Viability and Arbitrage under Knightian Uncertainty
aa r X i v : . [ q -f i n . E C ] F e b Viability and Arbitrage under KnightianUncertainty ∗ Matteo Burzoni † , Frank Riedel ‡ , and Mete Soner ETH Zurich, Switzerland Bielefeld University, Germany and University of Johannesburg, South Africa ETH Zurich and Swiss Finance Institute, Switzerland
February 26, 2019
Abstract
We reconsider the microeconomic foundations of financial economicsunder Knightian Uncertainty. We remove the (implicit) assumption of acommon prior and base our analysis on a common order instead. Eco-nomic viability of asset prices and the absence of arbitrage are equivalent.We show how the different versions of the Efficient Market Hypothesis arerelated to the assumptions one is willing to impose on the common order.We also obtain a version of the Fundamental Theorem of Asset Pricing us-ing the notion of sublinear pricing measures. Our approach unifies recentversions of the Fundamental Theorem under a common framework.
Keywords:
Viability, Knightian Uncertainty, No Arbitrage, Robust Finance
JEL subject classification: D53, G10AMS 2010 subject classification.
Primary 91B02; secondary 91B52, 60H30
Recently, a large and increasing body of literature has focused on decisions,markets, and economic interactions under uncertainty. Frank Knight’s pioneer-ing work (Knight (1921)) distinguishes risk – a situation that allows for anobjective probabilistic description – from uncertainty – a situation that cannotbe modeled by a single probability distribution. ∗ Comments by Rose–Anne Dana, Filipe Martins-da-Rocha, and seminar and workshopparticipants at Oxford University, the Institut Henri Poincar´e Paris, and Padova University,are gratefully acknowledged. † Matteo Burzoni and Mete Soner acknowledge the support of the ETH Foundation, theSwiss Finance Institute and the Swiss National Foundation through SNF 200020 − ‡ Frank Riedel gratefully acknowledges the financial support of the German Research Foun-dation (DFG) via CRC 1283.
1n this paper, we discuss the foundations of no–arbitrage pricing and itsrelation to economic equilibrium under Knightian Uncertainty.Asset pricing models typically take a basic set of securities as given anddetermine the range of option prices that is consistent with the absence ofarbitrage. From an economic point of view, it is crucial to know if modelingsecurity prices directly is justified; an asset pricing model is called viable if itssecurity prices can be thought of as (endogenous) equilibrium outcomes of acompetitive economy.Under risk, this question has been investigated in Harrison and Kreps’ sem-inal work (Harrison and Kreps (1979)). Their approach is based on a commonprior (or reference probability) that determines the null sets, the topology, andthe order of the model. The common prior assumption is made in almost allasset pricing models.In recent years, it has become clear that many of the standard financialmodels used in practice face Knightian uncertainty, the most salient examplesbeing stochastic volatility, term structure and credit risk models. If Knightianuncertainty is recognized in these models, a reference measure need not exist,see, e.g., Epstein and Ji (2013). Under Knightian uncertainty, we thus have toforego the assumption of a common prior.We replace the common prior with a common order with respect to whichagents’ preferences are monotone. We thus assume that market participantsshare a common view of when one contract is better than another. This as-sumption is far weaker than the assumption of a common prior. In particular,it allows to include Knightian uncertainty.Our main result shows that the absence of arbitrage and the (properly de-fined) economic viability of the model are equivalent. In equilibrium, there areno arbitrage opportunities; conversely, for arbitrage–free asset pricing models,it is possible to construct a heterogeneous agent economy such that the assetprices are equilibrium prices of that economy.The main result is based on a number of other results that are of inde-pendent interest. To start with, in contrast to risk, it is no longer possi-ble to characterize viability through the existence of a single linear pricingmeasure (or equivalent martingale measure). Instead, it is necessary to usea suitable nonlinear pricing expectation, that we call a sublinear martingaleexpectation. A sublinear expectation has the common properties of an ex-pectation including monotonicity, preservation of constants, and positive ho-mogeneity, yet it is no longer additive. Indeed, sublinear expectations canbe represented as the supremum of a class of (linear) expectations, an op-eration that does not preserve linearity . Nonlinear expectations arise nat-urally for preferences in decision–theoretic models of ambiguity–averse pref-erences (Gilboa and Schmeidler (1989), Maccheroni, Marinacci, and Rustichini(2006)). It is interesting to see that a similar nonlinearity arises here for the In economics, such a representation theorem appears first in Gilboa and Schmeidler(1989). Sublinear expectations also arise in Robust Statistics, compare Huber (1981),and they play a fundamental role in theory of risk measures in Finance, seeArtzner, Delbaen, Eber, and Heath (1999) and F¨ollmer and Schied (2011). ricing functional .The common order shapes equilibrium asset prices. We study various com-mon orders and how they are related to versions of the Efficient Market Hy-pothesis . The original (strong) version of the Efficient Market Hypothesis (Fama(1970)) states that properly discounted expected returns of assets are equal un-der the common prior. We show that the original Efficient Market Hypothesisholds true under the very strong assumption that the common order is inducedby expected payoffs under a common prior, i.e. when agents’ preferences aremonotone with respect to expected payoffs under a common prior.When the common order is given by the almost sure ordering under a givenprior, we obtain the weak form of the Efficient Market Hypothesis; it states thatexpected returns are equal under some probability measure that is equivalent tothe common prior. This order allows agents to use risk–adjusted probabilities (orstochastic discount factors based on the marginal rate of substitution) to pricefinancial claims (compare Cochrane (2001) and Rigotti and Shannon (2005)).We thus obtain the classic version of the Fundamental Theorem of Asset Pric-ing (Harrison and Kreps (1979); Harrison and Pliska (1981); Duffie and Huang(1985); Dalang, Morton, and Willinger (1990); Delbaen and Schachermayer (1998)).Under conditions of Knightian uncertainty, our main results lead to newversions of the Efficient Market Hypothesis.If the market orders payoffs by considering a family of expected payoffs fora set of possible priors in the spirit of the incomplete expected utility model ofBewley (2002), we obtain a generalization of the strong Efficient Market Hy-pothesis under Knightian uncertainty. In viable markets, a sublinear martingaleexpectation exists that is linear on the subspace of mean–ambiguity–free pay-offs (securities that have the same expectation under all priors). For the marketrestricted to this subspace we still get the classical strong form of the EMH. Formean–ambiguous securities, this conclusion need not be true.Under Knightian uncertainty, one is naturally led to study sets of probabilitymeasures that are not dominated by one common prior (Epstein and Ji (2014),Vorbrink (2014), e.g.). It is then natural to take the quasi–sure ordering asthe common order of the market. A claim dominates quasi–surely anotherclaim if it is almost surely greater or equal under all considered probabilitymeasures. If the class of probability measures describing Knightian uncertaintyis not dominated by a single probability measure, the quasi–sure ordering ismore incomplete than any almost sure ordering. If we merely assume that thecommon ordering is the quasi–sure order induced by a set of priors, we obtaina weak version of the efficient market hypothesis under Knightian uncertainty.Bouchard and Nutz (2015) and Burzoni, Frittelli, and Maggis (2016) discuss theabsence of arbitrage in such a setting. We complement their analysis by givinga precise economic equilibrium foundation.We also study the consequences for asset pricing when the common order isinduced by smooth ambiguity preferences as introduced by Klibanoff, Marinacci, and Mukerji Beißner and Riedel (2016)) develop a general theory of equilibria with such nonlinearprices under Knightian uncertainty. average discounted asset prices are equal where the average is taken overthe expected returns under all priors with the help of the second-order prior.The above examples show that asset pricing under Knightian uncertainty(or a common order) leads to weaker conclusions for expected returns. In thissense, our paper shows that the EMH has to be interpreted in a careful way.In the early days, the EMH has frequently been identified with the
RandomWalk Hypothesis , the claim that asset returns are independent and identicallydistributed, maybe even normally distributed. It has been amply demonstratedin empirical research that the random walk hypothesis is rejected by data.Even the weak form of the EMH based on a common prior does allow for time-varying and correlated expected returns, though. Our paper shows that withouta common prior, an even wider variety of “expected returns” is consistent withequilibrium and the absence of arbitrage .We conclude this introduction by discussing the relation to some furtherliterature.The relation of arbitrage and viability has been discussed in various con-texts. For example, Jouini and Kallal (1995) and Jouini and Kallal (1999) dis-cuss models with transaction cost and other frictions. Werner (1987) andDana, Le Van, and Magnien (1999) discuss the absence of arbitrage in its re-lation to equilibrium when a finite set of agents is fixed a priori. Our approachis based on the notion of a common weak order. Cassese (2017) considers theabsence of arbitrage in an order-theoretic framework derived from coherent riskmeasures. It is shown that a price system is coherent if and only if pricing byexpectation is possible.Knightian uncertainty is also closely related to robustness concerns that playan important role in macroeconomics. Rational expectations models have re-cently been extended to take the fear of model misspecification into accountsee, e.g., Hansen and Sargent (2001, 2008). In this literature, tools from robustcontrol are adapted to analyze how agents should cope with fear of model mis-specification, that is modelled by putting a penalty term on the discrepancybetween the agent’s and the true model. Such fear of model misspecification isa special case of ambiguity aversion. Our analysis thus sheds also new light onthe foundations of asset pricing in robust macroeconomic models.Riedel (2015) works in a setting of complete Knightian uncertainty undersuitable topological assumptions. Absence of arbitrage is equivalent to the exis-tence of full support martingale measures in this context. We show that one canobtain this result from our main theorem when all agents use the pointwise orderand consider contracts as relevant if they are nonnegative and positive in some Compare Lo and MacKinlay (2002) and Akerlof and Shiller (2010) for empirical resultson deviations from the random walk hypothesis. Malkiel (2003) reviews the EMH and itscritics. The exact determination of the impact of Knightian uncertainty on asset returns is beyondthe scope of this study.
