 4.4, the utility functional in (4.9)isFFSD-family-consistent on Yx0 (WP)P∈Pwith respect to P∗, and Theorem 4.8 shows that X∗ μ,σ solves the robust rank-dependent utility problem (4.9). Finally, assume that there is ambiguity on the volatility. If μ1−r s2∈(0,1], Proposition 3.9 shows that the least favourable measure with respect to FSSD is also P∗. If γ<0, then X∗ μ,σ is a concave and nondecreasing function of ∗. By Example 4.4, the utility functional in (4.9)isFSSD-family-consistent on Yx0 (WP)P∈Pwith respect to P∗. Apply Theorem 4.10 to show that also in this case, X∗ μ,σ solves the robust rank-dependent utility problem (4.9).  To better understand the solution in Example 4.14, we “rationalise” the solution as in Bernard et al. [6], i.e., we show that the optimal investment strategy in the robust rank-dependent setting also solves an expected utility maximisation problem. Example 4.15 shows that the solution to the expected rank-dependent utility problems (4.7) and (4.9) involving a Wang transform with parameter γand a CRRA utility function with parameter ηcan be rationalised by a CRRA utility with parameter ηθ γ+θ. Example 4.15 Let (μ, σ ) ∈Dμ1,μ2,σ1,s ,X∗ μ,σ ,θ,x0and Yx0 WPμ,σ as in Example 4.14, and such that γ>−θand ηθ = γ+θ. Then X∗ μ,σ solves the expected utility maximisation problem max X∈Yx0 WPμ,σ ∞ 0u(x)dF Pμ,σ X(x) for the utility function u(x) =1 1−ηθ γ+θ x1−ηθ γ+θ.(4.13) The function u:R+→Ris nondecreasing and concave. Proof Note that γ>−θimplies γ θη +1 η= 0. Let P=Pμ,σ and ξP:= e−rT P.As in [6], let c>0 and define ˜u(x) =x c (F P ξP)−11−FP X∗ μ,σ (y)dy, x ∈R+. Because X∗ μ,σ is lognormally distributed, it follows that (F P ξP)−11−FP X∗ μ,σ (x)=κx−ηθ γ+θ, Cost-efficient payoffs under model ambiguity 995 Lemma B.4 Assume F=FFSD.Under Assumptions 2.17–2.19,suppose that the robust maximisation problem (4.3)has a unique solution ˜ Xand that (WP)P∈Pis law-invariant and FFSD-family-consistent on Yx0 (WP)P∈Pwith respect to P∗.Then ˜ Xis P∗-cost-efficient. Proof The proof is similar to the one for Lemma B.3.Let X∗=(F P∗ ˜ X)−1FP∗ ∗(∗). It holds by law-invariance that max X∈Yx0 (WP)P∈P inf P∈PWP(X) =inf P∈PWP(˜ X) =inf P∈PWP(X∗).  Acknowledgements We thank two referees, an Associate Editor and Professor Alexander Schied for many helpful comments that significantly improved the paper. We also thank Ruodu Wang for constructive comments on an earlier version. The views expressed in this manuscript do not represent the opinions of Helvetia Insurance Group. Funding Open Access funding enabled and organized by Projekt DEAL. This work was supported by the FWO projects G015320N and S006721N. Declarations Competing Interests The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/ 4.0/. 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