Robust asymptotic insurance-finance arbitrage
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Oberpriller, Katharina; Ritter, Moritz; Schmidt, Thorsten Article — Published Version Robust asymptotic insurance-finance arbitrage European Actuarial Journal Provided in Cooperation with: Springer Nature Suggested Citation: Oberpriller, Katharina; Ritter, Moritz; Schmidt, Thorsten (2024) : Robust asymptotic insurance-finance arbitrage, European Actuarial Journal, ISSN 2190-9741, Springer, Berlin, Heidelberg, Vol. 14, Iss. 3, pp. 929-963, https://doi.org/10.1007/s13385-024-00389-1 This Version is available at: https://hdl.handle.net/10419/315858 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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/
European Actuarial Journal (2024) 14:929–963 https://doi.org/10.1007/s13385-024-00389-1 ORIGINAL RESEARCH PAPER Robust asymptotic insurance-finance arbitrage Katharina Oberpriller1·Moritz Ritter2·Thorsten Schmidt2 Received: 9 December 2022 / Revised: 19 February 2024 / Accepted: 10 June 2024 / Published online: 8 August 2024 © The Author(s) 2024 Abstract This paper studies the valuation of insurance contracts linked to financial markets, for example through interest rates or in equity-linked insurance products. We build upon the concept of insurance-finance arbitrage as introduced by Artzner et al. (Math Financ, 2024), extending their work by incorporating model uncertainty. This is achieved by introducing statistical uncertainty in the underlying dynamics to be represented by a set of priors P. Within this framework we propose the notion of robust asymptotic insurance-finance arbitrage (RIFA) and characterize the absence of such strategies in terms of the new concept of QP-evaluations. This nonlinear two-step evaluation ensures absence of RIFA. Moreover, it dominates all two-step evaluations, as long as we agree on the set of priors P. Our analysis highlights the role of QP-evaluations in terms of showing that all two-step evaluations are free of RIFA. Furthermore, we introduce a doubly stochastic model to address uncertainty for surrender and survival, utilizing copulas to define conditional dependence. This setting illustrates how the QP-evaluation can be applied for the pricing of hybrid insurance products, highlighting the flexibility and potential of the proposed approach. Keywords Insurance-finance arbitrage under uncertainty ·Robust QP-rule · Enlargement of filtration ·Absence of robust insurance-finance arbitrage The authors gratefully acknowledge the support of the German Research Foundation (DFG) through grant SCHM 2160/15-1, the CRC SFB 1597/1 (Small Data) and of the Freiburg Center for Data Analysis and Modeling (FDM). BThorsten Schmidt [email protected]g.de Katharina Oberpriller [email protected] Moritz Ritter [email protected]g.de 1University of Munich, Theresienstr. 39, 80333 München, Germany 2University of Freiburg, Ernst-Zermelo-Str. 1, 79104 Freiburg, Germany 123
930 K. Oberpriller et al. 1 Introduction This paper develops and characterizes the absence of insurance-finance arbitrage under model uncertainty. Our starting point is the observation that most insurance contracts are linked to financial markets, for example through interest rates or via direct links of the contractual benefits to stocks or indices. However, the modeling and the valuation of insurance contracts and products on financial markets are fundamentally different due to their distinct characteristics: insurance contracts are static and personalized products, whereas products on financial markets are standardized and traded frequently. Various approaches have been proposed in the literature as to how insurance and financial markets can be treated in a coherent manner, see e.g., [10,11,21, 23] and references therein. More recently, 2-step and 3-step approaches have been proposed as for example in [2,7] and in [19]. In this paper we aim for a fundamental analysis of arbitrage in these markets and rely on the notion of insurance-finance arbitrage (IFA) introduced by [1]. In this approach the insurance company may issue contracts to a large number of clients on the one hand. On the other hand, it can simultaneously hedge its positions by trading on the financial market. In order to model the two information flows to which the insurance company has access, we work with two filtrations. The smaller filtration represents the publicly available information on the financial market, denoted by F=(Ft)t≤T, while the larger filtration G=(Gt)t≤Tadditionally contains the insurer’s information. Given a pricing measure Qon the financial market (, F)and a statistical measure P on (, G)a characterization of the absence of IFA in terms of the QP-rule is derived in [1]. Even if a large set of homogeneous data is available, statistical uncertainties in predicting the future evolution of insurance losses in the considered portfolio remain a problem that needs to be addressed. In this paper we therefore take this uncertainty into account. To do so, we fix the nullsets Non the financial market (, F)which together with the traded assests Sdetermine the set of equivalent martingale measures Q. Second, we consider a class of probabilistic models Pon (, G), such that the measures in P restricted to Fhave exactly the nullsets Nand study the associated QP-rule. Most notably, this framework allows us to model uncertainty on the insurance market under the assumption that we do not face any model risk on the financial market. Working with a class of potential models Pis in line with the growing literature on model risk and uncertainty, see e.g., [5,6,8,9,24,28]. To the best of our knowledge, this is the first study of insurance-finance arbitrage under model uncertainty. More specifically, we prove a characterization of robust insurance-finance arbitrage (RIFA) by using the QP-evaluation in Theorem 2.12. Furthermore, we show that this result provides a theoretical foundation for a class of two-step evaluations introduced in [23], which is applied for the pricing of hybrid products depending on the financial market, as well as on other random sources. In particular, we prove that every two-step evaluation, which is the combination of a risk-free measure Qand a coherent F-conditional risk measure, being continuous from below, equals a QP-evaluation for a suitable subset Pon (, G). Thus, every 123