A non-empty set Ω contains the states of the world; the σ –field F on Ω collectsthe possible events.The commodity space (of contingent claims) H is a vector space of F -measurable real-valued functions containing all constant functions. We willuse the symbol c both for real numbers as well as for constant functions. H isendowed with a metrizable topology τ and a pre-order ≤ that are compatiblewith the vector space operations.The pre-order ≤ is interpreted as the common order of all agents in the econ-omy; it replaces the assumption of a common prior. We assume throughout thatthe preorder ≤ is consistent with the order on the reals for constant functionsand with the pointwise order for measurable functions. A consumption plan Z ∈ H is negligible if we have 0 ≤ Z and Z ≤ C ∈ H is nonnegative if 0 ≤ C and positive if in addition not C ≤
0. We denote by Z , P and P + the class ofnegligible, nonnegative and positive contingent claims, respectively.We also introduce a class of relevant contingent claims R , a convex subset of P + . We think of the relevant claims as being the contracts that allow to financedesirable consumption plans. In most examples below, we will take R = P + .The introduction of R allows to subsume various notions of arbitrage that werediscussed in the literature.The financial market is modelled by the set of net trades I ⊂ H , a convexcone containing 0. I is the set of payoffs that the agents can achieve from zeroinitial wealth by trading in the financial market.An agent in this economy is described by a preference relation (i.e. a com-plete and transitive binary relation) on H that is5 weakly monotone with respect to ≤ , i.e. X ≤ Y implies X (cid:22) Y for every X, Y ∈ H ; • convex , i.e. the upper contour sets { Z ∈ H : Z (cid:23) X } are convex; • τ -lower semi–continuous , i.e. for every sequence { X n } ∞ n =1 ⊂ H convergingto X in τ with X n (cid:22) Y for n ∈ N , we have X (cid:22) Y .The set of all agents is denoted by A .A financial market ( H , τ, ≤ , I , R ) is viable if there is a family of agents {(cid:22) a } a ∈ A ⊂ A such that • a ∈ A , i.e. ∀ ℓ ∈ I ℓ (cid:22) a , (2.1) • for every relevant contract R ∈ R there exists an agent a ∈ A such that0 ≺ a R . (2.2)We say that {(cid:22) a } a ∈ A supports the financial market ( H , τ, ≤ , I , R ).A net trade ℓ ∈ I is an arbitrage if there exists a relevant contract R ∗ ∈ R such that ℓ ≥ R ∗ . More generally, a sequence of net trades { ℓ n } ∞ n =1 ⊂ I is a free lunch with vanishing risk if there exists a relevant contract R ∗ ∈ R and asequence { e n } ∞ n =1 ⊂ H of nonnegative consumption plans with e n τ → e n + ℓ n ≥ R ∗ for all n ∈ N . We say that the financial market is strongly free ofarbitrage if there is no free lunch with vanishing risk.Our first main theorem establishes the equivalence of viability and absenceof arbitrage. Theorem 2.1.
A financial market is strongly free of arbitrage if and only if itis viable.
If there is a common prior P on (Ω , F ), a financial market is viable if and onlyif there exists a linear pricing measure in the form of a risk-neutral probabilitymeasure P ∗ that is equivalent to P . In the absence of a common prior, we haveto work with a more general, sublinear notion of pricing. A functional E : H → R ∪ {∞} is a sublinear expectation if it is monotone with respect to ≤ , translation-invariant, i.e. E ( X + c ) = E ( X ) + c for all constant contracts c ∈ H and X ∈ H ,and sublinear, i.e. for all X, Y ∈ H and λ >
0, we have E ( X + Y ) ≤ E ( X )+ E ( Y )and E ( λX ) = λ E ( X ). E has full support if E ( R ) > R ∈ R . Last notleast, E has the martingale property if E ( ℓ ) ≤ ℓ ∈ I . We say in shortthat E is a sublinear martingale expectation with full support if all the previousproperties are satisfied.It is well known from decision theory that sublinear expectations can bewritten as upper expectations over a set of probability measures. In our more6bstract framework, probability measures are replaced by suitably normalizedfunctionals. We say that ϕ ∈ H ′ +5 is a martingale functional if it satisfies ϕ (1) = 1 (normalization) and ϕ ( ℓ ) ≤ ℓ ∈ I . In the spirit of theprobabilistic language, we call a linear functional absolutely continuous if itassigns the value zero to all negligible claims. We denote by Q ac the set ofabsolutely continuous martingale functionals.The notions that we introduced now allow us to state the general version ofthe fundamental theorem of asset pricing in our order-theoretic context. Theorem 2.2 (Fundamental Theorem of Asset Pricing) . The financial marketis viable if and only if there exists a lower semi–continuous sublinear martingaleexpectation with full support.In this case, the set of absolutely continuous martingale functionals Q ac isnot empty and E Q ac ( X ) := sup φ ∈Q ac φ ( X ) is the maximal lower semi–continuous sublinear martingale expectation with fullsupport. Remark 2.3.
1. In the above theorem, we call E Q ac ( X ) maximal in thesense that any other lower semi–continuous sublinear martingale expecta-tion with full support E satisfies E ( X ) ≤ E Q ac ( X ) for all X ∈ H .2. Under nonlinear expectations, one has to distinguish martingales fromsymmetric martingales; a symmetric martingale has the property that theprocess itself and its negative are martingales. When the set of net trades I is a linear space as in the case of frictionless markets, a net trade ℓ andits negative − ℓ belong to I . In this case, sublinearity and the condition E Q ac ( ℓ ) ≤ ℓ ∈ I imply E Q ac ( ℓ ) = 0 for all net trades ℓ ∈ I . Thus,the net trades ℓ are symmetric E Q ac -martingales. Common order instead of common prior
Under risk it is natural to as-sume that all market participants consider a payoff X better than another payoff Y if X is greater or equal Y almost surely under a certain reference measure P .As we aim to discuss financial markets under Knightian uncertainty, we foregoany explicit or implicit assumption of a common prior P . Instead, we base ouranalysis on a common order ≤ , a far weaker assumption. As the preferences ofagents are monotone with respect to the common order, we assume that marketparticipants share a common view of when one contract is better than another . H ′ is the topological dual of H and H ′ + is the set of positive elements in H ′ . In this generality the terminology functional is more appropriate. When the dual space H ′ can be identified with a space of measures, we will use the terminology martingale measure . In a multiple prior setting, one is naturally led to the distinction of objective and subjec-tive rationality discussed by Gilboa, Maccheroni, Marinacci, and Schmeidler (2010); in theirpaper, the common order is given by Bewley’s incomplete expected utility model whereas thesingle agent has a complete multiple prior utility function. ≤ is consistent with the pointwise order, it follows that agents’ preferences aremonotone with respect to the pointwise order. If more is commonly knownabout the environment, one might want to impose stronger assumptions on thepre-order. For example, the common order could be generated by the expectedvalue of payoffs under a common prior P , a situation that corresponds to the“risk–neutral” world as we show below. Or we can use the almost–sure orderingunder that prior, the standard (implicit) assumption in finance models. In thesituation of a common set of priors , the so–called quasi–sure ordering is a nat-ural choice that is induced by the family of (potentially non–equivalent) priors.More examples are discussed below in Section 4.It might be interesting to note that it is possible to derive the commonpre-order ≤ from a given set of admissible agents by using the uniform orderderived from a set of preference relations ˆ A which are convex and τ -lower semi-continuous. Let Z (cid:22) := { Z ∈ H : X (cid:22) Z + X (cid:22) X, ∀ X ∈ H} , be the set of negligible (or null) contracts for the preference relation (cid:22)∈ ˆ A . Wecall Z uni := T (cid:22)∈ ˆ A Z (cid:22) the set of unanimously negligible contracts. We definethe uniform pre-order ≤ uni on H by setting X ≤ uni Y if and only if there exists Z ∈ Z uni such that X ( ω ) ≤ Y ( ω ) + Z ( ω ) for all ω ∈ Ω. ( H , ≤ uni ) is then apre-ordered vector space , and agents’ preferences are monotone with respectto the uniform pre-order. Relevant Claims
Our notion of arbitrage uses the concept of relevant claims,a subset of the set of positive claims. The interpretation is that a claim isrelevant if some agent views it as a desirable gain without any downside risk.For most models, it is perfectly natural to identify the set of relevant claims withthe set of positive claims, and the reader is invited to make this identification atfirst reading. In fact, if we think of the traded claims as consumption bundles,then for each contingent consumption plan R that is non-zero, there will bean agent who strictly prefers R to zero. This class of contracts describes thedirection of strict monotonicity that are identified by the market participants.In some finance applications, it makes sense to work with a smaller setof relevant claims. For example, if some positive claims cannot be liquidatedwithout cost, agents would not consider them as free lunches. When turningcomplex derivatives into cash involves high transaction costs, it is reasonable toconsider as relevant only a restricted class of positive claims, possibly only cash. Agents
We aim to clarify the relation between arbitrage–free financial mar-kets and equilibrium. For a given arbitrage–free market, we ask if it can be In the same spirit, one could define a pre-order X ≤ ′ u Y ⇔ X (cid:22) Y for all (cid:22) ∈ A . Ingeneral this will not define a pre-ordered vector space ( H , ≤ ′ ). The analysis of the papercarries over with minor modifications. common pre-order. Moreover, we imposesome weak form of continuity with respect to some topology. Convexity reflectsa preference for diversification. The Financial Market
We model the financial market in a rather reducedform with the help of the convex cone I . This abstract approach is sufficientfor our purpose of discussing the relation of arbitrage and viability. In thenext example, we show how the usual models of static and dynamic trading areembedded. Example 3.1.
We consider four markets with increasing complexity.1. In a one period setting with finitely many states Ω = { , . . . , N } , a finan-cial market with J + 1 securities can be described by its initial prices x j ≥ , j =0 , . . . , J and a ( J + 1) × N –payoff matrix F , compare LeRoy and Werner (2014).A portfolio ¯ H = ( H , . . . , H J ) ∈ R J +1 has the payoff ¯ HF = (cid:16)P Jj =0 H j F jω (cid:17) ω =1 ,...,N ;its initial cost satisfies H · x = P Jj =0 H j x j . If the zeroth asset is riskless witha price x = 1 and pays off 1 in all states of the world, then a net tradewith zero initial cost can be expressed in terms of the portfolio of risky assets H = ( H , . . . , H J ) ∈ R J and the return matrix ˆ F = ( F jω − x j ) j =1 ,...,J,ω =1 ,...,N . I is given by the image of the J × N return matrix ˆ F , i.e. I = { H ˆ F : H ∈ R J } .
