No arbitrage in insurance and the QP-rule
aa r X i v : . [ q -f i n . M F ] M a y NO ARBITRAGE IN INSURANCE AND THE QP-RULE
PHILIPPE ARTZNER, KARL-THEODOR EISELE, AND THORSTEN SCHMIDT
Abstract.
This paper is an attempt to study fundamentally the valuation of insurance contracts.We start from the observation that insurance contracts are inherently linked to financial markets, beit via interest rates, or – as in hybrid products, equity-linked life insurance and variable annuities –directly to stocks or indices. By defining portfolio strategies on an insurance portfolio and combiningthem with financial trading strategies we arrive at the notion of insurance-finance arbitrage (IFA). Afundamental theorem provides two sufficient conditions for presence or absence of IFA, respectively.For the first one it utilizes the conditional law of large numbers and risk-neutral valuation. As a keyresult we obtain a simple valuation rule, called QP-rule, which is market consistent and excludesIFA.Utilizing the theory of enlargements of filtrations we construct a tractable framework for generalvaluation results, working under weak assumptions. The generality of the approach allows to incor-porate many important aspects, like mortality risk or dependence of mortality and stock marketswhich is of utmost importance in the recent corona crisis. For practical applications, we provide anaffine formulation which leads to explicit valuation formulas for a large class of hybrid products.
Keywords: arbitrage, fundamental theorem of asset pricing, insurance valuation, variable an-nuities, hybrid products, equity-linked life insurance, enlargement of filtration, affine processes,longevity risk, corona crisis, QP-rule. Introduction
This paper is a first step into the direction of defining and finding arbitrage-free premiums ofinsurance products. We start from the observation that insurance contracts are inherently linked tofinancial markets, be it via interest rates, or via direct links of the contractual benefits to stocks orindices. Insurance companies hence naturally form a portfolio of insurance contracts for pooling risksand combine them with trading strategies on financial markets. We relate insurance arbitrage to thesize of premiums and possible insurance-finance arbitrages.1.1.
Effect of institutional differences.
If it is true that a share, a call option and an insurancecontract all qualify as securities defined as the legal representation of the right to receive prospective fu-ture benefits under stated conditions , see Sharpe et al. (1999), the environment of current trade varieswith the security to be sold or bought. While the share is anonymous in terms of buyer and seller, a call carries the information of the party who wrote it, an insurance contract can be sold only by anoffice and to a person who explicitly appears in the contract, see for example Boudreault and Renaud(2019). In contrast to financial markets, participants in primary insurance markets are a priori sepa-rated into two groups: potential sellers (insurance companies) who are to provide contracts contingentto insurance events and potential buyers (insurance seekers) who will provide premiums as currentcash. Secondary markets for insurance are rare and therefore not considered here.1.2.
Necessity of a definition of insurance arbitrage.
The definition of (financial) arbitrageas the simultaneous purchase and sale of the same, or essentially similar, security in two differentmarkets for advantageously different prices , see Sharpe et al. (1999), relies on despite whatever financerelated event happens -type statements and may require the insurance seeker to act as a seller ofcontracts, a rather rarely allowed case. Hence we have to find an appropriate definition of insurancearbitrage as a (first) step towards valuation. It is natural to try to express arbitraging insurancepayments (cash in-flows and out-flows) by looking at the with probability one insurance related events
Date : May 29, 2020.The authors gratefully acknowledge the financial support from the Freiburg Institute for Advanced Studies(FRIAS) and the University of Strasbourg Institute of Advances Studies (USIAS) within the FRIAS-USIAS ResearchProject ‘Linking Finance and Insurance: Theory and Applications (2017–2019). Support from the DFG is gratefullyacknowledged. resulting from the strong law of large numbers. Notice that we face an asymmetry between seller(s)and buyers of insurance, as insurance contracts are securities which are not liquidly traded receivables.To deal with them we first construct a particular linear measure, which values financial securities byrisk-neutral pricing while using the private view of insurance claims.From a more fundamental viewpoint, we aim at judging a candidate premium p for a consideredinsurance contract. To this end, we first define absence of insurance arbitrage starting from p . Then,we introduce additional trading on a financial discrete-time market by relying on classical self-financedtrading strategies and the combination leads to our notion of insurance-finance arbitrage (IFA). As anext step, we show that bounding p suitably from above guarantees the absence of insurance arbitrage(NIFA) which we find a reasonable condition to the status of valuation. To the best of our knowledgethis requirement is new in the literature.The mathematical foundation of our approach lies in considering two filtrations: the publiclyavailable information and the (additional) private information of the insurance company. We thereforeutilize the theory of enlargements of filtration to provide tractable environments for the valuation ofgeneral insurance claims. In an immersion setting we constitute a general affine framework whichis highly tractable and allows for the valuation of insurance products related to financial marketsexcluding insurance-finance arbitrage.Since our setting is very general, we are able to generalize over existing approaches: first, weconsider a financial market without restricting to the complete case. Second, we allow for arbitrarydependence between the financial market and the insurance quantities - this is important in manyinsurance products, for example when considering surrender behavior or stochastic mortality. Therecent corona outbreak highlights the necessity of allowing dependence between mortality and stockmarkets. Third, we provide a fundamental justification of our valuation rule by relying on the(conditional) strong law of large numbers on the insurance side and on risk-neutral pricing on thefinancial side. Finally, we provide a general affine framework for insurance products related to financialmarkets. Among many other effects, this allows to incorporate longevity risk via stochastic mortalityrates.1.3. Related literature.
The valuation of insurance contracts is a problem which has been inten-sively studied in the literature, see for example W¨uthrich and Merz (2013), Asmussen and Albrecher(2010) and references therein. The existing approaches to the valuation of insurance products linkedto financial market can be divided into five classes.First, it is common in practice to value directly under some ad-hoc chosen risk-neutral measure , seefor example Brennan and Schwartz (1976), Dai et al. (2008), Krayzler et al. (2015), Cui et al. (2017)and references therein. This approach is quite popular, as it often leads to explicit results in a directmatter.Second, the benchmark approach with its application to the problem at hand introduced inB¨uhlmann and Platen (2003) and its application to financial markets from Platen (2006). It usesthe growth-optimal portfolio (GOP) as num´eraire and evaluates products by taking expectationsunder P , see e.g. Biagini et al. (2015) for an insurance application.Third, the local risk-minimization approach , specifying the risk-neutral measure on the insuranceside with a risk-minimizing procedure, see for example Møller (2001), Pansera (2012). It is assumedin these approaches that insurance risks, like mortality risk, can be diversified away.Fourth, indifference pricing leads to a non-linear pricing rule and we refer to Blanchet-Scalliet et al.(2015), Chevalier et al. (2016) for details and further literature. Finally, there are valuation method-ologies utilizing risk measures, often based on an axiomatic view, see Tsanakas and Desli (2005),Pelsser and Stadje (2014), or on hedging, see Chen et al. (2020).Some insurance products we have in mind are variable annuities, which explicitly link the insurancebenefits to the performance of financial markets. We refer to Bacinello et al. (2011) for an extensiveoverview of related literature.1.4. Structure of the paper.
In Section 2 we introduce a linear valuation rule for insurance claimswhich we call QP-rule. This rule combines risk neutral pricing (under Q ) with actuarial valuation(under P ). In Section 3 we provide a fundamental theorem which relates NIFA on insurance-financemarkets to a majorization of premiums. The QP-rule arises as a natural valuation tool excluding IFA,see in particular Remarks 3.8 and 3.10. In Section 4 we introduce the connection to enlargements O ARBITRAGE IN INSURANCE 3 of filtration and in Section 5 we provide an affine framework. Section 6 discusses longevity risk andcomputes values explicitly for some examples. Section 7 concludes the paper.2.
Valuation of non-traded wealth
For insurance companies, the valuation of non-traded wealth is a core topic: indeed, as mentionedin the introduction, insurance contracts are not traded. In addition, they have direct links to finan-cial markets (interest rates, variable annuities, equity-linked life insurance etc.), hence a consistentvaluation rule linking traded assets with non-traded insurance contracts is of highest importance.In a first step, this leads to the following measure theoretic problem: given two σ -algebras F Ă G on a set Ω, a probability Q on p Ω , F q , and a probability P on p Ω , G q , how should we extend Q to p Ω , G q in a P -reasonable way. We assume that Q and P | F are equivalent; here P | F denotesthe restriction of P to F . In Section 2.2 we extend this problem to the dynamic setting with twofiltrations.2.1. Extending a measure to a larger σ -algebra. The following proposition gives a uniqueextension of Q with P . Proposition 2.1.