Robust insurance-finance arbitrage... 931 two-step evaluation of this kind leads to a robust arbitrage-free price in the asymptotic insurance-finance setting. We conclude the paper by suggesting possible applications in Sect. 4. First, we consider the case of two conditional independent random times such that the conditional distribution functions face a certain degree of uncertainty. Here, the random times represent the surrender time and the time of death of an insurance seeker. In this setting, we introduce a financial market via a Cox-Ross-Rubinstein model and consider finance-linked insurance benefits with surrender options. We compute the robust arbitrage-free price of these products numerically and find that the uncertainty in this example justifies that the robust arbitrage-free price is higher than the supremum of all arbitrage-free prices under each possible model. This underlines the importance of treating uncertainty in a systematic manner. Furthermore, we show that the wellknown Cox model under uncertainty is contained in the outlined setting. Moreover, we generalize the setting by allowing dependence between the random times described by a copula. The paper is structured as follows. In Sect.2we introduce the definition of a robust insurance-finance arbitrage and provide a characterization in our main result. After that, Sect.3studies the relation of the QP-rule to two-step evaluations. Then, in Sect.4we study an insurance-finance market with two conditionally independent random times and numerically provide the robust arbitrage-free prices for certain hybrid products. In addition, we consider a copula framework for the two random times under uncertainty. 2 Robust asymptotic insurance-finance arbitrage Let (, G)be a measurable space and denote by P(, G)the set of all probability measures on this space. We consider a discrete time model with times t=0,...,Tand introduce two different kinds of information flows described by the filtrations Fand G on (, G). The filtration F=(Ft)t≤Trepresents publicly available information and contains all information available on the financial market. The filtration G=(Gt)t≤T contains additional private information of the considered insurance company, which includes, for example, several datasets on its clients. In particular, F⊆G. Moreover, let F0=G0={∅,}and Fbe a σ-field such that FT⊆F⊆G. Given P⊆P(, G),asetA⊆is called P-polar if A⊆Nfor some N∈G satisfies P(N)=0 for all P∈P. Moreover, a property holds P-quasi surely (P-q.s.) if it holds outside a P-polar set. For a fixed σ-ideal of F-nullsets N, i.e., there exists a measure P0on (, F)such that Nare the nullsets of P0, we define the following sets of priors PN(, F):=P∈P(, F)|Nare the nullsets of P(2.1) and PN(, G):=P∈P(, G)|Nare the nullsets of P|F.(2.2) 123
932 K. Oberpriller et al. Hereafter, we fix a probability measure P0∈P(, F), which determines the nullset Non (, F), and a subset of priors P⊆PN(, G), which specifies the uncertainty about the model. In the following, we introduce the concept of insurance-finance arbitrage. For simplicity, we consider only a single insurer. We also assume that the insurer can contract with an arbitrarily high number of clients to reduce its risks, as specified in Assumption 2.1. To establish this, we consider a finite number of insurance seekers and study the limits of portfolio allocations - a technique inspired by large financial markets, see e.g., [15,17,18]. Such strategies may lead to insurance arbitrages and it is partly our aim to characterize when and how such arbitrages can be achieved and under which conditions they can be avoided. At the same time, the insurance company also trades on the financial market, potentially leading to a financial arbitrage. Thus, the combination of these concepts results in an insurance-finance arbitrage. 2.1 The insurance contracts Insurance contracts offer a variety of benefits at future times in exchange for a single premium or a premium stream. We work with discounted quantities and, without loss of generality, we consider a single premium paid at time 0 and an aggregated benefit received at future time T. More precisely, we denote by p∈Rthe premium to be paid at time 0 and a GT-measurable (discounted) benefit Xito be received by the ith client at time T. This allows to cover a wide range of contracts, particularly contracts depending on financial markets, such as variable annuities. We assume that all insurance seekers under consideration can be treated as homogeneous (under each model P∈P) and each insurance seeker pays the same premium p in order to receive his or her personal benefit Xi. This idea is formalized in the following assumption, which is a generalization of the framework of actuarial mathematics to stochastic assets, as discussed in [1]. Assumption 2.1 For all P∈P, the following holds: (i) X1,X2,...∈L2 +(, G,P)are F-conditionally independent. (ii) EP[Xi|F]=EP[X1|F]for all i∈N. (iii) VarP[Xi|F]=Var P[X1|F]for all i∈N. Remark 2.2 (The implications of Assumption 2.1) Assumption 2.1 is a fairly weak assumption and includes a wide range of existing models: (i) Approaches where insurance benefits are i.i.d. and independent of the publicly available information. (ii) Any kind of variable annuities including F-independent and i.i.d. survival (or surrender) times τi. This covers for example survival benefits of the form Xi=1{τi>T}e−rT F(˜ ST)or payments at surrender times like Xi= 1{τi=T}e−rτiF(˜ Sτi), where ˜ Sis the F-adapted (undiscounted) stock price process and e−rt is the deterministic discounting factor at time t. Similarly, stochastic interest rates or path-dependent payoffs can be included. 123