2. Our model includes the case of finitely many trading periods. Let F :=( F t ) Tt =0 be a filtration on (Ω , F ) and S = ( S t ) Tt =0 be an adapted stochasticprocess with values in R J + for some J ≥ S models the uncertain assets. Weassume that a riskless bond with interest rate zero is also given. Then, theset of net trades can be described by the gains from trade processes: ℓ ∈ H isin I provided that there exists predictable integrands H t ∈ (cid:0) L (Ω , F t − ) (cid:1) J for t = 1 , . . . , T such that, ℓ = ( H · S ) T := T X t =1 H t · ∆ S t , where ∆ S t := ( S t − S t − ) . In the frictionless case, the set of net trades is a subspace of H . In general,one might impose restrictions on the set of admissible trading strategies. Forexample, one might exclude short-selling of risky assets. More generally, tradingstrategies might belong to a suitably defined convex cone; in these cases, themarketed subspace I is a convex cone, too.9. In Harrison and Kreps (1979), the market is described by a marketedspace M ⊂ L (Ω , F , P ) and a (continuous) linear functional π on M . In thiscase, I is the kernel of the price system, i.e. I = { X ∈ M : π ( X ) = 0 } .
4. In continuous time, the set of net trades consists of stochastic integralsof the form I = (Z T θ u · dS u : θ ∈ A adm ) , for a suitable set of admissible strategies A adm . There are several possiblechoices of such a set. When the stock price process S is a semi-martingaleone example of A adm is the set of all S -integrable, predictable processes whoseintegral is bounded from below. Other natural choices for A adm would consistof simple integrands only; when S is a continuous process and A adm is the setof process with finite variation then the above integral can be defined throughintegration by parts (see Dolinsky and Soner (2014a, 2015)). Viability
Knightian uncertainty requires a careful adaptation of the notion ofeconomic viability. Harrison and Kreps (1979) and Kreps (1981) show that theabsence of arbitrage is equivalent to a representative agent equilibrium. UnderKnightian uncertainty, such a single agent construction is generally impossibleas the next example shows . Example 3.2.
Suppose that we have a situation of full Knightian uncertainty .Take Ω = [0 , F the Borel sets, let H be the set of all bounded, measurablefunctions on Ω. Let the common pre-order be given by the pointwise order. Takethe relevant contracts R = P + . A bounded measurable function is thus relevantif it is nonnegative everywhere and is strictly positive for at least one ω ∈ Ω.Suppose that the set of net trades is given by multiples of ℓ ( ω ) = 1 (0 , ( ω ). Notethat the contract ℓ itself is both relevant and achievable with zero wealth (a nettrade), hence is an arbitrage. Consequently, by Theorem 2.1, this market is notviable in the sense of Section 2.Consider the Gilboa–Schmeidler utility function U ( X ) = inf ω ∈ Ω u ( ω + X ( ω )) In continuous time, to avoid doubling strategies a lower bound (maybe more general thanabove) has to be imposed on the stochastic integrals. In such cases, the set I is not a linearspace. Compare Example 3 in Kreps (1981). The chosen example is only for illustrative purposes. The arguments that follow carry overto more sophisticated models of Knightian uncertainty that are described by a non-dominatedset of priors M , as in Epstein and Ji (2013), Vorbrink (2014), or Beissner and Denis (2018).In that setting, Ω is the set of continuous functions on [0 , ∞ ) that represent the possibletrajectories of financial prices. In the market, there is uncertainty about the true volatility ofthe price process, yet there is a unanimous agreement that it lies in a certain interval [ σ, σ ].The class of probability measures { P σ } σ ∈ [ σ,σ ] such that the price process has volatility σ under P σ defines a non-dominated set of priors. u : R → R . This particularagent weakly prefers the zero trade to any multiple of ℓ . Indeed, we have U (0) = u (0) ≥ U ( λℓ ) for all λ ∈ R . For positive λ , the agent cares only aboutthe worst state ω = 0 in which the claim λℓ has a payoff of zero. The agentdoes not want to short-sell the claim either because he would then lose moneyin each state of the world except at ω = 0.For this single agent economy, the equilibrium condition (2.1) is satisfiedas the zero trade is optimal; however, the relevance condition (2.2) does nothold true because the agent is indifferent between 0 and the relevant contract1 (0 , ( ω ). Condition (2.2) excludes situations in which the agents of the economydo not desire relevant payoffs. Note that the above agent is not really “represent-ing” the market ( H , τ, ≤ , I , R ) because he does not desire the relevant contract ℓ . The notion of economic equilibrium does not require the existence of a repre-sentative agent, of course. It is natural, and – in fact – closer to reality, to allowfor a sufficiently rich set of heterogeneous agents in an economy. Our abovedefinition of viability thus allows for an economy populated by heterogeneousagents.As the above example has shown, under Knightian uncertainty, reasonablepreferences need not be strictly monotone with respect to every positive (orrelevant) trade nor rule out arbitrage. One might just aim to replace the aboveGilboa–Schmeidler utility function by a strictly monotone utility. While thisapproach is feasible in a probabilistic setting, it does not work under Knightianuncertainty because strictly monotone utility functions do not exist, in general .In our definition of viability, we thus generalize the definition by Harrison andKreps by replacing the assumption of strict monotonicity with the property(2.2) that requires that each relevant contract is desirable for some agent inequilibrium .Our notion of equilibrium does not model endowments explicitly as we as-sume that the zero trade is optimal for each agent. Let us explain why thisreduced approach comes without loss of generality. In general, an agent is givenby a preference relation (cid:23)∈ A and an endowment e ∈ H . Given the set of nettrades, the agent chooses ℓ ∗ ∈ I such that e + ℓ ∗ (cid:23) e + ℓ for all ℓ ∈ I . By suitablymodifying the preference relation, this can be reduced to the optimality of thezero trade at the zero endowment for a suitably modified preference relation.Let X (cid:23) ′ Y if and only if X + e + ℓ ∗ (cid:23) Y + e + ℓ ∗ . It is easy to check that (cid:23) ′ is On the mathematical side, this is related to the absence of strictly positive linear function-als. For example, it is well known that there is no linear functional on the space of boundedmeasurable functions on [0 ,
1] that assigns a strictly positive value to { ω } for every ω ∈ Ω(Aliprantis and Border (1999)). It is thus impossible to construct a probability measure thatassigns a positive mass to a continuum of singletons. This fact carries over to more complexmodels of Knightian uncertainty as the uncertain volatility model. Note that the definition with R = P + is equivalent to the one given by Harrison and Krepsin the probabilistic setup. At the same time, it allows to overcome the problem identified inthe previous Example 3.2 because the single agent economy of the example violates (2.2).Our definition is equivalent to the definition by Harrison and Kreps in the probabilistic setupbecause strictly monotone preferences satisfy (2.2). (cid:23) ′ , wethen have 0 (cid:23) ′ ℓ if and only if e + ℓ ∗ (cid:23) e + ℓ ∗ + ℓ . As I is a cone, ℓ + ℓ ∗ ∈ I ,and we conclude that we have indeed 0 (cid:23) ′ ℓ for all ℓ ∈ I . Sublinear Expectations
Our fundamental theorem of asset pricing char-acterizes the absence of arbitrage with the help of a non–additive expecta-tion E . In decision theory, non–additive probabilities have a long history;Schmeidler (1989) introduces an extension of expected utility theory based onnon–additive probabilities. The widely used max-min expected utility modelof Gilboa and Schmeidler (1989) is another instance. If we define the subjec-tive expectation of a payoff to be the minimal expected payoff over a class ofpriors, then the resulting notion of expectation has the common properties ofan expectation like monotonicity, preservation of constants, but is no longeradditive.In our case, the non–additive expectation has a more objective than subjec-tive flavor because it describes the pricing functional of the market. Whereasan additive probability measure is sufficient to characterize viable markets inmodels with a common prior, in general, such a construction is no longer fea-sible. Indeed, Harrison and Kreps (1979) prove that viability implies that thelinear market pricing functional can be extended from the marketed subspaceto a strictly positive linear functional on the whole space of contingent claims.Under Knightian uncertainty, however, the fact that strictly positive linear func-tionals do not exist is the rule rather than the exception (compare Footnote 12).We thus rely on a non–additive notion of expectation .The pricing functional assigns a nonpositive value to all net trades; in thissense, net trades have the (super)martingale property under this expectation.If we assume for the sake of the discussion that the set of net trades is a linearsubspace, then the pricing functional has to be additive over that subspace. As aconsequence, the value of all net trades under the sublinear pricing expectationis zero. For contingent claims that lie outside the marketed subspace, the pricingoperation of the market is sub–additive.The following two examples illustrate the issue. We start with the simplecase of complete financial markets within finite state spaces. Here, an additiveprobability is sufficient to characterise the absence of arbitrage, as is well known. Example 3.3 (The atom of finance and complete markets) . The basic one–stepbinomial model, that we like to call the atom of finance, consists of two statesof the world, Ω = { , } . An element X ∈ H can be identified with a vector in R . Let ≤ be the usual partial order of R . Then Z = { } and X ∈ P if andonly if X ≥ Ω
0. The relevant contracts are the positive ones, R = P + .There is a riskless asset B and a risky asset S . At time zero, both assetshave value B = S = 1. The riskless asset yields B = 1 + r for an interestrate r > − u in state 1and respectively d in state 2 with u > d . Beissner and Riedel (2019) develop a general equilibrium model based on such non–additive pricing functionals.