There is a unique probability measure, denoted by Q d P , on p Ω , G q such that Q d P “ Q on F and for all G P G it holds that Q d P p G | F q “ P p G | F q . This measure Q d P satisfies for any random variable X ě that(i) for a σ -algebra H Ă F , E Q d P r X | H s “ E Q “ E P r X | F s| H ‰ , (ii) for a σ -algebra H satisfying F Ă H Ă G , E Q d P r X | H s “ E P “ X | H ‰ . Letting H “ tH , Ω u , we obtain from (i) the one-period QP-rule : E Q d P r X s “ E Q “ E P r X | F s ‰ . (1) Proof.
We construct the measure Q d P as follows: let L denote the Radon-Nikodym derivative of Q with respect to P | F , such that dQ “ LdP . Define Q d P on p Ω , G q by d p Q d P q “ LdP. (2)such that Q d P p G q “ E P r G L s . Obviously Q d P and Q coincide on F by construction. Moreover,for G P G we have that ż F G d p Q d P q “ ż F G LdP “ ż F LP p G | F q dP “ ż F P p G | F q d p Q d P q , F P F , (3)such that Q d P p G | F q “ P p G | F q . In view of (i), we obtain with H P H Ă F , that H P F , and italso follows that Q d P p G | H q “ P p G | H q . Moreover, letting H “ Ω we obtain ż G d p Q d P q “ ż LP p G | H q dP “ ż P p G | H q dQ (4)and an application of the monotone class theorem yields (i).For uniqueness, consider a further measure R , such that R “ Q on F and such that R p G | F q “ P p G | F q . Then, for G P G it holds that R p G q “ ż R p G | F q dR “ ż P p G | F q dR “ ż P p G | F q dQ “ Q d P p G q (5)and we obtain that Q d P “ R .Finally, in view of (ii) we observe that by Bayes’ rule, it follows from F Ă H Ă G that E Q d P r X | H s “ E P r L T X | H s E P r L T | H s “ E P r X | H s , since L T is F -measurable and hence H -measurable. (cid:3) ARTZNER, EISELE, AND SCHMIDT
The idea of choosing the Radon-Nikodym derivative on the smaller σ -algebra as density on thelarger probability space may already be found in Plachky and R¨uschendorf (1984) who are studyingmeasure extensions in a statistical context. In the financial context, already Dybvig (1992) studiedthis extension in the framework of state-price densities.For the application we have in mind, the valuation of a non-traded insurance contract affectedby financial markets has to be solved. In this case, the σ -algebra F consists of publicly availableinformation on financial markets, and in particular on the traded assets. The larger σ -algebra G contains private information available to the insurance company and the non-traded insurance liabilityis considered as a random variable on the larger probability space p Ω , G q .We assume that the financial market p F , S q is free of arbitrage with respect to publicly availableinformation F (in particular trading strategies adapted to F are being used). This is equivalent toexistence of a risk-neutral measure Q on p Ω , F q . Hence, Proposition 2.1 applies and allows, in thespirit of Dybvig (1992), to value the untraded insurance liability X : equation (1), which we call QP-rule , gives a value of X which does exclude financial arbitrage. We will also show that the QP-ruleis market-consistent in Section 2.3. Example 2.2 (A discrete example) . Consider a one-period model with three states on the financialmarket: good, medium, and bad ( g, m, b ). We assume two states on the insurance side: payment andno payment ( p, ¯ p ), corresponding toΩ “ tp g, p q , p g, ¯ p q , p m, p q , p m, ¯ p q , p b, p q , p b, ¯ p qu . Denote the good, medium and bad states by g “ tp g, p q , p g, ¯ p qu , m “ tp m, p q , p m, ¯ p qu , and b “tp b, p q , p b, ¯ p qu . The information on the financial market is given by F “ σ ` g, m, b ˘ , while G “ P p Ω q .The risky asset S satisfies S “ S p g q “ . S p m q “
1, and S p b q “ .
5. The num´eraire isset constant and equal to 1.We assume that the measure P is given by p . h, . h, . k, . k, . l, . l q with h ` k ` l “ h, k, l P r , s . Complete market: if k “ h, l P p , q , the financial market is binomial and therefore complete.The unique risk-neutral measure is given by Q p g q “ { .For the application of (1) we compute the conditional probabilities P p¨ | g q “ p . , . , , , , q ,P p¨ | b q “ p , , , , . , . q . (6)The measure Q d P is given as the convex combinationQ d P p A q “ t g u ¨ Q p g q ¨ P p A | g q ` t b u ¨ Q p b q ¨ P p A | b q , s.t. Q d P equals p . , . , , , . , . q , independently of the value of h and l . Incomplete market: if h, k, l P p , q , the market is incomplete. The set of risk-neutral measuresis given by Q s p g q “ s , Q s p m q “ ´ s , and Q s p b q “ s , with s P p , { q . In addition to (6) we have P p¨ | m q “ p , , . , . , , q . We obtain Q s d P p A q “ t g u ¨ Q s p g q ¨ P p A | g q ` t m u ¨ Q s p m q ¨ P p A | m q ` t b u ¨ Q s p b q ¨ P p A | b q , which gives Q s d P as p . s, . s, . p ´ s q , . p ´ s q , . s, . s q . ♦ The case of two filtrations.
In this section we extend Proposition 2.1 to multiple periods.The time space T “ t , , . . . , T u is considered to be discrete and finite with T ă 8 . We consider afiltration F “ p F t q t P T such that F T Ă G .We start with a stylized example which illustrates the most important feature of the insurancecontracts we intend to valuate. In this regard, we call a stochastic process p ˜ A t q t P T a payment flow , if˜ A t ě t P T . Let p ˜ S t q t P T ą A t “ ˜ A t { ˜ S t , t P T theassociated discounted payment flow. Negative cash flows can be incorporated by considering positiveand negative parts separately. Example 2.3 (Stochastic mortality) . Consider an insurance contract whose payments may be sep-arated into two components: on the one side a random time τ and on the other side a discountedpayment flow A ; the insurance offers the payment of ˜ A t at t if τ “ t . We assume that the paymentflow A is F -adapted. The random time τ can be used to model the random life time in the case All our results hold also for T “ 8 under the condition that there exists a martingale measure Q on p Ω , F q . O ARBITRAGE IN INSURANCE 5 of a life insurance, or the time of surrender in a variable annuity or the time of a catastrophe inCAT-bonds.Consider an F -adapted, increasing process Λ “ p Λ t q t P T with Λ “
0. Assume that there exists astandard exponential random variable E which is independent of the financial market (i.e. independentof F “ F T ) and let τ “ inf t t P T : Λ t ě E u , (7)with the convention that inf H “ T . Then, τ is not an F -stopping time, but it is a stopping time inthe progressive enlargement given by G t “ F t _ p τ ^ t q , t P T . Let G “ F T _ τ .We are interested in the market-consistent valuation of the insurance contract. Its value at time 0computes to ÿ t P T E Q d P r A t t t “ τ u s . (8)Using Proposition 2.1 and exploiting the independence between E and F we obtain thatQ d P p τ ą t | F q “ P p τ ą t | F q “ P p Λ t ă E | F q “ e ´ Λ t “ : G t , (9)and, hence, P p τ “ t | F q “ G t ´ ´ G t . By (1) we obtain that E Q d P r A τ s “ ÿ t P T E Q r A t P p τ “ t | F qs“ ÿ t P T E Q r A t p G t ´ ´ G t qs and the remaining computations can be done as usual in a risk-neutral manner. Stochastic mortality is achieved in this example by the (random) cumulated intensity processΛ. This is important for the modeling of longevity risk, see Section 6.3, or to incorporate depen-dencies between mortality and the stock market. The recent outbreak of the corona virus and therelated infections had a dramatic impact on the stock market together with increased mortality; seeFeng and Garrido (2011) for incorporating pandemic modeling into an actuarial context. The settingconsidered in this example allows to model this. ♦ The example above has numerous applications in insurance: any insurance contract whose payoffspaid at t when τ “ t and which are depending only on publicly available information (interest rates,stock markets, etc.). From Proposition 2.1 we obtain that for 0 ď t ď T , E Q d P r X | F t s “ E Q “ E P r X | F T s| F t ‰ , (10)for any X P L or bounded from below.While it might in general be difficult to compute the conditional expectations in Equation (10),under additional assumptions like immersion one is able to obtain tractable results. We exploit thisin Section 4.2.3. Market-consistent evaluation.