Robust insurance-finance arbitrage... 933 (iii) Doubly-stochastic random times are also part of the framework, i.e., random times which are driven by a hazard rate λwhich is F-progressively measurable such that the process 1{τi≤t}−1[0,t∧τi]λsds is an F-martingale. This allows to model important aspects like systemic risk or longevity risk, for example by incorporating factors which affect λand hence all insurance seekers. It seems important to point out, that Fcontains all publicly available information, and hence might be significantly larger then the filtration created by stock prices only. In particular, publicly available mortality tables would be included in F. Intuitively, λallows to cover risks which are are observable by the public, such as a general crisis giving rise to systemic risk or increasing longevity reflected in publicly available life tables. The individual risk, referring to the risk beyond systematic risk that aligns with the individual characteristics of the policyholder, is subsequently captured by the model’s additional stochastic component. We refer to Remark 4.3 which shows how to include increasing life expectations in the Gompertz model under Assumption 2.1. However, our current framework does not include different cohorts. To incorporate for example different ages of the insurance seekers, a separate model for each age must be considered. The insurance portfolio is obtained as a limit of allocations of contracts with a finite number of clients. An allocation at time 0, ψ=(ψi)i∈N,is c00-valued, deterministic and non-negative, where c00 denotes the space of sequences with a finite number of non-zero elements. For i∈N,ψi∈R+denotes the size of the contract with the ith policyholder. The accumulated benefits and premiums associated with the allocation ψare given by i∈NψiXiand i∈Nψip, while the associated profits and losses are denoted by VT(ψ) := i∈N ψi(p−Xi). An insurance portfolio strategy := (ψn)n∈N∈cN 00 is modeled as a sequence of allocations ψn=(ψn,i)i∈N. Moreover, the profit and loss of an insurance portfolio strategy is given (if it exists) by VT() := lim n→∞ VT(ψn)=lim n→∞ i∈N ψn,i(p−Xi). We introduce the following admissibility conditions for an insurance portfolio strategy =(ψn)n∈N. Assumption 2.3 (i) Uniform boundedness: There exists C>0,such that ψn:= i∈N ψn,i≤Cfor all n≥1. 123
934 K. Oberpriller et al. (ii) Convergence of the total mass: There exists γ≥0 such that γ=lim n→∞ ψn. (iii) Convergence of the total wealth: There exists a R-valued random variable VT(), such that VT() =lim n→∞ VT(ψn)P-q.s. An insurance portfolio strategy =(ψn)n∈Nwhich satisfies Assumption 2.3 is called P-admissible. 2.2 The financial market We introduce a financial market model in discrete time consisting of drisky assets S=(S1 t,...,Sd t)t=0,...,Ton (, F). For i=1,...,d,the discounted price of the ith risky asset at time tis given by the Ft-measurable random variable Si tand the bank account is given by S0≡1.We assume that the insurance company trades with F-trading strategies on the financial market, where a F-trading strategy is a ddimensional F-predictable process ξ=(ξt)t=1,...,Twith ξt=(ξ1 t,...,ξd t). Note that each strategy ξcan be extended by ξ0=(ξ0 t)t=1,...,Tto a self-financing trading strategy ¯ ξ=(ξ0 t,...,ξd t)t=1,...,T,cf.[14, Remark 5.8]. The associated gain process at time t=1,...,Tis then given by the discrete stochastic integral (ξ ·S)t:= t s=1 ξs(Ss−Ss−1)= t s=1 d i=1 ξi sSi s−Si s−1. The absence of arbitrage in this market with respect to one and thus any restricted measure P|Ffor P∈P⊆PN(, G)can be characterized by the existence of an equivalent martingale measure, cf. [14, Theorem 5.16]. The set of all equivalent martingale measures is denoted by Me(F)and defined by Me(F):= {Q∈PN(, F)|Sis a (Q,F)-martingale}.(2.3) Remark 2.4 By taking into account a set of priors P⊆PN(, G)and according to the definition of PN(, G)in (2.2), all nullsets on (, F)are already determined by N.Thus, the uncertainty of our model refers only to the additional insurance part defined on (, G). 2.3 Robust insurance-finance arbitrage Finally, we introduce the insurance-finance market (S,X,p)on (, G)consisting of the benefits X=(Xi)i∈N, the premium p, and the discounted asset prices S= (St)t=0,...,T. 123
Robust insurance-finance arbitrage... 935 In order to define a robust arbitrage in our setting, we use the concept of a robust arbitrage strategy introduced in [6]. Note, however, that we do not need to require convexity of Pfor the characterization of the absence of a robust arbitrage strategy. Definition 2.5 AP-robust asymptotic insurance-finance arbitrage RIFA(P)on the insurance-finance market (S,X,p)is a pair (ξ, ) consisting of an F-predictable trading strategy ξand a P-admissible insurance portfolio strategy such that (ξ ·S)T+VT() ≥0P-q.s. and EP[(ξ ·S)T+VT()]>0forsomeP∈P. (2.4) If there exists no such pair (ξ, ) satisfying (2.4), then there is no P-robust asymptotic insurance-finance arbitrage, which we denote by NRIFA(P). Remark 2.6 If for all P∈Pit holds NRIFA({P})then NRIFA(P)is also satisfied. However, the converse statement does not hold in general. In the case of no model uncertainty, i.e., when P={P}for a measure P∈ PN(, G), it is shown in Corollary 5.2 in [1] that if there exists Q∈Me(F)such that p≤EQP[X1]:=EQEP[X1|F],(2.5) then there is no asymptotic insurance-finance arbitrage. Thus, according to Remark 2.6, an insurance-finance market (S,X,p)fulfills NRIFA(P)if (2.5) holds for each P∈P. However, it should be noted that these conditions are not necessary. 2.4 Uniform essential supremum In order to characterize the absence of a P-robust asymptotic insurance-finance arbitrage, we aim to identify assumptions that allow us to take into account a robust version of the conditional expectations EP[X|F]for P∈Pin (2.5). Given a set of priors P⊆P(, G), it is in general not possible to consider supP∈PEP[X|F],asthe conditional expectation is only defined P-a.s. and the priors in Pmay have different nullsets. Furthermore, the supremum no longer needs to be measurable. The natural approach to solve the measurability issue is to work with the essential supremum instead of the supremum. For a fixed probability measure P∈P(, G) and a set of random variables on (, G)there is a random variable Y=: ess supP such that (i) Y≥ϕP-a.s. for all ϕ∈, and (ii) Y≤ψP-a.s. for every random variable ψsatisfying ψ≥ϕP-a.s. for all ϕ∈. We refer to [14, Theorem A.37] for a proof of the existence of the essential supremum and to [3], in whose work a general construction of such a nonlinear conditional expectation is studied. 123