12e use the riskless asset B as num´eraire. The discounted net return on therisky asset is ˆ ℓ := S / (1 + r ) − I is the linear space spanned by ˆ ℓ . There isno arbitrage if and only if the unique candidate for a full support martingaleprobability of state one p ∗ = 1 + r − du − d belongs to (0 ,
1) which is equivalent to u > r > d . p ∗ induces the uniquemartingale measure P ∗ with expectation E ∗ [ X ] = p ∗ X (1) + (1 − p ∗ ) X (2) . P ∗ is a linear measure with full support. The market is viable with A = {(cid:22) ∗ } ,the preference relation given by the linear expectation P ∗ , i.e. X (cid:22) ∗ Y if andonly if E ∗ [ X ] ≤ E ∗ [ Y ]. Indeed, under this preference ℓ ∼ ∗ ℓ ∈ I and X ≺ ∗ X + R for any X ∈ H and R ∈ P + . In particular, any ℓ ∈ I is an optimalportfolio and the market is viable.The preceding analysis carries over to all finite Ω and complete financialmarkets.We now turn to a somewhat artificial one period model with uncountablymany states. It serves well the purpose to illustrate the need for sublinearexpectations and thus stands in an exemplary way for more complex modelsinvolving continuous time and uncertain violatility, e.g. Example 3.4 (Highly incomplete one-period models) . This example shows thatsublinear expectations are necessary to characterize the absence of arbitrageunder Knightian uncertainty and with incomplete markets.Let Ω = [0 ,
1] and ≤ be the usual pointwise partial order. Payoffs X arebounded Borel measurable functions on Ω. As in the previous example we have Z = { } and X ∈ P if and only if X ≥ Ω
0. Let the relevant contracts beagain R = P + . Assume that there is a riskless asset with interest rate r ≥ S = 1 at time 0 and assume it pays off S ( ω ) = 2 ω at time 1. As in the previous example, I is spanned by the netreturn ˆ ℓ := S / (1 + r ) − Q satisfying R Ω ω Q ( dω ) = 1 + r is a martingale measure. Denote by Q ac the set of all martingale measures.No single martingale measure is sufficient to characterize the absence ofarbitrage because there is no single linear martingale probability measure withfull support. Indeed, such a measure would have to assign a non-zero value toevery point in Ω, an impossibility for uncountable Ω. Hence, the equivalence “noarbitrage” to “there is a martingale measure with some monotonicity property”does not hold true if one insists on having a linear martingale measure. Instead,one needs to work with the nonlinear expectation E ( X ) := sup Q ∈Q ac E Q [ X ]13or X ∈ H . We claim that E has full support and characterizes the absence ofarbitrage in the sense of Theorem 2.1 and Theorem 2.2.To see that E has full support, note that R ∈ P + if and only if R ≥ ω ∗ ∈ Ω so that R ( ω ∗ ) >
0. Define Q ∗ by Q ∗ := 12 (cid:0) δ { ω ∗ } + δ { − ω ∗ } (cid:1) . Then Q ∗ ∈ Q ac , and we have E ( R ) ≥ E Q ∗ [ R ] = 12 R ( ω ∗ ) + 12 R (1 − ω ∗ ) > E (0) . This example shows that the heterogeneity of agents needed in equilibriumto support an arbitrage-free financial market and the necessity to allow forsublinear expectations are two complementary faces of the same issue.
The Efficient Market Hypothesis (EMH) plays a fundamental role in the historyof Financial Economics. Fama (1970) suggests that expected returns of allsecurities are equal to the safe return of a suitable bond. This conjecture of thefinancial market’s being a “fair game” dates back to Bachelier (1900) and wasrediscovered by Paul Samuelson (1965; 1973).Our framework allows for a discussion of the various forms of the EMH froma general point of view. We show that the EMH is a result of the strength ofassumptions one is willing to make on the common order of the market. If weare convinced that agents’ preferences are monotone with respect to expectedpayoffs (respectively, the almost sure ordering) under a common prior, we ob-tain the strong (respectively, weak) form of the EMH. If we are not willing tomake such a strong assumption on agents’ probabilistic sophistication, weakerKnightian analogs of the EMH result.Throughout this section, let us assume that we have a frictionless one–periodor discrete–time multiple period financial market as in Example 3.1, 1. and 2.In particular, the set of net trades I is a subspace of H . In its original version, the efficient market hypothesis postulates that the “realworld probability” or historical measure P is itself a martingale measure. Wecan reach this conclusion if the common order is given by the expectation undera common prior.Let P be a probability measure on (Ω , F ). Set H = L (Ω , F , P ). Let thecommon order by given by X ≤ Y if and only if the expected payoffs under thecommon prior P satisfy E P [ X ] ≤ E P [ Y ] . (4.1)14n this case, negligible contracts coincide with the contracts with mean zerounder P . Moreover, X ∈ P if E P [ X ] ≥
0. We take R = P + . Proposition 4.1.
Under the assumptions of this subsection, the financial mar-ket is viable if and only if the common prior P is a martingale measure. In thiscase, P is the unique martingale measure.Proof. Note that the common order as given by (4.1) is complete. If P is a mar-tingale measure, the common order ≤ itself defines a linear preference relationunder which the market is viable with A = {≤} .On the other hand if the market is viable, Theorem 2.2 ensure that thereexists a sublinear martingale expectation with full support. By the Riesz dualitytheorem, a martingale functional φ ∈ Q ac can be identified with a probabilitymeasure Q on (Ω , F ). It is absolutely continuous (in our sense defined above)if and only if it assigns the value 0 to all negligible claims. As a consequence,we have E Q [ X ] = 0 whenever E P [ X ] = 0. Then Q = P follows .Hence the only absolutely continuous martingale measure is the commonprior itself. As a consequence, all traded assets have zero net expected returnunder the common prior. A financial market is thus viable if and only if thestrong form of the expectations hypothesis holds true. In its weak form, the efficient market hypothesis states that expected returnsare equal under some (pricing) probability measure P ∗ that is equivalent to thecommon prior (or “real world” probability) P . This hypothesis can be derivedin our framework as follows.Let P be a probability on (Ω , F ) and H = L (Ω , F , P ). In this example, thecommon order is given by the almost surely order under the common prior P ,i.e., X ≤ Y ⇔ P ( X ≤ Y ) = 1 . A payoff is negligible if it vanishes P –almost surely and is positive if it is P –almost surely nonnegative. The typical choice for relevant contracts R are the P –almost surely nonnegative payoffs that are strictly positive with positive P –probability R = (cid:8) R ∈ L (Ω , F , P ) + : P ( R > > (cid:9) . If Q = P , there is an event A ∈ F with Q ( A ) < P ( A ). Set X = 1 A − P ( A ). Then0 = E P [ X ] > Q ( A ) − P ( A ) = E Q [ X ]. Note that it is possible to derive these sets of positive and relevant contracts from theassumption that preferences in A consist of risk averse von Neumann-Morgenstern expectedutility maximizers with strictly increasing Bernoulli utility function. Define X (cid:22) Y if andonly if E P [ U ( X )] ≤ E P [ U ( Y )] for all strictly increasing and concave real functions U . It is wellknown that this order coincides with second order stochastic dominance under P . A randomvariable Y dominates 0 in the sense of second order stochastic dominance if and only if it is P –almost surely nonnegative. Moreover, the set of relevant contracts R corresponds to theset of contracts that are uniformly desirable for all agents.
15 functional φ ∈ H ′ + is an absolutely continuous martingale functional ifand only if it can be identified with a probability measure Q that is absolutelycontinuous with respect to P and if all net trades have expectation zero under φ . In other words, discounted asset prices are Q -martingales. We thus obtaina version of the Fundamental Theorem of Asset Pricing under risk, similar toHarrison and Kreps (1979) and Dalang, Morton, and Willinger (1990). Proposition 4.2.
Under the assumptions of this subsection, the financial mar-ket is viable if and only if there is a martingale measure Q that has a boundeddensity with respect to P .Proof. If P ∗ is a martingale measure equivalent to P , define X (cid:22) ∗ Y if and onlyif E P ∗ [ X ] ≤ E P ∗ [ Y ]. Then the market is viable with A = {(cid:22) ∗ } .If the market is viable, Theorem 2.2 ensures that there exists a sublinearmartingale expectation with full support. By the Riesz duality theorem, amartingale functional φ ∈ Q ac can be identified with a probability measure Q φ that is absolutely continuous with respect to P , has a bounded densitywith respect to P , and all net trades have zero expectation zero under Q φ . Inother words, discounted asset prices are Q φ -martingales. From the full supportproperty, the family { Q φ } φ ∈Q ac has the same null sets as P . By the Halmos-Savage Theorem, there exists an equivalent martingale measure P ∗ . We turn our attention to the EMH under Knightian uncertainty. We considerfirst the case when the common order is derived from a common set of pri-ors, inspired by the multiple prior approach in decision theory (Bewley (2002);Gilboa and Schmeidler (1989)). We then discuss a second-order Bayesian ap-proach that is inspired by the smooth ambiguity model (Klibanoff, Marinacci, and Mukerji(2005)).
We consider a generalization of the original EMH to Knightian uncertaintythat shares a certain analogy with Bewley’s incomplete expected utility model(Bewley (2002)) and Gilboa and Schmeidler’s maxmin expected utility (Gilboa and Schmeidler(1989)) . Agents might have different subjective perceptions, but they share acommon set of priors M . Their preferences are weakly monotone with respectto the uniform order induced by expectations over the set of priors.More formally, let Ω be a metric space and M be a convex, weak ∗ -closed setof common priors on (Ω , F ). Define a semi-norm k X k M := sup P ∈M E P | X | . For the relation between the two approaches, compare also the discussion of objective andsubjective ambiguity in Gilboa, Maccheroni, Marinacci, and Schmeidler (2010). L (Ω , F , M ) be the closure of continuous and bounded functions on Ω underthe semi-norm k · k M . If we identify the functions which are P -almost surelyequal for every P ∈ M , then H = L (Ω , F , M ) is a Banach space. Furthermore,the topological dual of L (Ω , F , M ) can be identified with probability measuresthat admit a bounded density with respect to some measure in M , compareBion-Nadal, Kervarec, et al. (2012); Beissner and Denis (2018). Therefore, anyabsolutely continuous martingale functional Q ∈ Q ac is a probability measureand M is closed in the weak ∗ topology induced by L (Ω , F , M ).Consider the uniform order induced by expectations over M , X ≤ Y ⇔ ∀ P ∈ M E P [ X ] ≤ E P [ Y ] . Then, Z ∈ Z if E P [ Z ] = 0 for every P ∈ M . A contract X is positive if E P [ X ] ≥ P ∈ M . A natural choice for the relevant contracts consistsof nonnegative contracts with a positive return under some prior belief, i.e. R = { R ∈ H : 0 ≤ inf P ∈M E P [ R ] and 0 < sup P ∈M E P [ R ] } . Proposition 4.3.