In this section we briefly discuss the important property ofmarket-consistency, see for example Cheridito et al. (2008), Artzner and Eisele (2010), Pelsser and Stadje(2014). Using equation (10) we extend (1) and obtain the
QP-rule for the insurance contract intro-duced in Example 2.3: the value at time t of the discounted payment flow A on p t, T s isΠ t p A q : “ T ÿ s “ t ` E Q d P r A s t τ “ s u | F t s . (11)The QP-rule ensures market-consistent evaluation , a necessary requirement under Solvency II:a valuation rule Π “ p Π t q t P T is a family of mappings with Π t : L p G q Ñ L p F t q . Here L p G q denotes the space of G -measurable random variables, and L p F t q the space of F t -measurable randomvariables, respectively. The valuation at time t , Π t , maps a discounted payoff H , i.e. a G -measurablerandom variable, to its value at time t , i.e. to an F t -measurable random variable. Such a construction is called doubly stochastic . We refer to Bielecki and Rutkowski (2002), Br´emaud (1981),Gehmlich and Schmidt (2018) for details and references. If Λ t “ ş t λ s ds , then λ is called the intensity of τ . If theintensity is deterministic, it is called hazard rate . For further details see Section 4. For the moment we simply apply Proposition 2.1, fundamental justifications will follow in Sections 2.3 and 3.
ARTZNER, EISELE, AND SCHMIDT
The valuation rule Π is called market-consistent if the following holds: the value of a discountedfinancial payoff of the form H ` H consisting of a financial part H P L p F T q and an actuarial part H P L p G q with H and H being bounded satisfiesΠ t p H ` H q “ E Q r H | F t s ` Π t p H q (12)for some Q P M e p F q . It follows immediately that the QP-rule and, in particular, the valuation ruleΠ given in (11), is market-consistent.In addition to market consistency, the fundamental valuation theorem proved in the following sec-tion will show that the QP-rule excludes financial and insurance arbitrages. For the precise statementwe refer to Theorem 3.6 and Remark 3.10.The approach in Chen et al. (2020) introduces a concept of valuation, called fair , if it is marketconsistent and in addition coincided with valuation under P for payoffs which are independent of thefuture stock evolution (called t -orthogonal). The QP-rule satisfies also this property.3. Arbitrage in Insurance
In this section we start from a valuation rule p , like in Section 2.3, and study whether p is indeedfit for the job. For this we introduce the notion of insurance arbitrage . Insurance arbitrage is definedunder the assumption that the insurer has the possibility to contract with a large number of clientsso that he may reduce risk substantially. If this leads to a profit without the risk of loosing money itwill be called, in analogy to financial markets, an insurance arbitrage.In addition we will allow for trading in a finite number of financial assets which will possibly leadto financial arbitrage and a combination of both approaches will be called insurance-finance arbitrage (IFA).We use the QP-rule of Section 2, and prove that certain inequalities between p and the QP-valueensure the absence of insurance-finance arbitrage.3.1. Definition of the insurance-finance arbitrage.
In this paper we consider only the case ofa single insurer. We assume that he can contract with possibly infinitely many insurance seekers .Insurance contracts are always of the form that a future claim is covered in exchange of a singlepremium paid at initiation of the insurance contract.The insurance company has additional information (e.g. survival times or health state) which wedescribe by the filtration G . Since the insurance company has also access to the publicly availableinformation F , we assume that F t Ă G t for all t P T .3.1.1. The insurance contracts.
At each time t P t , , ..., T ´ u , the insurer has the possibility toissue contracts of the same type (depending on t ). Each insurance contract exchanges (future) benefitsagainst a premium paid at initiation . Without loss of generality, we assume that all claims are settledat the maturity date T . The benefits of the insurance contract issued at time t are described by a G T -measurable non-negative random variable X t,T (already discounted). This could be claims arisingfrom life insurance in the time interval p t, T s or from a variable annuity. The family X “ p X t,T q ď t ă T is called the family of benefits .The premium of the insurance contract X t,T issued at time t is denoted by p t (already discounted);it is G t -measurable. We set p “ p p t q ď t ă T . The value p t has to be regarded as a basic part of premium at time t for the risk X t,T . On top, the insurance company adds a commercial part which containsadditional costs, risk margins, etc. Risk margins are enforced by regulation and are non-linear rules:these topics will be treated with in a forthcoming paper.To minimize risk on the insurance side, the insurer will exploit the possibility to form a portfolio ofinsurance contracts out of a population of insurance seekers . In this regard, we assume that at eachtime t the insurance can contract with possibly infinitely many insurance seekers. The associated G T -measurable benefits are denoted by X t,T , X t,T , . . . It will turn out that the σ -algebra G t,T “ G t _ F T containing the insurance information up totime t and all the publicly available information F T plays a distinctive role. We make the followingassumptions which generalize the classical framework of actuarial mathematics to stochastic assetsand are supposed to correspond to the actual structure of the concrete commitments agreed upon bysellers and buyers of contracts: See section 3.3 on how to deal with scheduled premiums
O ARBITRAGE IN INSURANCE 7
Assumption 3.1.
For each t P T ,(i) X t,T , X t,T , . . . are G t,T -conditionally independent,(ii) E r X it,T | G t,T s “ E r X t,T | G t,T s , i “ , , . . . , and(iii) Var p X it,T | G t,T q “ Var p X t,T | G t,T q ă 8 , i “ , , . . . . Inspired by large financial markets (see e.g. Klein (2000), Kabanov and Kramkov (1995), Klein et al.(2016)), we model the insurance portfolio as a limit of allocations of contracts with finitely many insur-ance seekers: an allocation ψ t “ p ψ it q i ě for time t is a finite-dimensional G t -measurable, non-negativerandom variable, i.e. there exists an m such that ψ it “ i ě m . Here, ψ it P R ` denotes the sizeof the contract with the i th insurance seeker. In contrast to a financial trading strategy there isno rebalancing: if the insurance chooses an allocation at time t , then it is entailed to pay to thetime- t -customers the associated claims at maturity.This, together with the assumptions of homogeneity of premiums and benefits, induces the followingnotions: while the commitment of the insurer with allocation ψ t at date t is given by the accumulatedbenefits, i.e. ÿ i ě ψ it X it,T , its associated value , given by the accumulated premiums, equals ÿ i ě ψ it p t . The profits and losses (P&L) of the allocation sequence ψ “ p ψ t q t ă T amounts to the accumulatedpremiums minus the accumulated benefits. Summing up over all time instances t “ , . . . , T ´ ψ , which we denote by V IT p ψ q : “ T ´ ÿ t “ ÿ i ě ψ it ` p t ´ X it,T ˘ . (13)Finally, an insurance portfolio strategy ψ is a sequence p ψ n q n ě of (finite) allocations. We imposethe following admissibility conditions :(i) uniform boundedness : there exists C ą
0, such that k ψ nt k : “ ÿ i ě ψ n,it ď C (14)for all n ě ď t ă T ,(ii) convergence of the total mass : there exists 0 ă γ t P F t , such that k ψ nt k Ñ γ t a.s. for all t ă T, (15)(iii) convergence of the total wealth : there exists a real-valued random variable V “ V ψ , such thatlim n Ñ8 V IT p ψ n q “ V,P -almost surely.Let us remark that condition (14) implies uniform integrability for the sum of the G t,T -conditionallosses: indeed, if p A m q is a decreasing sequence of G t,T -measurable sets with A m Œ H , then E ” A m ÿ t ă T ÿ i ě ψ n,it E “ X it,T | G t,T ‰ı “ ÿ t ă T E „ ÿ i ě ψ n,it E “ A m X it,T | G t,T ‰ ď C ÿ t ă T E “ A m X t,T ‰ Ñ m Ñ 8 , uniformly in n .The assumption on γ t P F t is weak: indeed, we often will use a total mass of γ t “ P F t andconsider allocations satisfying k ψ nt k “
1. These allocations distribute 1 in a G t -measurable waybetween the insurance seekers. The assumption on convergence could be replaced by weaker ones,see Remark 3.4. ARTZNER, EISELE, AND SCHMIDT
The financial market.