936 K. Oberpriller et al. However, for a general set of priors P⊆P(, G)and a set of random variables on (, G)there may be no uniform essential supremum, i.e., a random variable Y such that Y=ess supPP-a.s. for all P∈P.(2.6) The definition of the set PN(, G)in (2.2) allows us to consider for P⊆ PN(, G)the set of random variables =(ϕP)P∈Psuch that ϕP=EP[X|F]P-a.s. for all P∈P.(2.7) Indeed, we observe that by F-measurability together with the fact that P|F∼P|F for all P,P∈P,the conditional expectation EP[X|F]is not only P-a.s. uniquely determined but also P-q.s.. Moreover, there exists an uniform essential supremum which fulfills (2.6), as the following result demonstrates. Lemma 2.7 Let Nbe the nullsets generated by the probability measure P0∈ P(, F),P⊆PN(, G)and a set of F-measurable random variables. Then ess supP0=ess supPP-a.s. for all P ∈P.(2.8) Proof We show that ess supP0fulfills condition (i) and (ii) from the definition of the essential supremum for all P∈P. For the first part, using the definition of ess supP0, we obtain that ess supP0≥ϕP0-a.s. for all ϕ∈. (2.9) Since {ess supP0≥ϕ}is F-measurable and P|F∼P0for all P∈P,(2.9)also holds P-a.s. for all P∈P. Relying on the construction of the essential supremum, see, for example [14, Theorem A.37], there exists a countable subset ∗⊆such that ess supP0(ω) =sup ∗(ω) for all ω∈. Fix any P∈P, then for each random variable ψon (, G)such that ψ≥ϕP-a.s. for all ϕ∈, we get ψ≥sup ∗=ess supPP-a.s.. This shows the second part and the result is proven. In the following, for the fixed measure P0, which generates the F-nullsets Nand use the notation ess sup := ess supP0. 123
Robust insurance-finance arbitrage... 943 where we use [1, Proposition B.1] for the fifth equality and (2.20) for the inequality. We now define the process Z=(Zt)t=0,...,Tby Zt=ER[VT()|Ft],for all t=0,...T. Zis a (F,R)-martingale with Z0<0by(2.21). Let ξbe some F-predictable strategy. Then, as in [14, Remark 9.5] the value process (ξ ·S)is a local (F,R)-martingale and thus (ξ ·S)+Zis a local (F,R)-martingale. For the sake of contradiction we assume that (ξ, ) is a P-robust asymptotic insurance-finance arbitrage such that has positive total mass γ>0. Thus, it holds that (ξ ·S)T+ZT≥0P-a.s. for all P∈P.(2.22) Given that R|F∼P|F1 N n i=1 Pi|F, equation (2.22)isalsotrueR-a.s., i.e., (ξ ·S)T+ZT≥0R-a.s.. (2.23) Thus, according to [14, Proposition 9.6] the process (ξ ·S)+Zis a (F,R)- supermartingale and we obtain ER[(ξ ·S)T+ZT]≤(ξ ·S)0+Z0<0. This contradicts (2.23) and there cannot exist a P-robust asymptotic insurance-finance arbitrage (ξ, ) such that has positive total mass γ>0. However, given that the pure financial market is arbitrage-free with respect to F-predictable trading strategies ξ, there could also not be an arbitrage (ξ, ) such that has total mass γ=0. Overall, this leads to a contradiction and thus the result is proven. Remark 2.13 Let us compare Theorem 2.12 in the case of P={P}with Theorem 1 in [26]. According to Assumptions 2.1 and 2.3, each condition (i)and (ii)in Theorem 2.12 implies the absence of arbitrage in the sense of Definition 2.5 and there is no need for an additional boundedness assumption on the density of the corresponding martingale measures. By contrast, in [26] the boundedness assumption on the density of the equivalent martingale measure is essential and cannot be substituted by means of Lemma 2.10, cf. Example 2 in [26]. Motivated by Theorem 2.12 and the previous discussion (see e.g., Equation (2.5)), we define the following robust version of the QP-rule. Definition 2.14 Let P⊆PN(, G)and Q∈Me(F). Then, for X≥0P-q.s., we define the QP-evaluation of X by EQP[X]:=EQess sup P∈P EP[X|F].(2.24) Note that QPdoes not define a probability measure as it is the case for QP. 123
944 K. Oberpriller et al. Remark 2.15 (On the choice of Q) If the financial market is complete, Me(F)={Q} and there is only one possibly choice for Q. If the market is incomplete, the choice of Qbecomes more difficult. When sufficiently many traded derivatives are available, a market-consistent Qcan be obtained by calibrating the model to those derivatives. In the insurance context, the most challenging calibration problems occur when long maturities (like 10 to 30 years) are considered. Here, there are few or even no tradeable instruments and a good calibration requires much more effort, in particular to exclude model risk. Remark 2.16 If the set {EP[X|F]|P∈P}is directed upward, i.e., for all P,P∈P there exists ˜ P∈Psuch that max{EP[X|F],EP[X|F]} ≤ E˜ P[X|F]P0-a.s.,(2.25) then, as demonstrated by [14, Theorem A.37], there exists a sequence of measures (Pn)n∈N⊂Psuch that EPn[X|F]ess sup P∈P EP[X|F]P0-a.s. for n→∞. Using monotone convergence we find that EQess sup P∈P EP[X|F]=lim n→∞ EQEPn[X|F] ≤sup P∈P EQ[EP[X|F]] ≤EQess sup P∈P EP[X|F] and that EQP[X]= sup P∈P EQ[EP[X|F]]. Consequently, for a set of priors Pthat is directed upwards in the sense of Equation (2.25), there is no P-robust asymptotic insurance-finance arbitrage if and only if there is no asymptotic insurance-finance arbitrage with respect to Pfor all P∈P. Remark 2.17 We now briefly consider the case of G-trading strategies on the financial market introduced in Sect.2.2, i.e., ξis a d-dimensional G-predictable process. This reflects the fact that the insurer has access to information on the financial market as well as on the insurance market. In order to define a no-arbitrage condition for the financial market, we introduce some more notation. Let P⊆PN(, G). We say that a measure Q∈P(, G)is dominated by Pif there exists P∈Psuch that QP, and in this case we write Q≪P. Next, we define the set M(G):= {Q∈P(, G)|Q≪Pand Sis a (Q,G)-martingale}.(2.26) 123