Under the assumptions of this subsection, if the financialmarket is viable, then the set of absolutely continuous martingale functionals Q ac is a subset of the set of priors M .Proof. Set E M ( X ) := sup P ∈M E P [ X ]. Then, Y ≤ E M ( Y ) ≤ Q ∈ Q ac with the preference relation given by X (cid:22) Q Y if E Q [ X − Y ] ≤ Q / ∈ M . Since M is a weak ∗ -closed and convex subsetof the topological dual of L (Ω , F , M ), there exists X ∗ ∈ L (Ω , F , M ) with E M ( X ∗ ) < < E Q [ X ∗ ] by the Hahn-Banach theorem. In particular, X ∗ ∈L (Ω , F , M ) and X ∗ ≤
0. Since (cid:22) Q is weakly monotone with respect to ≤ , X ∗ (cid:22) Q
0. Hence, E Q [ X ∗ ] ≤ X ∗ . Therefore, Q ac ⊂ M .Expected returns of traded securities are thus not necessarily the same underall P ∈ M . However, for a smaller class Q ac of M they remain the same andthus the strong form of EMH holds for this subset of the priors.Let H M be the subspace of claims that have no ambiguity in the mean in thesense that E P [ X ] is the same constant for all P ∈ M . Consider the submarket( H M , τ, ≤ , I M , R M ) with I M := I ∩ H M and R M := R ∩ H M . Restricted tothis market, the measures Q ac and M are identical and the strong EMH holdstrue.The following simple example illustrates these points. Example 4.4.
Let Ω = { , } , H be all functions on Ω. Then, H = R andwe write X = ( x, y, v, w ) for any X ∈ H . Let I = { ( x, y, ,
0) : x + y = 0 } .Consider the priors given by M := (cid:26)(cid:18) p, − p, , (cid:19) : p ∈ (cid:20) , (cid:21)(cid:27) . Q ac = { Q ∗ } = { ( , , , ) } . Notice that Q ∗ ∈ M .In this case, H M = { X = ( x, y, v, w ) ∈ H : x = y } . In particular, all priorsin M coincide with Q ∗ when restricted to H M . Hence, for the claims that aremean-ambiguity-free, the strong efficient market hypothesis holds true. Let M be a common set of priors on (Ω , F ). Set H := B b . Let the commonorder be given by the quasi-sure ordering under the common set of priors M ,i.e. X ≤ Y ⇔ P ( X ≤ Y ) = 1 , ∀ P ∈ M . In this case, a contract X is negligible if it vanishes M –quasi surely , i.e. withprobability one for all P ∈ M . An indicator function 1 A is thus negligible if theset A is polar , i.e. a null set with respect to every probability in M . Take theset of relevant contracts to be R = { R ∈ P : ∃ P ∈ M such that P ( R > > } . Proposition 4.5.
Under the assumptions of this subsection, the financial mar-ket is viable if and only if there is a set of finitely additive measures Q thatare martingale measures and that has the same polar sets as the common set ofpriors P .Proof. Suppose that the market is viable. We show that the class Q ac fromTheorem 2.2 satisfies the desired properties. The martingale property followsby definition and from the fact that I is a linear space. Suppose that A is polar.Then, 1 A is negligible and from the absolute continuity property, it follows φ ( A ) = 0 for any φ ∈ Q ac . On the other hand, if A is not polar, 1 A ∈ R andfrom the full support property, it follows that there exists φ A ∈ Q ac such that φ A ( A ) >
0. Thus, A is not Q ac -polar. We conclude that M and Q ac sharethe same polar sets. For the converse implication, define E ( · ) := sup φ ∈Q E φ [ · ].Using the same argument as above, E is a sublinear martingale expectation withfull support. From Theorem 2.2 the market is viable.Under Knightian uncertainty, there is indeterminacy in arbitrage–free pricesas there is always a range of economically justifiable arbitrage–free prices. Suchindeterminacy has been observed in full general equilibrium analysis as well(Rigotti and Shannon (2005); Dana and Riedel (2013); Beissner and Riedel (2019)). These sets of positive and relevant contracts can be derived from Gilboa–Schmeidlerutilities. Define X (cid:22) Y if and only if E M [ U ( X )] := inf P ∈M E P [ U ( X )] ≤ E M [ U ( Y )] for allstrictly increasing and concave real functions U . The 0 (cid:22) Y is equivalent to Y dominatingthe zero contract in the sense of second order stochastic dominance under all P ∈ M . Hence, Y is nonnegative almost surely for all P ∈ M .
18n this sense, Knightian uncertainty shares a similarity with incomplete mar-kets and other frictions like transaction costs, but the economic reason for theindeterminacy is different.
Let us next turn to the case that arises when the order is modeled by correspondsto a second–order Bayesian approach, in the spirit of the smooth ambiguitymodel (Klibanoff, Marinacci, and Mukerji (2005)),Let F be a sigma algebra on Ω and P = P (Ω) the set of all probabilitymeasures on (Ω , F ). Let µ be a second order prior, i.e. a probability measure on P . We define a common prior in this setting as follows. The set functionˆ P : F → [0 ,
1] defined as ˆ P ( A ) = R P P ( A ) µ ( d P ) is a probability measure on(Ω , F ). Let H = L (Ω , F , ˆ P ).The common order is given by X ≤ Y ⇔ µ ( { P ∈ P : P ( X ≤ Y ) = 1 } ) = 1 . A contract is positive if it is P –almost surely nonnegative for all priors in thesupport of the second order prior µ . A natural choice for relevant contracts is R = { R ∈ P : µ [ P ∈ P : P ( R > > > } , i.e. the set of beliefs P under which the contract is strictly positive with positiveprobability is not negligible according to the second order prior Proposition 4.6.
Under the assumptions of this subsection, the financial mar-ket is viable if and only if there is a martingale measure Q that has the form Q ( A ) = Z P Z A D d P µ ( d P ) for some state price density D .Proof. The set function ˆ P : F → [0 ,
1] defined as ˆ P ( A ) = R P P ( A ) µ ( d P ) is aprobability measure on (Ω , F ). The induced ˆ P -a.s. order coincides with ≤ ofthis subsection. The result thus follows from Proposition 4.2. From Theorem 15.18 of Aliprantis and Border (1999), the space of probability measure isa Borel space if and only if Ω is a Borel space. This allows to define second order priors. These sets of positive and relevant contracts can be derived from smooth ambiguity utilityfunctions. Define X (cid:22) Y if and only if Z P ψ ( E P [ U ( X )]) µ ( d P ) ≤ Z P ψ ( E P [ U ( X )]) µ ( d P )for all strictly increasing and concave real functions U and ψ . Recall that ψ reflects uncertaintyaversion.The 0 (cid:22) Y is equivalent to Y dominating the zero contract in the sense of second orderstochastic dominance for µ –almost all P ∈ P , i.e. when Y ≥ Q . We conclude this section by relating our work to recent results in MathematicalFinance. Our approach gives a microeconomic foundation to the characteriza-tion of absence of arbitrage in “robust” or “model–free” finance.In this subsection, Ω is a metric space. We say X ≤ Y if X ≤ Ω Y, (4.2)which implies Z = { } .In the finance literature, this approach is called model-independent as it doesnot rely on any probability measure. There is still a model, of course, given byΩ. A contract is nonnegative, X ∈ P , if X ( ω ) ≥ ω ∈ Ω and R ∈ P + if R ∈ P and there exists ω ∈ Ω such that R ( ω ) > .We start with the following large set of relevant contracts R op := P + = { R ∈ P : ∃ ω ∈ Ω such that R ( ω ) > } . With this notion of relevance, an investment opportunity ℓ is an arbitrage if ℓ ( ω ) ≥ ω with a strict inequality for some ω , corresponding to thenotion of one point arbitrage considered in Riedel (2015). In this setting, noarbitrage is equivalent to the existence a set of martingale measures Q op so thatfor each point there exists Q ∈ Q op putting positive mass to that point.In a second example, one requires the relevant contracts to be continuous,i.e., R open := { R ∈ C b (Ω) ∩ P : ∃ ω ∈ Ω such that R ( ω ) > } . It is clear that when R ∈ R then it is non-zero on an open set. Hence, in thisexample the empty set is the only small set and the large sets are the ones thatcontain a non-empty open set.Then, ℓ ∈ I is an arbitrage opportunity if it is nonnegative and is strictly pos-itive on an open set, corresponding to the notion of open arbitrage that appearsin Burzoni, Frittelli, and Maggis (2016); Riedel (2015); Dolinsky and Soner (2014b).Acciaio, Beiglb¨ock, Penkner, and Schachermayer (2016) defines a contractto be an arbitrage when it is positive everywhere. In our context, this definesthe relevant contracts as those that are positive everywhere, i.e., R + := { R ∈ P : R ( ω ) > , ∀ ω ∈ Ω } . One might also compare the similar approach in Burzoni, Frittelli, and Maggis (2016). R u = { R ∈ P : ∃ c ∈ (0 , ∞ ) such that R ≡ c } . (4.3)Hence, ℓ ∈ I is an arbitrage if is uniformly positive, which is sometimes called uniform arbitrage . Notice that with the choice R u , the notions of arbitrage andfree lunch with vanishing risk are equivalent.The no arbitrage condition with R u is the weakest while the one with R op isthe strongest. The first one is equivalent to the existence of one sublinear mar-tingale expectation. The latter one is equivalent to the existence of a sublinearexpectation that puts positive measure to all points.In general, the no-arbitrage condition based on R + is not equivalent to theabsence of uniform arbitrage. However, absence of uniform arbitrage impliesthe existence of a linear bounded functional that is consistent with the market.In particular, risk neutral functionals are positive on R u . Moreover, if the set I is “large” enough then one can show that the risk neutral functionals give rise tocountably additive measures. In Acciaio, Beiglb¨ock, Penkner, and Schachermayer(2016), this conclusion is achieved by using the so-called “power-option” placedin the set I as a static hedging possibility, compare also Bartl, Cheridito, Kupper, and Tangpi(2017). Let ( H , τ, ≤ I , R ) be a given financial market. Recall that ( H , τ ) is a metrizabletopological vector space; we write H ′ for its topological dual. We let H ′ + be theset of all positive functionals, i.e., ϕ ∈ H ′ + provided that ϕ ( X ) ≥ X ≥ X ∈ H .The following functional generalizes the notion of super-replication func-tional from the probabilistic to our order-theoretic framework. It plays a centralrole in our analysis. For X ∈ H , let D ( X ) := inf { c ∈ R : ∃{ ℓ n } ∞ n =1 ⊂ I , { e n } ∞ n =1 ⊂ H + , e n τ → , (5.1)such that c + e n + ℓ n ≥ X } . Following the standard convention, we set D ( X ) to plus infinity, when the aboveset is empty. Note that D is extended real valued. In particular, it takes thevalue + ∞ when there are no super-replicating portfolios. It might also take thevalue −∞ if there is no lower bound.We observe first that the absence of free lunches with vanishing risk can beequivalently described by the statement that the super-replication functional D assigns a strictly positive value to all relevant contracts. Proposition 5.1.