The financial market consists of d ` F -adaptedprice process ˜ S “ p ˜ S , ˜ S , . . . , ˜ S d q . Traded assets could be bonds with and without credit risk, stocks,indices, etc. The information F will typically be strictly larger than the filtration generated by thetraded assets: economic variables like employment rates or national mortality rates are examples ofsuch publicly available information.The num´eraire ˜ S with ˜ S “ S “ p S , . . . , S d q where S i “ ˜ Si { ˜ S , i “ , . . . , d , and S ” d -dimensional, F -adapted process ξ “ p ξ t q ď t ď T ´ with ξ t “ p ξ t , . . . , ξ dt q . For a self-financing trading strategy ξ (see for example F¨ollmer and Schied(2011), Proposition 5.7), its P&L until T is given by V FT p ξ q : “ T ´ ÿ t “ ξ t ¨ ∆ S t “ T ´ ÿ t “ d ÿ i “ ξ it ¨ ∆ S it , with ∆ S t “ S t ` ´ S t . We assume that the market does not allow arbitrage so that there exist atleast one equivalent martingale (or market-consistent) measure, i.e. M e p F q ‰ H . (17)We emphasize that we do not assume completeness of the financial market p F , S q and that we allowfor (financial) arbitrages when trading is done with the filtration G . However, we assume that tradingof the insurance company on the financial market is only done with respect to the public informationin F , thus coping with the law against insider-trading . On the other side, the insurance company ofcourse uses the information G for the valuation and the allocation on the insurance contracts.Summarizing, the insurance-finance market p X, p, S q consist of the three ingredients: the benefits X , the premiums p and the financial securities prices S . Definition 3.2. On p X, p, S q we have an insurance-finance arbitrage , if there exist an admissibleinsurance portfolio strategy p ψ n q n ě , and a self-financing trading strategy ξ such thatlim n Ñ8 V IT p ψ n q ` V FT p ξ q P L ` zt u . (18)Otherwise, we say that there does not exist an insurance-finance arbitrage (NIFA) .It seems to be worthwhile pointing out again the asymmetry between the non-negative G -adaptedprocesses ψ n and the F -predictable strategy ξ without sign restrictions. Remark 3.3 (On the role of the information) . Depending on the application we will exploit theflexibility of choosing F in a suitable manner. In particular, it may be useful to assume that F contains also insurance information, like life tables or general factors which drive the evolution. Remark 3.4 (On the convergence assumption) . It may be noted that the existence of the limesin (18) is guaranteed by the third admissibility condition, convergence of the total wealth. Thisassumption could be weakened by replacing the limit in (18) by a limes inferior and then applyingour results to the insurance portfolio strategy converging to this lower limit.At the current stage, we provide a simple example which may serve as basis building block for avariable annuity. The valuation of a variable annuity will be described in detail in Section 6.
Example 3.5 (A simple example) . Consider d “ S “ S the price of the stock. Let τ denote the lifetime of the insured person and consider the benefits X t,T “ t τ ą T u S T , i.e. a contractpromising S T given survival. Now if we have conditional independent copies of t τ ą T u with identicalmean, assumed to satisfy P p τ ą T | F T q “ p p S T q , averaging by the strong law of large numbers (tobe made precise below) leads to the F T -conditional expectation p p S T q S T , which serves as a basis for valuation of this claim. Note that p p S T q S T is F T -measurable and henceturns out to be a European contingent claim which can be valued by standard risk-neutral valuation.In this regard, for valuing this product we would consider F to be generated by S while theinsurance information G is the progressive enlargement of F with τ . ♦ O ARBITRAGE IN INSURANCE 9
Theorem on the absence of insurance-finance arbitrage.
We begin by introducing certainupper bounds. Since the premium p t is G t -measurable, already the knowledge of p t for an insuranceseeker might offer arbitrage possibilities for her on the financial market. To avoid this, we introducethe F t -conditional essential supremum (infimum) of p t : for a family Ξ of random variables we define ess sup F Ξ : “ ess inf t q | q is F -measurable and q ě ξ for all ξ P Ξ u . (19)Furthermore, denote p Ò t “ ess sup F t p t , and p Ó t “ ess inf F t p t . (20)Our main goal is the following fundamental theorem about the absence of insurance-finance arbi-trage. Theorem 3.6.
Let the insurance-finance market be described by p X, p, S q and assume that Assump-tion 3.1 holds.(i) If there exists Q P M e p F q such that for all t ă Tp t ď E Q d P r X t,T | F t s , P -a.s. , (21) then there is no insurance-finance arbitrage (NIFA).(ii) If there exists t ă T , such that P ˆ č Q P M e p F q p Ó t ą E Q d P “ X t,T ˇˇ F t ‰)˙ ą then there exists an insurance-finance arbitrage (IFA). For the proof of Theorem 3.6 we need some preliminary results.
Proposition 3.7.
Under the Assumption 3.1 and the admissibility conditions (14) and (15) we havefor all admissible allocation ψ and all Q P M e p F q that E Q d P ” lim n Ñ8 ÿ i ě ψ n,it p t ı “ E Q d P r γ t p t s , for all t ă T and (23) E Q d P ” lim n Ñ8 T ´ ÿ t “ ÿ i ě ψ n,it X it,T ı “ ÿ t ă T E Q d P “ γ t X t,T ‰ . (24) Proof.
Indeed, (23) follows from the convergence of the total mass in (15) together with dominatedconvergence. Recall that V IT p ψ n q “ ř t ă T ř i ě ψ n,it X it,T . As in Proposition 2.1, we denote theRadon-Nikodym density of Q with respect to P | F T by L T . Then, E Q d P ” lim n Ñ8 V IT p ψ n q ı “ E P ” L T lim n Ñ8 V IT p ψ n q ı “ E P ” L T E P “ lim n Ñ8 V IT p ψ n q| G t,T ‰ı . For any G P G t,T , we have ż G E P “ lim n Ñ8 V IT p ψ n q| G t,T ‰ dP “ ż G lim n Ñ8 V IT p ψ n q dP “ ż lim inf n Ñ8 G V IT p ψ n q dP ď lim inf n Ñ8 ż G V IT p ψ n q dP “ lim inf n Ñ8 ż G E P “ V IT p ψ n q| G t,T ‰ dP. Furthermore, E P “ V IT p ψ n q| G t,T ‰ “ ÿ t ă T ÿ i ě ψ n,it E P “ X it,T | G t,T ‰ “ ÿ t ă T ÿ i ě ψ n,it E P “ X t,T | G t,T ‰ “ ÿ t ă T E P “ X t,T | G t,T ‰ k ψ nt k . Compare Proposition 2.6 in Barron et al. (2003). In the following application Ξ will consist of only one randomvariable; we however take the freedom to define the conditional essential supremum for the more general case.