Robust insurance-finance arbitrage... 945 In this case we assume that for all P∈Pthere exists Q∈M(G)such that PQ. Then, according to [6, Theorem 4.5], there is no P-robust arbitrage, denoted by NA(P,G), on the financial market, which means that for all G-trading strategies (ξ ·S)T≥0P-q.s. implies (ξ ·S)T=0P-q.s. (2.27) Given that every F-trading strategy is also a G-trading strategy, it is clear that NA(P,G)implies NA(P,F), where Gand Frefers here to the G-trading strategies and F-trading strategies, respectively. It thus follows that NRIFA(P,G)implies NRIFA(P,F)and thus (i) or (ii) in Theorem 2.12 is satisfied. Here, NRIFA(P,G) is defined as in Definition 2.5, but with a G-trading strategy ξ. However, the corresponding if direction of Theorem 2.12 is more delicate and could therefore serve as a topic for future research. 3 Robust two-step evaluation In this section we show that Theorem 2.12 provides a theoretical foundation for the socalled two-step evaluation,cf.[11,23]. This kind of evaluation is used for the pricing of hybrid products depending on the financial market, as well as on other random sources, e.g., individual risks depending on the policy holder of an insurance contract. The idea of a two-step evaluation is to combine actuarial techniques with concepts from financial mathematics. In the following, we recap the idea and the concept. However, note that in contrast to the existing literature we do not fix any probability measure on (, G), but only a prior P0on the measurable space (, F). Let Xbe a G-measurable random variable, representing the discounted payoff of an insurance product. In a first step we consider the F-conditional risk of X, i.e. ρF(X), where ρFis a suitable F-conditional risk measure defined on the space of bounded random variables on (, G), which is denoted by Lb(, G). This corresponds to an actuarial evaluation resulting in a F-measurable random variable ρF(X)defined on the financial market (, F,P0). In the second step we price ρF(X)on the financial market under an equivalent risk-neutral measure Q∈Me(F). Combining these, we arrive at the following two-step evaluation: π:Lb(, G)→R,π(X)=EQ[ρF(−X)].(3.1) As already mentioned in [1], the QP-evaluation in (2.5) is a two-step evaluation with the F-conditional risk measure ρF(X)=EP[X|F]as well as the new concept of the robust QP-evaluation from Definition 2.14. In the following we recap some well-known facts for conditional risk measures in order to highlight that for specific coherent F-conditional risk measures ρFthe two-step evaluation in (3.1) can be rewritten by a QP-evaluation for a suitable choice of P. 123
946 K. Oberpriller et al. 3.1 Robust representation of conditional risk measures For the reader’s convenience we recall the definition of conditional risk measures (see e.g., [14]). Definition 3.1 AmapρF:Lb(, G)→L∞(, F,P0)is called a convex Fconditional risk measure, if for all X,Y∈Lb(, G)the following holds P0-a.s.: (i) Conditional cash invariance: ρF(X+¯ X)=ρF(X)−¯ Xfor any ¯ X∈Lb(, F). (ii) Monotonicity: X ≤Yimplies ρF(X)≥ρF(Y). (iii) Normalization: ρF(0)=0. (iv) Conditional convexity: ρF(λX+(1−λ)Y)≤λρF(X)+(1−λ)ρF(Y)for λ∈Lb(, F)with 0 ≤λ≤1. A convex F-conditional risk measure ρFis called coherent if it also satisfies the following condition: (v) Conditional positive homogeneity: ρF(λX)=λρF(X)for λ∈Lb(, F)with λ≥0. We say that ρFis continuous from below if (vi) Continuity from below: XnXpointwise on implies ρF(Xn)ρF(X). Moreover, we say that ρ:Lb(, G)→Ris a (coherent) convex risk measure if F={,∅}. Definition 3.2 Let Sbe a financial market on (, F,F,P0)and denote by Me(F)the set of all martingale measures which are equivalent to P0.Amapπ:Lb(, G)→R is called two-step evaluation if there is a F-conditional risk measure ρFand an equivalent martingale measure Q∈Me(F)such that π(X)=EQ[ρF(−X)]for all X∈Lb(, G). We now show that Theorem 2.12 provides an economic foundation for the pricing of finance-linked insurance products via two-step evaluations. Indeed, by using results from [14], we formulate sufficient condition for a conditional risk measure ρFin a two-step evaluation π(X)=EQ[ρF(−X)]such that πis a QP-evaluation. In this way, the two-step evaluation πcharacterizes the P-robust asymptotic insurancefinance arbitrage-free price as presented in Theorem 2.12. Lemma 3.3 Let ρF:Lb(, G)→L∞(, F,P0)be a convex F-conditional risk measure which is continuous from below. Then ρFis represented by ρF(X)=ess sup P∈PP0(,G)EP[−X|F]−αmin F(P),(3.2) where the acceptance set AF, the penalty function αmin Fand the set of priors PP0(, G)are defined by AF:= {X∈Lb(, G)|ρF(X)≤0}, 123
Robust insurance-finance arbitrage... 947 αmin F(P):= ess sup X∈AF EP[−X|F], and PP0(, G):= {P∈P(, G)|P|F=P0}⊆PN(, G). (3.3) If, in addition, ρFis a coherent F-conditional risk measure, then there is a subset P⊆PP0(, G)such that ρF(X)=ess sup P∈P EP[−X|F].