The financial market is strongly free of arbitrage if and onlyif D ( R ) > for every R ∈ R . roof. Suppose { ℓ n } ∞ n =1 ⊂ I is a free lunch with vanishing risk. Then, there is R ∗ ∈ R and { e n } ∞ n =1 ⊂ H + with e n τ → e n + ℓ n ≥ R ∗ . In view of thedefinition, we obtain D ( R ∗ ) ≤ D ( R ∗ ) ≤ R ∗ ∈ R . Then,the definition of D ( R ∗ ) implies that there is a sequence of real numbers { c k } ∞ k =1 with c k ↓ D ( R ∗ ), net trades { ℓ k,n } ∞ n =1 ⊂ I , and { e k,n } ∞ n =1 ⊂ H + with e k,n τ → n → ∞ such that c k + e k,n + ℓ k,n ≥ R ∗ , ∀ n, k ∈ N . Let B r (0) be the ball with radius r centered at zero with the metric compatiblewith τ . For every k , choose n = n ( k ) such that e k,n ∈ B k (0). Set ˜ ℓ k := ℓ k,n ( k ) and ˜ e k := e k,n ( k ) + ( c k ∨ e k + ˜ ℓ k ≥ R ∗ for every k . Since ˜ e k τ → { ˜ ℓ k } ∞ k =1 is a free lunch with vanishing risk.It is clear that D is convex and we now use the tools of convex duality tocharacterize this functional in more detail. Recall the set of absolutely contin-uous martingale functionals Q ac defined in Section 2. Proposition 5.2.
Assume that the financial market is strongly free of arbi-trage. Then, the super-replication functional D defined in (5.1) is a lower semi-continuous, sublinear martingale expectation with full support. Moreover, D ( X ) = sup ϕ ∈Q ac ϕ ( X ) , X ∈ H . The technical proof of this statement can be found in Appendix A. Theimportant insight is that the super-replication functional can be described by afamily of linear functionals. In the probabilistic setup, they correspond to thefamily of (absolutely continuous) martingale measures. With the help of thisduality, we are now able to carry out the proof of our first main theorem.
Proof of Theorem 2.1.
Suppose first that the market is viable and for some R ∗ ∈R , there are sequences { e n } ∞ n =1 ⊂ H + and { ℓ n } ∞ n =1 ⊂ I with e n τ →
0, and e n + ℓ n ≥ R ∗ . By viability, there is a family of agents {(cid:22) a } a ∈ A ⊂ A suchthat ℓ ∗ a = 0 is optimal for each agent a ∈ A and for some a ∈ A we have R ∗ ≻ a
0. Since ≤ is a pre-order compatible with the vector space operations,we have − e n + R ∗ ≤ ℓ n . As (cid:22) a ∈ A is monotone with respect to ≤ , we have − e n + R ∗ (cid:22) a ℓ n . Since ℓ ∗ a = 0 is optimal, we get − e n + R ∗ (cid:22) a
0. By lowersemi–continuity of (cid:22) , we conclude that R ∗ (cid:22) a
0, a contradiction.Suppose now that the market is strongly free of arbitrage. By Proposition5.1, D ( R ) >
0, for every R ∈ R . In particular, this implies that the family Q ac is non-empty, as otherwise the supremum over Q ac would be −∞ . For each ϕ ∈ Q ac , define (cid:22) ϕ by, X (cid:22) ϕ Y, ⇔ ϕ ( X ) ≤ ϕ ( Y ) . One directly verifies that (cid:22) ϕ ∈ A . Moreover, ϕ ( ℓ ) ≤ ϕ (0) = 0 for any ℓ ∈ I implies that ℓ ∗ ϕ = 0 is optimal for (cid:22) ϕ and (2.1) is satisfied. Finally, Proposition22.1 and Proposition 5.2 imply that for any R ∈ R , there exists ϕ ∈ Q ac suchthat ϕ ( R ) >
0; thus, (2.2) is satisfied. We deduce that {(cid:22) ϕ } ϕ ∈Q ac supports thefinancial market ( H , τ, ≤ , I , R ).The previous arguments also imply our version of the fundamental theoremof asset pricing. In fact, with absence of arbitrage, the super-replication functionis a lower semi-continuous sublinear martingale expectation with full support.Convex duality allows to prove the converse. Proof of Theorem 2.2.
Suppose the market is viable. From Theorem 2.1, it isstrongly free of arbitrage. From Proposition 5.2, the super-replication functionalis the desired lower semi-continuous sublinear martingale expectation with fullsupport.Suppose now that E is a lower semi-continuous sublinear martingale expec-tation with full support. In particular, E is a convex, lower semi-continuous,proper functional. Then, by the Fenchel-Moreau theorem, E ( X ) = sup ϕ ∈ dom ( D ∗ ) ϕ ( X ) , where dom ( D ∗ ) = { ϕ ∈ H ′ : ϕ ( X ) ≤ E ( X ) , ∀ X ∈ H } . We now proceed asin the proof of Theorem 2.1, to verify the viability of ( H , τ, ≤ , I , R ) using thepreference relations {(cid:22) ϕ } ϕ ∈D ∗ . This paper studies the economic viability of a given financial market withoutassuming a common prior of the state space. We show that it is possible tounderstand viability and the absence of arbitrage based on a common notionof “more” that is shared by all potential agents of the economy. A given finan-cial market is viable if and only if a sublinear pricing functional exists that isconsistent with the given asset prices.Our paper also shows how the properties of the common order are reflected inexpected equilibrium returns. When the common order is given by the expectedvalue under some common prior, expected returns under that prior have to beequal in equilibrium, and thus, Fama’s Efficient Market Hypothesis results. Ifthe common order is determined by the almost sure order under some commonprior, we obtain the weak form of the efficient market hypothesis that statesthat expected returns are equal under some (martingale) measure that sharesthe same null sets as the common prior.In situations of Knightian uncertainty, it might be too demanding to imposea common prior for all agents. When Knightian uncertainty is described by aclass of priors, it is necessary to replace the linear (martingale) expectation bya sublinear expectation. It is then no longer possible to reach the conclusionthat expected returns are equal under some probability measure. Knightianuncertainty might thus be an explanation for empirical violations of the Effi-cient Market Hypothesis. In particular, there is always a range of economically23ustifiable arbitrage–free prices. In this sense, Knightian uncertainty sharessimilarities with markets with friction or that are incomplete, but the economicreason for the price indeterminacy is different.
A Proof of Proposition 5.2
We separate the proof in several steps. Recall that the super-replication func-tional D is defined in (5.1). Lemma A.1.
Assume that the financial market is strongly free of arbitrage.Then, D is convex, lower semi–continuous and D ( X ) > −∞ for every X ∈ H .Proof. The convexity of D follows immediately from the definitions. To provelower semi-continuity, consider a sequence X k τ → X with D ( X k ) ≤ c . Then, bydefinition, for every k there exists a sequence { e k,n } ∞ n =1 ⊂ H + with e k,n τ → n → ∞ and a sequence { ℓ k,n } ∞ n =1 ⊂ I such that c + k + e k,n + ℓ k,n ≥ X k , forevery k, n . Let B r (0) be the ball of radius r centered around zero in the metriccompatible with τ . Choose n = n ( k ) such that e k,n ∈ B k and set ˜ e k := e k,n ( k ) ,˜ ℓ k := ℓ k,n ( k ) . Then, c + k + ˜ e k + ( X − X k ) + ˜ ℓ k ≥ X and k + ˜ e k + ( X − X k ) τ → k → ∞ . Hence, D ( X ) ≤ c . This proves that D is lower semi-continuous.The constant claim 1 is relevant and by Proposition 5.1, D (1) ∈ (0 , X ∈ H such that D ( X ) = −∞ . For λ ∈ [0 , X λ := X + λ (1 − X ). Theconvexity of D implies that D ( X λ ) = −∞ for every λ ∈ [0 , D is lowersemi-continuous, 0 < D (1) ≤ lim λ → D ( X λ ) = −∞ , a contradiction. Lemma A.2.
Assume that the financial market is strongly free of arbitrage.The super-replication functional D is a sublinear expectation with full-support.Moreover, D ( c ) = c for every c ∈ R , and D ( X + ℓ ) ≤ D ( X ) , ∀ ℓ ∈ I , X ∈ H . (A.1) In particular, D has the martingale property.Proof. We prove this result in two steps.