Hence, lim inf n Ñ8 ż G E P “ V IT p ψ n q| G t,T ‰ dP “ lim inf n Ñ8 ż G ÿ t ă T E P “ X t,T | G t,T ‰ k ψ nt k dP “ ż G E P “ ÿ t ă T γ t X t,T | G t,T ‰ dP, where we used uniform boundedness, (15), and dominated convergence for the last equality. Weobtain that E Q d P ” lim n Ñ8 V IT p ψ n q ı ď ÿ t ă T E Q d P “ γ t X t,T ‰ . (25)On the other side, for any G P G t,T , we have ż G E P “ lim n Ñ8 V IT p ψ n q| G t,T ‰ dP “ ż G lim n Ñ8 V IT p ψ n q dP “ ż lim sup n Ñ8 G V IT p ψ n q dP ě lim sup n Ñ8 ż G V IT p ψ n q dP “ lim sup n Ñ8 ż G E P “ V IT p ψ n q| G t,T ‰ dP. Hence, we obtain as above E Q d P ” lim n Ñ8 V IT p ψ n q ı ě ÿ t ă T E Q d P “ γ t X t,T ‰ (26)and the claim follows. (cid:3) Proof of Theorem 3.6. (i) Assume that (21) holds and that we had an insurance-finance arbitrage,i.e. lim n Ñ8 V IT p ψ n q ` V FT p ξ q P L ` zt u for some insurance portfolio strategy p ψ n q n ě “ p ψ nt q n ě ,t ă T and some financial strategy p ξ t q t ď T .First, for any Q P M e p F q , E Q d P r V FT p ξ qs “ E Q r V FT p ξ qs “ E Q d P ” lim n Ñ8 V IT p ψ n q ` V FT p ξ q ı “ E Q d P ” lim n Ñ8 V IT p ψ n q ı (27) “ E Q d P ” lim n Ñ8 T ´ ÿ t “ ÿ i ě ψ it ` p t ´ X it,T ˘ı . With the equations (23), (24) we obtain that(27) “ T ´ ÿ t “ E Q d P ” γ t ` p t ´ E Q d P r X t,T | F T s ˘ı . We consider the specific Q from (21). Then, p t ď E Q d P r X t,T | F t s , P -a.s., hence(27) ď T ´ ÿ t “ E Q d P ” γ t ´ E Q d P r X t,T | F T s ´ E Q d P r X t,T | F T s ¯ı “ , a contradiction to the assumption of an insurance-finance arbitrage.(ii) Assume that (22) holds. We set, A t : “ č Q P M e p F q p Ó t ą E Q d P “ X t,T ˇˇ F t ‰) P F t and, by assumption, P p A t q ą A t p t ě A t p Ó t ě A t ess sup Q P M e p F q E Q d P “ X t,T ˇˇ F t ‰ “ A t ess sup Q P M e p F q E Q ” E P “ X t,T ˇˇ G t,T ‰ˇˇˇ F t ı “ : π t . (28) O ARBITRAGE IN INSURANCE 11 At t we take the uniform allocation ψ nt “ A t p n ´ , ..., n ´ , , ... q over the first n insurance seekers,restricted to the set A t and ψ ns “ s ‰ t ă T . This is an admissible strategy and since ÿ i ě i Var p X it,T | G t,T q “ Var p X t,T | G t,T q ÿ i ě i ă 8 , we are entitled to apply the conditional strong law of large numbers given in Theorem 3.5 inMajerek et al. (2005). Hence with Assumption 3.1 and the fact that γ t “ ř i ě ψ n,it “ A t P F t , weget ÿ i ě ψ n,i X it,T Ñ A t E P r X t,T | G t,T s “ : H, (29) P -almost surely as n Ñ 8 . Thereforelim n Ñ8 V IT p ψ n q “ A t ` p t ´ E P r X t,T | G t,T s ˘ ě A t ` p Ó t ´ H ˘ . As π t is the conditional superhedging price of H , we obtain from Theorem 7.2 in F¨ollmer and Schied(2004) that there is a superhedging strategy ξ “ A t ξ such that A t ` π t ` T ´ ÿ s “ t ξ s ¨ ∆ S s ˘ ě H. (30)Using this financial trading strategy ξ , we find from (28) thatlim n Ñ8 V IT p ψ n q ` V FT p ξ q ě A t ` p Ó t ´ H ` T ´ ÿ s “ t ξ s ¨ ∆ S s ˘ (31)almost surely. Now, using (28) and (30), we obtain(31) ě A t ` π t ´ π t q “ . (32)For the final step we have to distinguish if the claim H is replicable or not. For the first case let B t : “ ess sup Q P M e p F q E Q “ H ˇˇ F t ‰ “ ess inf Q P M e p F q E Q “ H ˇˇ F t ‰( and assume P p A t X B t q ą
0. By assumption (22), we have p Ó t ą π t on a set of positive probability.This allows to drop equality in (32): indeed, since p Ó t ą π t with positive probability, we obtain from(31) that (31) ě A t X B t ` p Ó t ´ H ` T ´ ÿ s “ t ξ s ¨ ∆ S s ˘ ą A t X B t ` π t ´ π t q “ , (33)with positive probability. Hence, lim n Ñ8 V IT p ψ n q ` V FT p ξ q ‰ n Ñ8 V IT p ψ n q ` V FT p ξ q P L ` zt u . For the second case let B t : “ A t X ess sup Q P M e p F q E Q “ H ˇˇ F t ‰ ą ess inf Q P M e p F q E Q “ H ˇˇ F t ‰( and assume P p A t X B t q ą
0. Again, we can drop equality in (32): indeed, we obtain analogously that(31) ě A t X B t ` p Ó t ´ H ` T ´ ÿ s “ t ξ s ¨ ∆ S s ˘ ě A t X B t ` π t ´ H ` T ´ ÿ s “ t ξ s ¨ ∆ S s ˘ ě . (34)Since on B t , the no-arbitrage interval of the European claim H is a true interval, the upper boundof the conditional no-arbitrage interval, π t , already yields a (financial) arbitrage (on B t ). Hence, A t X B t ` π t ´ H ` ξ ∆ S ˘ P L ` zt u ,and, in addition,lim n Ñ8 V IT p ψ n q ` V FT p ξ q P L ` zt u . The existence of an insurance-finance arbitrage is proved. (cid:3) Compare Theorem 2.4.4. in Delbaen and Schachermayer (2006), whose proof however requires a finite Ω.
Remark 3.8 (Using the QP-rule for arbitrage-free valuation) . First, the measure Q d P constructedin Proposition 2.1 can directly be used for valuation excluding financial arbitrage, since the restrictionQ d P | F T is an equivalent F -martingale measure.Second, in the context of Theorem 3.6, the QP-rule is the main tool to exclude insurance-financearbitrage. Since it is also market-consistent as observed in Section 2.3, it comes as a candidate forthe valuation of insurance contracts, in particular for the valuation of hybrid contracts depending onthe financial market. ♦ Example 3.9 (Valuation by the QP-rule) . We continue in the setting of example 2.2: consider thedifferent insurance claims X “ t p u with p “ tp g, p q , p m, p q , p b, p qu and Y “ XC where C is the callon S at strike 0 . Complete market: if k “ h, l P p , q , then Q p g q “ { and we obtain E Q d P r tp g,p qu s “ { ¨ . “ . . Hence, E Q d P r X s “ . ` . “ .
25 and every value p ď .
25 does not allow for an insurance-financearbitrage.
Incomplete market: if k, h, l P p , q , then E Q d P r tp g,p qu s “ s ¨ . . Then, E Q d P r X s “ . s ` . p ´ s q ` . s “ . ` . s P p . , . q . Hence, every value p ă . Y we find E Q d P r Y s “ . s p . ´ . q ` . p ´ s qp ´ . q “ . ´ . s . Thus, a premium for the hybrid contract with benefits Y mustbe less than 0.06 if insurance arbitrage were excluded. ♦ Remark 3.10 (On the role of p Ó ) . From a general perspective, the premium p t should be a G t -measurable random variable. However, the knowledge of p t might then lead to arbitrage possibilitieson the financial ( F -adapted) market. This is the reason why we introduced p Ó in the theorem above.In the practically most relevant cases the setting can be simplified as follows: Enlarge the filtration F progressively by p leading to the filtration ˜ F “ p ˜ F t q t P T with˜ F t “ F t _ σ t p s , s ď t u . (35)The assumption that the financial market with respect to ˜ F is free of financial arbitrage will be satisfiedin most applications: meaning that the knowledge of p t does not introduce arbitrages. Indeed, it isin most cases easy to find premiums of insurance contracts in the Internet. Then, F can be replacedby ˜ F and p can be considered to be adapted to F .3.3. Scheduled premiums.
Up to now we were assuming that the premium is paid at initiationof the contract. In practice, one typically pays premiums at scheduled time until a certain stoppingtime. We show in this subsection how to incorporate the scheduled premiums in our setting.In this regard, let A be a deterministic function of finite variation over r t, T s decoding the payments.For example, the premiums p , . . . , p n at times t , . . . , t n are captured via A p t q “ ř ni “ t t i ď t u p i .Consider a G -stopping time τ at which the payment of the premiums is terminated. The totalpremiums sum to A p T q , while the realized premiums are only A p τ ^ T ´q .Consider an insurance contract with benefits X t,T . Our goal is compute a suitable A or to check,for a given A , if NIFA holds. This can be traced back to the setting of a single premium by thefollowing two steps: first, the initial premium is set to p At “ A p T q . Second, the insured has the additional option to cease payments after τ , which we simply add to thebenefits. In this regard, define the modified benefits X A,τt,T “ X t,T ` p A p T q ´ A p τ ^ T qq . Now, Theorem 3.6 can be a applied to the premium p At and the benefits X A,τt,T . Then, the QP-rulein the context of Remarks 3.8 and 3.10 can be applied to compute the basic parts of the scheduledpremiums A . O ARBITRAGE IN INSURANCE 13
Adding a risk margin.