(3.4) Proof In the unconditional case the first statement follows from Theorem 4.16 and Theorem 4.22 as proven in [14]. Using the same idea as in the proof of Theorem 11.2 in [14] yields the conditional statement. For the second statement we refer to Corollary 4.19 in [14] for the unconditional case. Using similar arguments to those in Corollary 11.6 in [14], the conditional statement follows. Thus, Lemma 3.3 shows that every two-step evaluation π(X)=EQ[ρF(−X)] given by an equivalent martingale measure Q∈Me(F)and a coherent F-conditional risk measure ρFwhich is continuous from below can be written as QP-evaluation π(X)=EQ[ess sup P∈P EP[X|F]] for a suitable subset P⊆PP0(, G). Remark 3.4 If we a priori fix the nullsets on (, G)by a probability measure Pon (, G)such that P|F=P0, then we can define the conditional risk measure ρFon L∞(, G,P), instead of working with Lb(, G). In this case ρFonly needs to be continuous from above, as opposed to satisfying the stronger assumption of continuity from below, in order to have a representation as in Lemma 3.3, cf. Theorem 4.33 and Theorem 11.2 in [14]. The drawback of this approach is that we consider uncertainty in a narrow sense because we fix all relevant nullsets on (, G)using a single probability measure P. Nevertheless, this also leads to a robust pricing problem in the spirit of Sect.2, which will be discussed in more detail in Remark 3.8. Next, we provide some examples for the set P⊆PP0(, G)in Lemma 3.3 and the associated F-conditional risk measure in Equation (3.4). Example 3.5 Let Pbe a probability measure on (, G)such that P|F=P0.We consider the set of priors Pλ⊆PP0(, G)given by Pλ:=˜ P∈PP0(, G)|˜ PPwith d˜ P/dP ≤λ−1P-a.s.for λ∈(0,1). (3.5) In this case, the associated risk measure ρF(X)is the conditional average value at risk, denoted by AVRλ(X|F), see also Definition 11.8 in [14]. Note that the set Pλ is dominated by the probability measure P. 123
948 K. Oberpriller et al. Example 3.6 Let Pbe a probability measure on (, G)such that P|F=P0.We consider the set of priors Pc⊆PP0(, G)for c>0 given by P c:=˜ P∈PP0(, G)|H(˜ P|P)≤c,(3.6) where H(˜ P|P)denotes the relative entropy of ˜ P with respect to P and is defined by H(˜ P|P):= E˜ Plog d˜ P dP,if ˜ PP +∞,otherwise. Here, the associated risk measure ρF(X)is the coherent entropic risk measure,introduced in [13]. As in Example 3.5,thesetP cis dominated by the probability measureP. In the following proposition we show that pricing with two-step evaluations leads to arbitrage-free premiums in the sense of Sect.2. This remarkable result is a consequence of Lemma 3.3 and Theorem 2.12. Proposition 3.7 Let S be a financial market on (, F)and let πbeatwo-step evaluation with F-conditional convex risk measure of the form (3.4) for the set P={P∈PN(, G)|αmin F(P)<∞}. Assume that X=(Xi)i∈Nis a sequence of insurance benefits fulfilling Assumption 2.1 and assume that p <π(X1). Then there is NRIFA(P) with respect to the insurance-finance market (S,X,p). Proof The result is a direct consequence of Theorem 2.12, since the QP-evaluation is an upper bound for the two-step evaluation πand the chosen premium pis even smaller by assumption. 3.2 Construction of conditional iid copies Our next goal is to apply Theorem 2.12 in the context of a robust two-step evaluation. More specifically, given a random variable ˜ Xdescribing an insurance benefit, we determine a robust arbitrage-free premium pfor ˜ Xby using Theorem 2.12 and by taking into account some actuarial constraints, which are reflected by the set of priors P⊆PN(, G)(see e.g., the sets Pλand Pcin Example 3.5 and 3.6, respectively). To do so, two factors must to be considered. First, the assumptions of Theorem 2.12 must be satisfied. To this end, we construct a sequence of benefits (Xj)j∈Nwhich are copies of ˜ Xand which satisfy Assumption 2.1. Second, the set of priors Pmust be shifted to the product space where we model the benefits (Xj)j∈N. We observe that these steps contain some subtleties which we discuss in more detail in Remark 3.8 after formally introducing the setting. Let (F,FF)and (I,FI)be two measurable spaces on which we model purely financial and purely insurance events, respectively. On the product space (F×I,FF⊗FI)we introduce the stochastic process ˜ S=(˜ St)t=0,...,Tand the random variable ˜ Xdescribing the financial market and a single insurance benefit, respectively. Moreover, let P0be a measure on (F×I,FF⊗{∅,I})which 123
Robust insurance-finance arbitrage... 949 determines the nullsets in FF⊗{∅,}and Pbe a set of probability measures on (F×I,FF⊗FI)such that P|FF⊗{∅,I}∼P0for all P∈P. We now shift Pto a set of priors μPon (F×(I)N,FF⊗(FI)⊗N). Furthermore, on this space we copy the financial market ˜ Sto Sand construct insurance benefits (Xj)j∈Nwhich are iid conditionally on Ssuch that for all P∈Pthe law of (S,Xj) for j∈Nunder μPcoincides with the law of (˜ S,˜ X)under P. Remark 3.8 We also emphasize that if the set Pis dominated by a measure P∈P, as is the case in Example 3.5 and 3.6, this will no longer hold for the shifted set μP. The reason for this is that absolute continuity of measures is not stable under countable products. Therefore, the seemingly not robust problem in the dominated case on (F×I,FF⊗FI,P)is indeed a robust pricing problem on (F× (I)N,FF⊗(FI)⊗N,μ P). To be precise, we define ˜ := F×I,:= F×(I)N, ˜ F:= FF⊗{I,∅},F:= FF⊗{I,∅}⊗N, ˜ G:= FF⊗FI,G:= FF⊗(FI)⊗N. We denote by ˜ω=(˜ωF,˜ωI)an element in ˜ and by ω=(ωF,(ωI j)j∈N)an element in . Furthermore, we introduce the following projections on ˜ : ˜πF:˜ →F ,˜πF(˜ω) =˜ωF , ˜πI:˜ →I,˜πI j(˜ω) =˜ωI, as well as the following the projections on : πF:→F ,π F(ω) =ωF , πI j:→I,π I j(ω) =ωI j. Given a measure Pon (˜ , ˜ G), the aim is to define a probability measure μPon (, G) which fulfills the following properties: The law of (πF,π I j)under μPequals Pfor all j∈N,(3.7) and (πI j)j∈Nare F-conditionally independent under μP.(3.8) 123