Step 1.
In this step we prove that D is a sublinear expectation. Let X, Y ∈ H such that X ≤ Y . Suppose that there are c ∈ R , { ℓ n } ∞ n =1 ⊂ I and { e n } ∞ n =1 ⊂H + with e n τ → Y ≤ c + e n + ℓ n . Then, from the transitivity of ≤ ,we also have X ≤ c + e n + ℓ n . Hence, D ( X ) ≤ D ( Y ) and consequently D ismonotone with respect to ≤ .Translation-invariance, D ( c + g ) = c + D ( g ), follows directly from the defi-nitions.We next show that D is sub-additive. Fix X, Y ∈ H . If either D ( X ) = ∞ or D ( Y ) = ∞ . Then, since by Lemma A.1 D > −∞ , we have D ( X ) + D ( Y ) = ∞ and the sub-additivity follows directly. Now we consider the case24 ( X ) , D ( Y ) < ∞ . Hence, there are c X , c Y ∈ R , { ℓ Xn } ∞ n =1 , { ℓ Yn } ∞ n =1 ⊂ I and { e Xn } ∞ n =1 , { e Yn } ∞ n =1 ⊂ H + with e Xn , e Yn τ → c X + ℓ Xn + e Xn ≥ X, c Y + ℓ Yn + e Yn ≥ Y. Set ¯ c := c X + c Y , ¯ ℓ n := ℓ Xn + ℓ Yn , ¯ e n := ℓ Xn + ℓ Yn . Since I , P are positive cones, { ¯ ℓ n } ∞ n =1 ⊂ I , ¯ e n τ → c + ¯ e n + ¯ ℓ n ≥ X + Y ⇒ D ( X + Y ) ≤ ¯ c. Since this holds for any such c X , c Y , we conclude that D ( X + Y ) ≤ D ( X ) + D ( Y ) . Finally we show that D is positively homogeneous of degree one. Supposethat c + e n + ℓ n ≥ X for some constant c , { ℓ n } ∞ n =1 ⊂ I and { e n } ∞ n =1 ⊂ H + with e n τ →
0. Then, for any λ > n ∈ N , λc + λe n + λℓ n ≥ λX . Since λℓ n ∈ I and λe n τ →
0, this implies that D ( λX ) ≤ λ D ( X ) , λ > , X ∈ H . (A.2)Notice that above holds trivially when D ( X ) = + ∞ . Conversely, if D ( λX ) =+ ∞ we are done. Otherwise, we use (A.2) with λX and 1 /λ , D ( X ) = D (cid:18) λ λX (cid:19) ≤ λ D ( λX ) , ⇒ λ D ( X ) ≤ D ( λX ) . Hence, D positively homogeneous and it is a sublinear expectation. Step 2.
In this step, we assume that the financial market is strongly free ofarbitrages. Since 0 ∈ I , we have D (0) ≤
0. If the inequality is strict we obviouslyhave a free lunch with vanishing risk, hence D (0) = 0 and from translation-invariance the same applies to every c ∈ R . Moreover, by Proposition 5.1, D has full support. Thus, we only need to prove (A.1).Suppose that X ∈ H , ℓ ∈ I and c + e n + ℓ Xn ≥ X . Since I is a convex cone, ℓ Xn + ℓ ∈ I and c + e n + ( ℓ + ℓ Xn ) ≥ X + ℓ . Therefore, D ( X + ℓ ) ≤ c . Since thisholds for all such constants, we conclude that D ( X + ℓ ) ≤ D ( X ) for all X ∈ H .In particular D ( ℓ ) ≤ Remark A.3.
Note that for H = ( B b , k · k ∞ ), the definition of D reduces tothe classical one: D ( X ) := inf { c ∈ R : ∃ ℓ ∈ I , such that c + ℓ ≥ X } . (A.3)Indeed, if c + ℓ ≥ X for some c and ℓ , one can use the constant sequences ℓ n ≡ ℓ and e n ≡ D in (5.1) is less or equal than the one in (A.3).For the converse inequality observe that if c + e n + ℓ n ≥ X for some c, ℓ n and e n with k e n k ∞ →
0, then the infimum in (A.3) is less or equal than c . Thethesis follows. Lemma A.1 is in line with the well known fact that the classicalsuper-replication functional in B b is Lipschitz continuous with respect to thesup-norm topology. 25he results of Lemma A.2 and Lemma A.1 imply that the super-replicationfunctional defined in (5.1) is a regular convex function in the language of convexanalysis, compare, e.g., Rockafellar (2015). By the classical Fenchel-Moreautheorem, we have the following dual representation of D , D ( X ) = sup ϕ ∈H ′ { ϕ ( X ) − D ∗ ( ϕ ) } , X ∈ H , where D ∗ ( ϕ ) = sup Y ∈H { ϕ ( Y ) − D ( Y ) } , ϕ ∈ H ′ . Since ϕ (0) = D (0) = 0, D ∗ ( ϕ ) ≥ ϕ (0) − D (0) = 0 for every ϕ ∈ H ′ . However, itmay take the value plus infinity. Set, dom ( D ∗ ) := { ϕ ∈ H ′ : D ∗ ( ϕ ) < ∞} . Lemma A.4.
We have dom ( D ∗ ) = (cid:8) ϕ ∈ H ′ + : D ∗ ( ϕ ) = 0 (cid:9) = (cid:8) ϕ ∈ H ′ + : ϕ ( X ) ≤ D ( X ) , ∀ X ∈ H (cid:9) . (A.4) In particular, D ( X ) = sup ϕ ∈ dom ( D ∗ ) ϕ ( X ) , X ∈ H . Furthermore, there are free lunches with vanishing risk in the financial market,whenever dom ( D ∗ ) is empty.Proof. Clearly the two sets on the right of (A.4) are equal and included in dom ( D ∗ ). The definition of D ∗ implies that ϕ ( X ) ≤ D ( X ) + D ∗ ( ϕ ) , ∀ X ∈ H , ϕ ∈ H ′ . By homogeneity, ϕ ( λX ) ≤ D ( λX ) + D ∗ ( ϕ ) , ⇒ ϕ ( X ) ≤ D ( X ) + 1 λ D ∗ ( ϕ ) , for every λ > X ∈ H . Suppose that ϕ ∈ dom ( D ∗ ). We then let λ go toinfinity to arrive at ϕ ( X ) ≤ D ( X ) for all X ∈ B b . Hence, D ∗ ( ϕ ) = 0.Fix X ∈ H + . Since ≤ is monotone with respect to ≤ Ω , − X ≤
0. Then, bythe monotonicity of D , ϕ ( − X ) ≤ D ( − X ) ≤ D (0) ≤
0. Hence, ϕ ∈ H ′ + .Now suppose that dom ( D ∗ ) is empty or, equivalently, D ∗ ≡ ∞ . Then, thedual representation implies that D ≡ −∞ . In view of Proposition 5.1, there arefree lunches with vanishing risk in the financial market.We next show that, under the assumption of absence of free lunch withvanishing risk with respect to any R , the set dom ( D ∗ ) is equal to Q ac definedin Section 2. Since any relevant set R by hypothesis contains R u defined in(4.3), to obtain this conclusion it would be sufficient to assume the absence offree lunch with vanishing risk with respect to any R u .26 emma A.5. Suppose the financial market is strongly free of arbitrage withrespect to R . Then, dom ( D ∗ ) is equal to the set of absolutely continuous mar-tingale functionals Q ac .Proof. The fact that dom ( D ∗ ) is non-empty follows from Lemma A.2 and LemmaA.4. Fix an arbitrary ϕ ∈ dom ( D ∗ ). By Lemma A.2, D ( c ) = c for every constant c ∈ R . In view of the dual representation of Lemma A.4, cϕ (1) = ϕ ( c ) ≤ D ( c ) = c, ∀ c ∈ R . Hence, ϕ (1) = 1.We continue by proving the monotonicity property. Suppose that X ∈ P .Since 0 ∈ I , we obviously have D ( − X ) ≤
0. The dual representation impliesthat ϕ ( − X ) ≤ D ( − X ) ≤
0. Thus, ϕ ( X ) ≥ ℓ ∈ I . Obviously D ( ℓ ) ≤ ϕ ( ℓ ) ≤ D ( ℓ ) ≤
0. Hence ϕ is a martingale functional.The absolute continuity follows as in Lemma E.3. Hence, ϕ ∈ Q ac .To prove the converse, fix an arbitrary ϕ ∈ Q ac . Suppose that X ∈ H , c ∈ R , { ℓ n } ∞ n =1 ⊂ I and { e n } ∞ n =1 ⊂ H + with e n τ → c + e n + ℓ n ≥ X .From the properties of ϕ ,0 ≤ ϕ ( c + e n + ℓ n − X ) = ϕ ( c + e n − X ) + ϕ ( ℓ n ) ≤ c − ϕ ( X − e n ) . Since e n τ → ϕ is continuous, ϕ ( X ) ≤ D ( X ) for every X ∈ H . Therefore, ϕ ∈ dom ( D ∗ ). Proof of Proposition 5.2.
It follows directly from Lemma A.4 and Lemma A.5.We have the following immediate corollary, which states that is the first partof the Fundamental Theorem of Asset Pricing in this context.
Corollary A.6.