As already mentioned, premiums in practice consist of a basic partand a part covering risk margins (and possibly other costs), see also the discussion in Chen et al.(2020). We elaborate shortly on the non-linear valuation of the risk margin here. Consider the time0 ď t ă T and consider the allocation ψ nt insurance seekers leading to the commitment ÿ i ě ψ i,nt X it,T . We decompose the commitment ÿ i ě ψ i,nt X it,T “ γ t E Q d P r X t,T | G t s` γ t ´ E Q d P r X t,T | G t,T s ´ E Q d P r X t,T | G t s ¯ ` ÿ i ě ψ i,nt X it,T ´ γ t E Q d P r X t,T | G t,T s “ : γ t E Q d P r X t,T | G t s ` Y t ` Y ,nt . The risk in Y t can further be reduced by hedging on the financial market: assume that ξ is thechosen hedging strategy, then the remaining risk is given by˜ Y t “ Y t ` T ´ ÿ s “ t ξ s ¨ ∆ S s . When we have a complete market or a perfect hedge, ˜ Y t “ p ρ t q t P T . In this case, the security or supervisory margin is given by ρ t p ˜ Y t ` Y ,nt q (36)and the cost-of-capital part of it, i.e. the risk margin , has to be added to the basic part of the premium E Q d P r X t,T | G t s .In the case where a conditional strong law of large numbers applies, which is for example used inTheorem 3.6, then Y ,nt Ñ ρ t p ˜ Y t q .An alternative decomposition of the premium into the terms γ t E Q d P r X t,T | G t s ` ρ t p Y t q ` ρ t p Y ,nt q has recently been proposed in Deelstra et al. (2018), however under stronger assumptions on thefinancial market and the choice of the equivalent martingale measure.4. Progressive enlargement of the financial filtration
Besides the publicly available information F , the insurer has access to private information, givenby the filtration G . In this section we introduce additional structure on G by utilizing the theory ofprogressive enlargements.Consider a random time τ which could be used, e.g., to model the random life time of the insured,a claim arrival time, or a surrender time. We consider multiple times in Section 4.1.We require that t τ ą t u , t P T , are atoms in the filtration G ; more precisely, we assume that forall t P T , G t Ş t τ ą t u “ F t Ş t τ ą t u , (37)wich means that for each G P G t there exists an F P F t , such that G X t τ ą t u “ F X t τ ą t u andvice versa.A classical and well-known example where (37) holds is the progressive enlargement of F with therandom time τ . The enlarged filtration is the smallest filtration such that τ becomes a stopping time. Remark 4.1 (Progressive enlargement) . Assume that for all t P T , G t : “ σ ` t F Ş t τ ď s u : F P F t , s ď t u ˘ “ : F t _ t τ ^ t u , (38)then G “ p G t q t ě is called the progressive enlargement of F with τ . Then (37) holds: indeed, thisfollows from F Ş t τ ą s u Ş t τ ą t u “ F Ş t τ ą t u , for 0 ď s ď t ď T and F P F t . For a detailed study and many references to related literature seeAksamit and Jeanblanc (2017). Typically we will be interested in a filtration which contains moreinformation, like the employment status, health status, surrender behaviour, etc. ♦ We place ourselves in the context of Remark 3.10. To this end, fix t and assume that the information G t does not generate arbitrage, i.e. there exists an equivalent martingale measure Q on p Ω , G t,T q . TheQP-rule now works as follows: the insurance premium p t is computed via taking expectations underQ d P with respect to ˜ F t “ G t . Denote by G t “ Q d P p τ ą t | F t q , t ě Az´ema supermartingale . The Az´ema supermartingale decodes the survival probability of τ in thesmall filtration F . Proposition 4.2.
Assume F Ă G , τ being a G -stopping time, and that (37) holds. Then, for any G -adapted process A bounded from below, E Q d P r A τ | G t s “ t τ ď t u A τ ` t τ ą t, G t ą u G ´ t E Q ” E P r A τ t τ ą t u | F s| F t ı (40) holds for all t P T .Proof. First, we consider A τ “ A τ t τ ď t u ` A τ t τ ą t u . Since A τ t τ ď t u is G t -measurable, we obtain E Q d P r A τ t τ ď t u | G t s “ A τ t τ ď t u , the first addend of (40).Second, we consider A τ t τ ą t u . Note that E Q d P r A τ t τ ą t u | F t s vanishes on the set t τ ą t, G t “ u ,and so does the second addend in (40). So, we may assume G t ą t ě G t -measurable random variable ˜ A t we find a F t -measurable randomvariable Y t , such that ˜ A t t τ ą t u “ Y t t τ ą t u (41)by an application of the monotone class theorem. Using E Q d P r A τ t τ ą t u | G t s for ˜ A t in this equationand taking conditional expectation with respect to F t , we obtain that G ´ t E Q d P ” E Q d P r A τ | G t s t τ ą t u | F t ı “ G ´ t E Q d P ” Y t t τ ą t u | F t ı “ Y t . (42)On the other hand, E Q d P ” E Q d P r A τ | G t s t τ ą t u | F t ı “ E Q d P “ A τ t τ ą t u | F t ‰ “ E Q d P ” E P r A τ t τ ą t u | F s| F t ı “ E Q ” E P r A τ t τ ą t u | F s| F t ı , (43)where we have used that, according to Proposition 2.1, Q d P coincides with P , conditionally on F ,and, that it coincides with Q on p Ω , F q . This shows (40). (cid:3) Example 4.3.
We continue Example 2.3 and compute the above quantities in this setting. Recallthat the cumulated intensity Λ was an F -adapted, increasing process and P p τ ą t | F t q “ e ´ Λ t andthat A was F -adapted. We obtained that P p τ ą t | F q “ P p Λ t ă E | F q “ e ´ Λ t . On the other side, e ´ Λ t “ Q d P p τ ą t | F t q “ G t . Moreover, E Q d P r A τ t τ ą t u | G t s “ t τ ď t u A τ ` t τ ą t u G ´ t T ÿ s “ t ` E Q r A s P p τ “ s | F q| F t s“ t τ ď t u A τ ` t τ ą t u e Λ t T ÿ s “ t ` E Q “ A s ` e ´ Λ s ´ ´ e ´ Λ s ˘ | F t ‰ . (44)This formula is a key result for the valuation of a large number of hybrid products. ♦ O ARBITRAGE IN INSURANCE 15
An additional difficulty arises when A is not F -measurable. Here one is able to exploit the struc-ture of the progressive enlargement in the case where τ is honest. For example, Corollary 5.12 inAksamit and Jeanblanc (2017) allows to decompose X in several F -adapted components on randomintervals depending solely on τ . Proposition 4.4.
Assume F Ă G , τ being a G -stopping time, and that (37) holds. Then, for any G -adapted process A and any F T -measurable random variable F T , both bounded from below, it holdsthat E Q d P r F T A τ | G t s “ t τ ď t u A τ E Q d P r F T | G t s` t τ ą t, G t ą u G ´ t E Q ” F T E P r A τ t τ ą t u | F s| F t ı holds for all t P T . The proof follows as for Proposition 4.2.4.1.
Multiple stopping times.
For the insurance company it is of course important to considermore than one stopping time, and to allow for dependence between these stopping times. We gener-alize the previous framework by considering a countable number of atoms and provide the associatedgeneralizations of the previously obtained results.Consider for each t P T G t -measurable sets p P t , P t , . . . , P nt q such that P it X P jt “ H for i ‰ j .Assume that for all 1 ď i ď n and t P T , G t Ş P it “ F t Ş P it (45)and denote by G it “ Q d P p P it | F t q (46)the respective generalization of the Az´ema supermartingale. Note that without additional assump-tions G i does not need to be a supermartingale. Nevertheless, we have the following generalizationof Proposition 4.2. Set Ω t “ ř ni “ P it . Proposition 4.5.
Assume F Ď G , and that (45) and (46) hold. Then, for any G -measurable randomvariable A bounded from below, Ω t E Q d P r A | G t s “ n ÿ i “ P it Xt G it ą u p G it q ´ E Q ” E P r A P it | F s| F t ı (47) holds for all t P T .Proof. First, we decompose Ω t E Q d P r A | G t s “ n ÿ i “ P it E Q d P r A P it | G t s . (48)For the following, we fix i . From (45), a monotone class arguments gives the following: for a G t -measurable random variable X t we can find a F t -measurable random variable ˜ Y t , such that X t P it “ ˜ Y t P it . (49)Hence, there exists an F t -measurable random variable Y t such that E Q d P r A | G t s P it “ Y t P it . (50)Taking conditional expectations with respect to F t and multiplying with P it yields that P it E Q d P ” E Q d P r A P it | G t s| F t ı “ Y t G it P it . The left hand side equals P it E Q d P r A P it | F t s , which consequently vanishes when G it “
0. Inserting(50), we obtain that P it Xt G it ą u p G it q ´ E Q d P “ A P it | F t ‰ “ E Q d P “ A P it | G t ‰ P it . The claim now follows by combining this with Proposition 2.1 and (48). (cid:3)
Two stopping times.