950 K. Oberpriller et al. If Pis a product measure given by P=PF⊗PIfor measures PFon (F,FF) and PIon (I,FI), then the measure μPcan be defined by μP=PF⊗(PI)⊗N. Otherwise, we construct μPvia disintegration as follows. For some measure Pon (˜ , ˜ G)the measure μpis defined as μP(A×B):= 1˜π−1 F(A)(P˜πI|˜ F)⊗N(B)dP for A∈FFand B∈(FI)⊗N, (3.9) where P˜πI|˜ Fdenotes the regular version of the conditional probability of ˜πIgiven ˜ F(see e.g., [16, Chapter 8]). Note that we implicitly assume its existence. This is no restriction, however, because F, representing a financial market with d+1 assets and Ttime steps, can always be assumed to have the form F=R(d+1)×(T+1)and, thus, is a Borel space. Moreover, using a monotone class argument, it follows that (P˜πI|˜ F)⊗N:(FI)⊗Nט →[0,1] (B,˜ω) → (P˜πI|˜ F(·,˜ω))⊗N(B) is a probability kernel from (˜ , ˜ G)to (, G)and, thus, the measure μPis well defined. If Pis a measure on (˜ , ˜ F), then (3.9) defines a measure on (, F). In this case we have B=N. Note that by using (3.9) we get the following: μP(A×(I)N)=P(A×I)for all A∈FF and thus P|˜ F∼P0implies μP|F∼μP0for all P∈P. Let ˜ F=(˜ Ft)t≤Twith ˜ Ft:=FF t∨{I,∅}, be a filtration on (˜ , ˜ F). Hereafter, we assume that ˜ S=(˜ St)t≤Tis a ˜ F-adapted stochastic process on (˜ , ˜ F)describing the prices in a financial market. Moreover, let Me(˜ F)be the set of all martingale measures on (˜ , ˜ F)which are equivalent some fixed measure P0on (˜ , ˜ F). The insurance and financial filtration on (˜ , ˜ G)is denoted by ˜ G=(˜ Gt)t≤Tand the insurance benefit, a random variable on (˜ , ˜ G), is denoted by ˜ X. In order to shift all quantities to (, G) we define F=(Ft)t≤Twith Ft:=FF t∨{I,∅}⊗N, G=(Gt)t≤Twith Gt:=σ((πF,π I j):→(˜ , ˜ Gt)|j∈N), and St:= ˜ St◦(πF,π I 1)=˜ St◦(πF,π I j)for all j∈N,t=0,...,T, Xj:= ˜ X◦(πF,π I j)for all j∈N. 123
Robust insurance-finance arbitrage... 951 Note that ˜ Stis assumed to be measurable with respect to ˜ Ft⊆˜ Fand, thus, it does not depend on the second coordinate ˜ωI. We show that the measure μPdefined in (3.9) satisfies the desired properties in (3.7) and (3.8) and is the unique measure with this property. For the reader’s convenience we provide the proofs of these results in detail in the Appendix. Proposition 3.9 The measure μP, as defined by (3.9), is the unique measure on (, G) which fulfills (3.7)and (3.8). Next, we characterize the set of all equivalent martingales measures on (, F). Proposition 3.10 The set Me(F)of all measures on (, F)such that S is a Fmartingale and which are equivalent to μP0is given by Me(F)={μQ|Q∈Me(˜ F)}. Proposition 3.11 For any P ∈Pand Q ∈Me(˜ F)we have the following: EQ[EP[˜ X|˜ F]] = EμQ[EμP[X1|F]], EQ[ess sup P∈P EP[˜ X|˜ F]] = EμQess sup P∈P EμP[X1|F]. We emphasize that Theorem 2.12 and Proposition 3.11 build a foundation of two-step evaluations from a new perspective. We shift the insurance benefit to an insurance-finance market such that the assumptions for Theorem 2.12 are fulfilled and characterize the robust insurance-finance arbitrage-free prices therein. Then, Proposition 3.11 shows that the prices in the shifted insurance-finance market coincide with the two-step evaluation of the initial benefit. Summarizing, the QP-evaluation provides a valuation methodology which excludes a robust insurance-finance arbitrage in the above sense. While the argument involves the conditional strong law of large numbers, and thus infinitely many insurance contracts, the QP-evaluation also provides a highly reasonable valuation rule when only finitely many contractors are available. However, with only finitely many contracts, the no-arbitrage concept becomes less powerful: this phenomenon arises, because the theoretical limit, the conditional expectation, is not fully reached in this case. Consequently, this would permit higher prices without necessarily creating an insurance-finance arbitrage since always a small risk remains. 4 Modeling of insurance-finance markets In this section we provide models for insurance-finance markets and calculate the robust insurance-finance arbitrage-free premium by means of the QP-evaluation, cf. Theorem 2.12 and Definition 2.14. As in Sect.2.2,letS0be the bank account and denote by S1=(S1 t)t=0,...,Tthe non-discounted price process of a risky asset on (, F). We fix the F-nullsets N 123
952 K. Oberpriller et al. generated by a probability measure P0∈P(, F). We assume that the filtration Fis generated by S0and S1. Next, we introduce the N0-valued random variables τ1and τ2 representing the time of death and the time of surrender of a policy holder, respectively. Let the σ-algebra Gbe given by G=F∨σ(τ1)∨σ(τ2)and the filtration Ggiven by G=F∨H, where His the filtration generated by the processes (1{τ1≤t})t=0,...,T and (1{τ2≤t})t=0,...,T. Note that τ1and τ2are G-stopping times but, in general, they are not F-stopping times. Given a parameter set , we introduce the law of the stopping times (τ1,τ2)under the parameterized set of priors P=(Pθ)θ∈⊆PN(, G). In particular, we assume that for each Pθ∈Pthe conditional laws of τ1and τ2are given by Pθ(τ1≤t|F):=FF 1(θ, t)and Pθ(τ 2≤t|F):=FF 2(θ, t)for t∈N0,(4.1) where for fixed θ∈the mappings FF 1(θ, ·)and FF 2(θ, ·)are F-conditional distribution functions. 4.1 Modeling under conditional independence In this subsection we assume that under every Pθ∈Pthe random variables τ1and τ2are F-conditionally independent, i.e., Pθ(τ1≤s,τ2≤t|F):=FF 1(θ, s)FF 2(θ, t)for s,t∈N0.(4.2) In order to determine the law of (S,τ1,τ2)under Pθit now remains to introduce the restricted measures Pθ|F∼P0and use disintegration. However, we could assume that Pθ|F=P0for all θ∈since the QP-evaluation is invariant under the specific choice of the measures {Pθ|F|θ∈}as the set of equivalent martingale measures Me(F)only depends on the nullsets Ngenerated by P0. We introduce the discounted survival benefit Xsurvival and the discounted surrender benefit Xsurrender by Xsurvival:=1{τ1>T,τ2>T}Y1(S0 T)−1,(4.3) Xsurrender:= T−1 t=1 1{τ1>t,τ2=t}Y2 t(S0 t)−1,(4.4) where Y1is a F-measurable random variable and Y2:= (Y2 t)t=0,...,Tis a F-adapted process. The insurance benefit Xis then given by X:=Xsurvival +Xsurrender.(4.5) Policy holder with such a policy receive the payment Y1at maturity Tif they survives until Tand do not surrender before time T. If they surrender at time t<Tthey receive the payment Y2 t. 123