The for us most interesting case is the case of two stopping times, whichwill be the random life time of the insured and the surrender time. Typically it is assumed thatthey are conditionally independent. This can be a serious restriction for the applications we have inmind: indeed, dependence between remaining life time and surrender is of course possible and shouldbe taken into account. We may use the above result to do so. Motivated by this, we will developProposition 4.5 further in the case of two stopping times.Consider two F -adapted, increasing processes Λ , and Λ . Assume there exists two standardexponential random variables E , and E with continuous copula C p u , u q : “ P ` exp p´ E q ă u , exp p´ E q ă u ˘ . Let σ “ inf t t P T : Λ t ě E u ^ T, and τ “ inf t t P T : Λ t ě E u ^ T . Using independence of E i and F “ F T , we obtain that P p σ ą s, τ ą t | F q “ P p Λ s ă E , Λ t ă E | F q“ P ` exp p´ E q ă Λ s , exp p´ E q ă Λ t | F ˘ “ C ` e ´ Λ s , e ´ Λ t ˘ . (51) Example 4.6 (Conditional independence) . If in addition, E and E are independent, then σ and τ are independent conditional on F and C p u, v q “ u ¨ v . This simplifies the computations significantlysince then P p σ ą s, τ ą t | F q “ exp p´ Λ s ´ Λ t q . ♦ Fix t P T and consider the disjoint G t -measurable sets P , t “ t σ ą t, τ ą t u ,P ,it “ t σ ą t, τ “ i u , i “ , . . . , t ´ ,P ,it “ t σ “ i, τ ą t u , i “ , . . . , t ´ ,P ,i,jt “ t σ “ i, τ “ j u , i, j “ , . . . , t ´ , which will take the role of P t , . . . , P nt in the previous section. Let Γ p s, t q “ C p exp p´ Λ s q , exp p´ Λ t qq .Then we obtain from (51) that G , t “ Q d P p σ ą t, τ ą t | F t q “ Γ p t, t q ,G ,it “ Q d P p σ ą t, τ “ i | F t q “ Γ p t, i ´ q ´ Γ p t, i q ,G ,it “ Q d P p σ “ i, τ ą t | F t q “ Γ p i ´ , t q ´ Γ p i, t q ,G ,i,jt “ Q d P p σ “ i, τ “ j | F t q“ Γ p i ´ , j ´ q ´ Γ p i, j ´ q ´ Γ p i ´ , j q ` Γ p i, j q . Now, consider two F -adapted payment streams A and A . If the insurer dies at σ before surren-dering, he will receive A σ while if he first surrenders, he will receive A τ . Precisely, this defines thefollowing payoff: X t,T “ t σ ą t,τ ą t u p t σ ă T,σ ď τ u A σ ` t τ ă T,τ ă σ u A τ q . (52) Proposition 4.7.
For the payoff in (52) we have that E Q d P r X t,T | G t s “ t σ ą t,τ ą t u p G , t q ´ T ´ ÿ s “ t ` E Q r A s G ,ss ´ ` A s G ,ss | F t s . Proof.
For the part with A we obtain the following decomposition: t t ă σ ă T,σ ď τ u A σ “ T ´ ÿ s “ t ` t σ “ s,τ ě s u A s “ T ´ ÿ s “ t ` t σ “ s,τ ą s ´ u A s “ T ´ ÿ s “ t ` P ,ss ´ A s . Similarly, t τ ă T,τ ă σ u A τ “ T ´ ÿ s “ t ` P ,ss A s . The result now follows by applying Proposition 4.5. (cid:3)
O ARBITRAGE IN INSURANCE 17 An affine framework
Affine processes are a well-established tool for modeling term structures, in particular stochas-tic mortality term structures. We refer to Keller-Ressel et al. (2019) for a detailed treatment ofaffine processes, including affine processes in discrete time, and to Biffis (2005), Schrager (2006),Luciano et al. (2008), and Blackburn and Sherris (2013) for applications to stochastic mortality. Inthis section we propose a framework in discrete time for the valuation of insurance products linkedto financial markets.We now assume that there is a driving d -dimensional process Z , which is affine. The process Z is called affine , if it is a Markov process and its characteristic function is exponential affine. Weadditionally require the existence of exponential moments, such that E r exp p uZ t ` q| Z t s “ exp ` A p u q ` B p u q ¨ Z t ˘ for 0 ď t ă T and all u P R d ; with deterministic functions A : R d Ñ R and B : R d Ñ R d . We willdenote the state space of Z by Z . Typically Z “ R m ě ˆ R n with m ` n “ d .In the light of the QP-rule it will be important to distinguish between the F -adapted parts of Z and the parts which are only G -adapted. This will become important when we consider the set ofequivalent martingale measures M e p F q where the choice of F plays an important role. In this regard,let Z “ p X, Y q with Y generating the filtration F ; here X is d -dimensional and Y is d -dimensional,with d ` d “ d . We will allow that either d or d vanishes. In the light of Remark 3.10, X mightcontain insurance-related quantities which are not publicly available.We model the stock market as exponential driven by the affine process with an additional drift,i.e. we assume that discounted stock prices are given by S t “ exp p a t ` a ¨ Y t q , t ě , with p a , a q “ p a , a , . . . , a d q P R ` d . This modeling contains the Black-Scholes model and expo-nential L´evy models as a special case (in discrete time).5.1. The case of one stopping time.
For the modeling of τ , we follow the doubly stochasticapproach introduced in Example 2.3. Precisely, we consider a non-decreasing process Λ ě t “ b ` t ÿ s “ ´ b ¨ X s ` c ¨ Y s ¯ , t ě p b , b, c q P R ` d . Here, b is chosen such that Λ “
0, guaranteeing in particular P p τ “ q “ p b, c q are chosen such that Λ is increasing.Denote by F Z the filtration generated by Z . Together with Λ we assume that τ satisfies (7) andthat G is the progressive enlargement of F Z with τ , G t “ F Zt _ p τ ^ t q , for all t P T .To ensure absence of financial arbitrage, we assume that there exists an equivalent martingalemeasure Q . For tractability, we assume that Y is again affine under Q (with existing exponentialmoments), E Q r exp p uY t ` q| Y t s “ exp ` A Q p u q ` B Q p u q ¨ Y t ˘ (53)for all 0 ď t ă T and u P R d . Remark 5.1.
The Esscher change of measure is one well-known example which keeps the affineproperty during the measure change, but not the only one, see for example Kallsen and Muhle-Karbe(2010).Finally, we assume that the conditional distribution of X t , conditional on Y t and X t ´ has affineform and denote E P r exp p u ¨ X t q| Y t , X t ´ s “ exp ´ α p u q ` β p u q ¨ X t ´ ` γ p u q ¨ Y t ¯ , (54)for all 0 ă t ď T and u P R d . The coefficients α, β, and γ can be computed from A and B . We introduce the following recursive notation: define φ p T q “ α p´ b q ` A Q p a ´ c ` γ p´ b qq , ψ p T q “ β p´ b q , and ψ p T q “ B Q p a ´ c ` γ p´ b qq ; now let u p s q “ ψ p s ` q ´ b , v p s q “ ψ p s ` q ´ c and set φ p s q “ α p u p s qq ` A Q p v p s q ` γ p u p s qqq ,ψ p s q “ β p u p s qq , (55) ψ p s q “ B Q p v p s q ` γ p u p s qqq for s “ , . . . , T ´
1. Moreover, φ p T q “ A Q p a q , ψ p T q “
0, and ψ p T q “ B Q p a q ;, following the samerecursion rule (55). We are interested in the claim S τ t t ă τ ď T u , which we decompose to T ÿ s “ t ` S s p t τ ą s ´ u ´ t τ ą s u q . The following proposition now allows to value this claim as well as the claim S T t τ ą T u . DenoteΦ p t, T q “ T ÿ s “ t ` φ p s q , (56)and, analogously, Φ . Moreover we write ψ ¨ X ` ψ ¨ Y “ ψ ¨ Z , again analogously for ψ . Proposition 5.2.
We have the following valuation-results E Q d P r S T t τ ą T u | G t s “ t τ ą t u e a T ` Φ p t,T q` ψ p t ` q¨ Z t ,E Q d P r S T t τ ą T ´ u | G t s “ t τ ą t u e a T ` Φ p t,T q` ψ p t ` q¨ Z t . Proof.
Using Proposition 4.4, and the affine representations of S and Λ, we obtain that E Q d P r S T t τ ą T u | G t s “ t τ ą t u e Λ t E Q d P r S T e ´ Λ T | F Zt s“ t τ ą t u E Q d P ” exp ´ a T ` a ¨ Y T ´ T ÿ s “ t ` ` b ¨ X s ` c ¨ Y s ˘¯ | Z t ı . Now we proceed iteratively: first, consider s “ T . Then, E P r e ´ b ¨ X T | F ZT ´ _ Y T s “ e α p´ b q` β p´ b q¨ X T ´ ` γ p´ b q¨ Y T E Q r e p a ´ c ` γ p´ b qq¨ Y T | F ZT ´ s “ e A Q p a ´ c ` γ p´ b qq` B Q p a ´ c ` γ p´ b qq¨ Y T ´ . In the next step, for s “ T ´ E Q d P r e u ¨ X s ` v ¨ Y s | F Zs ´ s “ e α p u q` β p u q¨ X s ´ ` A Q p v ` γ p u qq` B Q p v ` γ p u qq¨ Y s ´ (57)with u “ β p´ b q ´ b and v “ B Q p a ´ c ` γ p´ b qq ´ c . Proceeding iteratively until s “ t ` E Q d P r S T t τ ą T u | G t s “ t τ ą t u e a T ` ř Ts “ t ` φ p s q` ψ p t ` q¨ X t ` ψ p t ` q¨ Y t , the first claim. The second claim follows in a similar way. (cid:3) Survival and surrender.
Since we are interested in modeling surrender and survival, we needto consider two stopping times. To exploit the full power of the affine framework, we will assumethat they are conditionally independent. In this regard, let σ, τ be doubly stochastic random timesas introduced in Section 4.2 with associated non-decreasing processes Λ and Λ and under theassumption of Example 4.6. We assume thatΛ it “ b i ` t ÿ s “ ` b i ¨ X s ` c i ¨ Y s ˘ , t ě p b i , b i , c i q P R ` d , i “ ,
2. Again, the coefficients are chosen such that the processes start in 0and are increasing.First, we obtain that under these assumptions G , t “ e ´ Λ t ´ Λ t ,G ,it “ e ´ Λ t ´ e ´ Λ i ´ ´ e ´ Λ i ¯ ,G ,it “ ´ e ´ Λ i ´ ´ e ´ Λ i ¯ e ´ Λ t O ARBITRAGE IN INSURANCE 19 and an analogous expression for G (which we will not use here).From these expressions it is clear that we need to generalize our previous notions of φ and ψ .Fix 0 ď s ă T and consider κ i “ p κ i , κ i q P R p T ` qˆ d ˆ R p T ` qˆ d , 0 ď i ă T (we use κ for thecoefficients associated with X and κ for those associated with Y ) defined by κ T “ p´ b , a ´ c q , κ t “ p´ b , ´ c q for s ` ď t ă T and κ t “ p´ b ´ b , ´ c ´ c q for t ď s ; moreover κ T “ p´ b , a ´ c q , κ t “ p´ b , ´ c q for s ` ď t ă T and κ t “ p´ b ´ b , ´ c ´ c q for t ď s .Define recursively φ p T q “ α p κ T q ` A Q p κ T ` γ p κ T qq ,ψ p T q “ β p κ T q ,ψ p T q “ B Q p κ T ` γ p κ T qq φ p i q “ α p u p i qq ` A Q p v p i q ` γ p u p i qqq ,ψ p i q “ β p u p i qq ,ψ p i q “ B Q p v p i q ` γ p u p i qqq , with u p i q “ ψ p i ` q ` κ i , v p i q “ ψ p i ` q ` κ i . Again, we writeΦ p t, T q “ T ÿ i “ t ` φ p i q and ψ ¨ X ` ψ ¨ Y “ ψ ¨ Z . The coefficients φ , ψ and Φ are obtained with the same recursion,replacing κ by κ . Proposition 5.3.
We have the following valuation-results for t ď s ă T , e Λ t ` Λ t E Q d P r S T e ´ Λ T ´ Λ s | G t s “ e a T ` Φ p t,T q` ψ p t ` q¨ Z t e Λ t ` Λ t E Q d P r S T e ´ Λ s ´ Λ T | G t s “ e a T ` Φ p t,T q` ψ p t ` q¨ Z t Proof.
We proceed iteratively as in the proof of Proposition 5.2. Note that e Λ t ` Λ t S T e ´ Λ T ´ Λ s “ e a T ` a ¨ Y T ´ ř Ti “ t ` p b X i ` c Y i q´ ř sj “ t ` p b X j ` c Y j q . For i “ T and i ą s we obtain with (57) E Q r e ´ b ¨ X T `p a ´ c q¨ Y T | F T ´ s “ e α p´ b q` β p´ b q¨ X T ` A Q p a ´ c ` γ p´ b qq` B Q p a ´ c ` γ p´ b qq¨ Y T ´ “ e φ p T q` ψ p T q¨ X T ´ ` ψ p T q¨ Y T ´ . For i ď T ´ i ą s , E Q r e p ψ p i ` q´ b q¨ X i `p ψ p i ` q´ c q¨ Y i | F i ´ s “ e φ p i q` ψ p i q¨ X i ´ ` ψ p i q¨ Y i ´ . For i ď s we have to compute E Q r e p ψ p i ` q´ b ´ b q¨ X i `p ψ p i ` q´ c ´ c q¨ Y i | F i ´ s “ e φ p i q` ψ p i q¨ X i ´ ` ψ p i q¨ Y i ´ and the first claim follows. The second claim is obtained similarly. (cid:3) Examples
In this section we provide several examples which illustrate the application of our results.6.1.
The surrender option.
A simple example where the full generality of our approach should beused is a surrender option. We ignore for simplicity survival. If the surrender options is exercised,the insured will receive the amount of the underlying portfolio. The surrender time is a random time,and the likelihood for surrender will depend on the performance of the underlying portfolio, see forexample Ballotta et al. (2019) and references therein.Let τ denote the surrender time and consider the affine framework in Section 5.1. Let t ă T denotethe current time. The payoff of the surrender option is t t ă τ ă T u S τ , payments in the case of no surrender can be priced in a similar way and are therefore ignored here.With the aid of Proposition 5.2 its value computes to E Q d P r t t ă τ ă T u S τ | G t s “ t τ ą t u T ´ ÿ i “ t ` E Q d P r S i p t τ ą i ´ u ´ t τ ą i u q| G t s“ t τ ą t u T ´ ÿ i “ t ` e a i ´ e Φ p t,i q` ψ p t ` q¨ Z t ´ e Φ p t,i q` ψ p t ` q¨ Z t ¯ . (58)6.2. Variable annuities.
For the case of a variable annuity we have to consider surrender andsurvival: let σ be the surrender time and τ the life time of the insured. As a typical payoff illustratingthe application of our methodology to the valuation of variable annuities we consider the surrenderoption in this context: at surrender at σ , the insured receives S σ , provided σ ď τ . We obtain that E Q d P r t t ă σ ă T,σ ď τ u S σ | G t s“ t σ ą t,τ ą t u e Λ t ` Λ t T ´ ÿ s “ t ` E Q r S s G ,ss ´ | F t s“ t σ ą t,τ ą t u e Λ t ` Λ t T ´ ÿ s “ t ` E Q ” S s ´ e ´ Λ s ´ ´ e ´ Λ s ¯ e ´ Λ s ´ | F t ı . (59)Analogously to Proposition 5.3 we obtain that E Q ” S s e ´ Λ s ´ ´ Λ s ´ ` Λ t ` Λ t | F t ı “ e a s ` Φ p t,T q` ψ p t ` q¨ Z t E Q ” S s e ´ Λ s ´ Λ s ´ ` Λ t ` Λ t | F t ı “ e a s ` Φ p t,T q` ψ p t ` q¨ Z t where we have κ s “ a , κ s ´ , . . . , κ t ` “ ´ b ´ b and κ s “ a , κ s ´ “ ´ b , and κ s ´ , . . . , κ t ` “´ b ´ b .6.3. Longevity.
Longevity risk has become increasingly important in insurance valuation, see forexample MacMinn et al. (2006), Blake et al. (2014), Blake et al. (2018), and references therein.From a general viewpoint, longevity comes as a driving systemic factor which affects all insuredcustomers. The generality of our approach allows to incorporate such factors. Moreover, affineprocesses have already been used to model longevity, see for example Biffis (2005).For illustrative purposes consider the following example: let θ i : T Ñ R ` , i “ , . . . , n denote n possible mortality curves with initial mixture probability p , . . . , p n . For valuation we use the mixturecurve n ÿ i “ p i θ i p t q , t P T. If we assume that the curves are ordered, i.e. θ n is the curve associated with the highest longevity, longevity risk describes the possible increase of p n through time.Consider a filtration G and denote by p it the update of the initial probabilities given G t , whichis typically achieved via filtering and Bayes rule, see Schmidt (2016) and Bain and Crisan (2009).Longevity risk arises if chance implies an upward trend in p n , and affine models are perfectly suitedfor capturing this: according to Biffis (2005), already random fluctuations in p model longevity risk,while Luciano et al. (2008) specifies an exponentially growing mean-reversion level, which generalizesthe Gompertz law, for modeling longevity risk in an affine setting. Both approaches can be embeddedin our affine framework. 7. Conclusion
This paper deals with the valuation of hybrid insurance products depending on insurance-relatedquantities like surrender, claim arrival or mortality and financial markets. Since interest rates typ-ically have to be taken into account, this applies to most insurance contracts. We fundamentallyanalyze the valuation of these products by introducing actuarial portfolio strategies combined withfinancial trading strategies and relate them to the absence of insurance-finance arbitrage (IFA).Moreover, we provide a simple valuation rule which avoids insurance and finance arbitrage. This isillustrated in an affine framework.
O ARBITRAGE IN INSURANCE 21
Our work can be seen as the first step towards a general study of insurance valuation and futurework will need to include non-linear valuation rules, allow for more than one insurance company andshould take possible ruin of insurance companies into account.
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