Robust insurance-finance arbitrage... 959 where we use the definition of the conditional probability and the tower property in the third equality. The second statement follows by μ(S,Xj) P=μ (πF,πI j ) P(˜ S,˜ X) =P(˜ S,˜ X)for all j∈N. Proposition A.2 Let j ∈Nand νPbe a measure on (, G)which fulfills (3.7), i.e., ν (πF,πI j ) P=P for all j ∈N. Then it holds for all A ∈FIthat EνP[1{πI j ∈A}|F] =EP[1{˜πI∈A}|˜ F]◦(πF,π I )ν P-a.s. forall A ∈FIand ∈N.(A.1) Proof Let B∈Fand A∈FI. Then we have 1B1{πI j∈A}dνP=(1πF(B)◦πF)(1A◦πI j)dνP =(1πF(B)◦˜πF)(1A◦˜πI)dP =(1πF(B)◦˜πF)EP[(1A◦˜πI)|˜ F]dP =1BEP[(1A◦˜πI)|˜ F]◦((πF,π I ))dνP, where we use (3.7) in the second equality and ˜π−1 F(πF(F)) ∈˜ Fin the third equality. Proposition A.3 The projections (πI j)j∈Nare F-conditionally iid under μPand consequently also the insurance benefits (Xj)j∈Nare F-conditionally iid under μP. Proof We must show that for every finite subset J⊂Nand Aj∈FIfor all j∈J that EμP⎡ ⎣! j∈J 1{πI j ∈Aj} F⎤ ⎦=! j∈J EμP1{πI j ∈Aj} F. 123
960 K. Oberpriller et al. This follows because for every B∈F, it holds that EμP⎡ ⎣1B! j∈J 1{πI j ∈Aj}⎤ ⎦=EP⎡ ⎣1˜π−1 F(πF(B)) ! j∈J P˜πI|˜ F(Aj)⎤ ⎦ =EP⎡ ⎣1˜π−1 F(πF(B)) ! j∈J EP1{˜πI∈Aj}|˜ F⎤ ⎦ =EμP⎡ ⎣1B! j∈J EP1{˜πI∈Aj}|˜ F]◦(πF,π I 1)⎤ ⎦ =1B! j∈J Eμp[1{πI j ∈Aj}|F]dμP, where we use the definition of μPgivenin(3.9) in the first equality and Proposition A.2 in the fourth equality. Proposition A.4 The measure μPdefined by (3.9)is the unique measure on (, G) which fulfills (3.7)and (3.8). Proof Using Proposition A.1 and Proposition A.3 the measure μPfulfills (3.7) and (3.8). Moreover, let νPbe a measure on (, G)which also satisfies (3.7) and (3.8) (where μPis replaced by νP). Let J⊂Nbe a finite subset and A∈FFand Cj∈FI for all j∈J. Then for Dgiven by D=π−1 F(A)∩$ j∈J π−1 I j (Cj)(A.2) we get the following νP(D)=EνP1π−1 F(A)EνP! j∈J 1πI j ∈Cj|F =EνP1π−1 F(A)! j∈J EνP1πI j ∈Cj|F =EνP1π−1 F(A)! j∈J EP1˜πI∈Cj|˜ F◦(πF,π I 1) =EP1˜π−1 F(A)! j∈J EP1˜πI∈Cj|˜ F, whereweuse(3.8) in the third equality and Proposition A.2 in the fourth equality. The same calculations can be done for μP. The sets in (A.2)forma∩-stable generator of Gand thus we obtain μP=νP. This demonstrates the uniqueness. 123
Robust insurance-finance arbitrage... 961 Proposition A.5 The set Me(F)of all measures on (, F)such that S is a Fmartingale and which are equivalent to μP0is given by Me(F)={μQ|Q∈Me(˜ F)}. Proof We show that for all t,s∈{0,...,T}it holds that EμQ[St|Fs]=EQ[˜ St|˜ Fs]◦(πF,π I 1). (A.3) Let Bs∈Fs,then we have EμQ[1BsSt]=EμQ1Bs˜ St◦(πF,π I 1) =EQ1˜π−1 F(πF(Bs)) ˜ St =EQ1˜π−1 F(πF(Bs)) EQ[˜ St|˜ Fs] =EμQ1BsEQ[˜ St|˜ Fs]◦(πF,π I 1) (A.4) Thus, it follows (A.3) and so we can conclude that Q∈Me(˜ F)implies μQ∈Me(F). The equivalence of μP0and μQfollows from the equivalence of P0and Q.Forthe other inclusion, let R∈Me(F). We now show that there exists Q∈Me(˜ F)such that R=μQ. Define Qby Q:= RπF(˜πF(·)). Then, by changing the roles of πFand ˜πFthe result follows using (A.4). Proposition A.6 For any P ∈Pand Q ∈Me(˜ F)we have the following: EQEP[˜ X|˜ F]=EμQ[EμP[X1|F]], EQess sup P∈P EP[˜ X|˜ F]=EμQess sup P∈P EμP[X1|F]. Proof By using the same arguments as in the proof of Proposition 3.10, we find that EμP[X1|F]=EP[˜ X|˜ F]◦(πF,π I 1). This implies that EμQEμP[X1|F]=EμQEP[˜ X|˜ F]◦(πF,π I 1)=EQEP[˜ X|˜ F]. The second statement follows analogously. 123
962 K. Oberpriller et al. Funding Open Access funding enabled and organized by Projekt DEAL. 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/. References 1. Artzner P, Eisele K-T, Schmidt T (2024) Insurance-finance arbitrage. Math Financ. 34(3):739–773 2. Barigou K, Linders D, Yang F (2023) Actuarial-consistency and two-step actuarial valuations: a new paradigm to insurance valuation. Scand Actuar J 2:191–217 3. Bartl D (2020) Conditional nonlinear expectations. Stoch Process Appl 130(2):785–805 4. Biagini F, Zhang Y (2019) Reduced-form framework under model uncertainty. Ann Appl Probab 29(4):2481–2522 5. Biagini S, Bouchard B, Kardaras C, Nutz M (2017) Robust fundamental theorem for continuous processes. Math Financ 27(4):963–987 6. Bouchard B, Nutz M (2015) Arbitrage and duality in nondominated discrete-time models. Ann Appl Probab 25(2):823–859 7. Deelstra G, Devolder P, Gnameho K, Hieber P (2020) Valuation of hybrid financial and actuarial products in life insurance by a novel three-step method. ASTIN Bull 50(3):709–742 8. Denis L, Martini C (2006) A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann Appl Probab 16(2):827–852 9. Denis L, Hu M, Peng S (2010) Function spaces and capacity related to a sublinear expectation:application to G-Brownian motion paths. Potent Analy 34:139–161 10. Dhaene J, Kukush A, Luciano E, Schoutens W, Stassen B (2013) A note on the (in-)dependence between financial and actuarial risks. Insur Math Econ 52(3):522–531 11. Dhaene J, Stassen B, Barigou K, Linders D, Chen Z (2017) Fair valuation of insurance liabilities: merging actuarial judgement and market-consistency. Insur Math Econ 76:14–27 12. Dybvig P (1992) Hedging non-traded wealth: when is there separation of hedging and investment. Opt Recent Adv Theory Pract 2:13–24 13. Föllmer H, Knispel T (2011) Entropic risk measures: coherence vs convexity, model ambiguity and robust large deviations. Stoch Dyn 11(02n23):333–351 14. Föllmer H, Schied A (2016) Stochastic finance. de Gruyter 15. Kabanov YM, Kramkov DO (1995) Large financial markets: asymptotic arbitrage and contiguity. Theory Probab Appl 39(1):182–187 16. Kallenberg O (1997) Foundations of modern probability, vol 2. Springer, Berlin 17. Klein I (2000) A fundamental theorem of asset pricing for large financial markets. Math Financ 10(4):443–458 18. Klein I, Schmidt T, Teichmann J (2016) No arbitrage theory for bond markets. Advanced modelling in mathematical finance. Springer, Berlin, pp 381–421 19. Linders D (2023) The 3-step hedge-based valuation: fair valuation in the presence of systematic risks. ASTIN Bull 53(2):418–442 20. Majerek D, Nowak W, Zieba W (2005) Conditional strong law of large number. Int J Pure Appl Math 20(2):143–156 21. Malamud S, Trubowitz E, Wüthrich MV (2008) Market consistent pricing of insurance products. ASTIN Bull 38(2):483–526 22. Nelsen RB (2007) An introduction to copulas. Springer, Berlin 23. Pelsser A, Stadje M (2014) Time-consistent and market-consistent evaluations. Math Financ 24(1):25– 65 123
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