The financial market is strongly free of arbitrage if and only Q ac = ∅ and for any R ∈ R , there exists ϕ R ∈ Q ac such that ϕ R ( R ) > .Proof. By contradiction, suppose that there exists R ∗ such that e n + ℓ n ≥ R ∗ with e n τ →
0. Take ϕ R ∗ such that ϕ R ∗ ( R ∗ ) > <ϕ R ∗ ( R ∗ ) ≤ ϕ ( e n + ℓ n ) ≤ ϕ ( e n ). Since ϕ ∈ H ′ + , ϕ ( e n ) → n → ∞ , which isa contradiction.In the other direction, assume that the financial market is strongly free ofarbitrage. By Lemma A.5, dom ( D ∗ ) = Q ac . Let R ∈ R and note that, byProposition 5.1, D ( R ) >
0. It follows that there exists ϕ R ∈ dom ( D ∗ ) = Q ac satisfying ϕ R ( R ) > Remark A.7.
The set of positive functionals Q ac ⊂ H ′ + is the analogue of theset of local martingale measures of the classical setting. Indeed, all elementsof ϕ ∈ Q ac can be regarded as supermartingale “measures”, since ϕ ( ℓ ) ≤ ℓ ∈ I . Moreover, the property ϕ ( Z ) = 0 for every Z ∈ Z can be regarded27s absolute continuity with respect to null sets. The full support property is ouranalog to the converse absolute continuity. However, the full-support propertycannot be achieved by a single element of Q ac .Bouchard and Nutz (2015) study arbitrage for a set of priors M . The ab-solute continuity and the full support properties then translate to the statementthat “ M and Q have the same polar sets”. In the paper by Burzoni, Frittelli, and Maggis(2016), a class of relevant sets S is given and the two properties can summarisedby the statement “the set S is not contained in the polar sets of Q ”.Also, when H = B b , H ′ is the class of bounded additive measures ba . It isa classical question whether one can restrict Q to the set of countable additivemeasures ca r (Ω). In several of the examples described in Section 3 and 4 thisis proved. However, there are examples for which this is not true. B Linearly Growing Claims
Let B (Ω , F ) be the set of all F measurable real-valued functions on Ω. AnyBanach space contained in B (Ω , F ) satisfies the requirements for H . In ourexamples, we used the spaces L (Ω , F , P ), L (Ω , F , P ), L (Ω , F , M ) (defined inthe subsection 4.3.1) and B b (Ω , F ), the set of all bounded functions in B (Ω , F ),with the supremum norm. In the latter case, the super-hedging functional enjoysseveral properties as discussed in Remark A.3.Since we require that I ⊂ H (see Section 2), in the case of H = B b (Ω , F ) thismeans that all the trading instruments are bounded. This could be restrictivein some applications and we now provide another example that overcomes thisdifficulty. To define this set, fix L ∗ ∈ B (Ω , F ) with L ∗ ≥ Ω
1. Consider the linearspace B ℓ := (cid:8) X ∈ B (Ω , F ) : ∃ α ∈ R + such that | X | ≤ Ω αL ∗ (cid:9) equipped with the norm, k X k ℓ := inf { α ∈ R + : | X | ≤ Ω αL ∗ } = (cid:13)(cid:13)(cid:13)(cid:13) XL ∗ (cid:13)(cid:13)(cid:13)(cid:13) ∞ . We denote the topology induced by this norm by τ ℓ . Then, B ℓ (Ω , F ) with τ ℓ is a Banach space and satisfies our assumptions. Note that if L ∗ = 1, then B ℓ (Ω , F ) = B b (Ω , F ).Now, suppose that L ∗ ( ω ) := c ∗ + ˆ ℓ ( ω ) , ω ∈ Ω , (B.1)for some c ∗ >
0, ˆ ℓ ∈ I . Then, one can define the super-replication functional asin (A.3). 28 No Arbitrage versus No Free-Lunch-with-Vanishing-Risk
We recall the definition of arbitrage. Let ( H , τ, ≤ , I , R ) be a financial market.We say that an achievable contract ℓ ∈ I , is an arbitrage if there exists a relevantcontract R ∗ ∈ R with ℓ ≥ R ∗ .When R = P + the definitions above become simpler. In this case ℓ ∈ I is an arbitrage if and only if ℓ ∈ P + . From the definition, it is clear that anarbitrage opportunity is always a free lunch with vanishing risk. The purposeof this section is to investigate when these two notions are equivalent. C.1 Attainment
We first show that the attainment property is useful in discussing the connectionbetween two different notions of arbitrage. We start with a definition.
Definition C.1.
We say that a financial market has the attainment property ,if for every X ∈ H there exists a minimizer in (5.1), i.e., there exists ℓ X ∈ I satisfying, D ( X ) + ℓ X ≥ X. Proposition C.2.
Suppose that a financial market has the attainment property.Then, it is strongly free of arbitrage if and only if it has no arbitrages.Proof.
Let R ∗ ∈ R . By hypothesis, there exist ℓ ∈ I ∗ so that D ( R ∗ ) + ℓ ∗ ≥ R ∗ .If the market has no arbitrage, then we conclude that D ( R ∗ ) >
0. In view ofProposition 5.1, this proves that the financial market is also strongly free ofarbitrage. Since no arbitrage is weaker condition, they are equivalent.
C.2 Finite discrete time markets
In this subsection and in the next section, we restrict ourselves to arbitrageconsiderations in finite discrete-time markets.We start by introducing a discrete filtration F := ( F t ) Tt =0 on subsets of Ω.Let S = ( S t ) Tt =0 be an adapted stochastic process , with values in R M + forsome M . For every ℓ ∈ I there exist predictable integrands H t ∈ B b (Ω , F t − )for all t = 1 , . . . , T such that, ℓ = ( H · S ) T := T X t =1 H t · ∆ S t , where ∆ S t := ( S t − S t − ) . Denote by ℓ t := ( H · S ) t for t ∈ I and ℓ := ℓ T . When working with N stocks, a canonical choice for Ω would beΩ = { ω = ( ω , . . . , ω T ) : ω i ∈ [0 , ∞ ) N , i = 0 , . . . , T } . Then, one may take S t ( ω ) = ω t and F to be the filtration generated by S . Note that we do not specify any probability measure. ℓ = P k,i S ik − S i . Then, one can directly show that with an appropriate c ∗ , we have L ∗ := c ∗ + ˆ ℓ ≥
1. Define B ℓ using ˆ ℓ , set H = B ℓ and denote by I ℓ the subset of I with H t bounded for every t = 1 , . . . , T .We next prescribe the equivalence relation and the relevant sets. Our start-ing point is the set of negligible sets Z which we assume is given. We also makethe following structural assumption. Assumption C.3.
Assume that the trading is allowed only at finite time pointslabeled through 1 , , . . . , T . Let I be given as above and let Z be a lattice whichis closed with respect to pointwise convergence.We also assume that R = P + and the pre-order is given by, X ≤ Y ⇔ ∃ Z ∈ Z such that X ≤ Ω Y + Z. In particular, X ∈ P if and only if there exists Z ∈ Z such that Z ≤ Ω X .An example of the above structure is the Example 4.3.2. In that example, Z is polar sets of a given class M of probabilities. Then, in this context allinequalities should be understood as M quasi-surely. Also note also that theassumptions on Z are trivially satisfied when Z = { } . In this latter case,inequalities are pointwise.Observe that in view of the definition of ≤ and the fact R = P + , ℓ ∈ I isan arbitrage if and only if there is R ∗ ∈ P + and Z ∗ ∈ Z , so that ℓ ≥ Ω R ∗ + Z ∗ .Hence, ℓ ∈ I is an arbitrage if and only is ℓ ∈ P + . We continue by showingthe equivalence of the existence of an arbitrage to the existence of a one-steparbitrage. Lemma C.4.
Suppose that Assumption C.3 holds. Then, there exists arbitrageif and only if there exists t ∈ { , . . . , T } , h ∈ B b (Ω , F t − ) such that ℓ := h · ∆ S t is an arbitrage.Proof. The sufficiency is clear. To prove the necessity, suppose that ℓ ∈ I is anarbitrage. Then, there is a predictable process H so that ℓ = ( H · S ) T . Also ℓ ∈ P + , hence, ℓ / ∈ Z and there exists Z ∈ Z such that ℓ ≥ Z . Defineˆ t := min { t ∈ { , . . . , T } : ( H · S ) t ∈ P + } ≤ T. First we study the case where ℓ ˆ t − ∈ Z . Define ℓ ∗ := H ˆ t · ∆ S ˆ t , and observe that ℓ ˆ t = ℓ ˆ t − + ℓ ∗ . Since ℓ ˆ t − ∈ Z , we have that ℓ ∗ ∈ P + iff ℓ ˆ t ∈ P + and consequently the lemma is proved.Suppose now ℓ ˆ t − / ∈ Z . If ℓ ˆ t − ≥ Ω
0, then ℓ ˆ t − ∈ P and, thus, also in P + ,which is not possible from the minimality of ˆ t . Hence the set A := { ℓ ˆ t − < Ω } is non empty and F ˆ t − -measurable. Define, h := H ˆ t χ A and ℓ ∗ := h · ∆ S ˆ t . Notethat, ℓ ∗ = χ A ( ℓ ˆ t − ℓ ˆ t − ) ≥ Ω χ A ℓ ˆ t ≥ Ω χ A Z ∈ Z . ℓ ∗ ∈ P . Towards a contradiction, suppose that ℓ ∗ ∈ Z . Then, ℓ ˆ t − ≥ Ω χ A ℓ ˆ t − ≥ χ A ( Z − ℓ ∗ ) ∈ Z , Since, by assumption, ℓ ˆ t − / ∈ Z we have ℓ ˆ t − ∈ P + from which ˆ t is not minimal.The following is the main result of this section. For the proof we follow theapproach of Kabanov and Stricker (2001) which is also used in Bouchard and Nutz(2015). We consider the financial market Θ ∗ = ( B ℓ , k · k ℓ , ≤ Ω , I , P + ) describedabove. Theorem C.5.
In a finite discrete time financial market satifying the Assump-tion C.3, the following are equivalent:1. The financial market Θ ∗ has no arbitrages.2. The attainment property holds and Θ ∗ is free of arbitrage.3. The financial market Θ ∗ is strongly free of arbitrages.Proof. In view of Proposition C.2 we only need to prove the implication 1 ⇒ X ∈ H such that D ( X ) is finite we have that c n + D ( H ) + ℓ n ≥ Ω X + Z n , for some c n ↓ ℓ n ∈ I and Z n ∈ Z . Note that since Z is a lattice we assume,without loss of generality, that Z n = ( Z n ) − and denote by Z − := { Z − | Z ∈ Z} .We show that C := I− ( L (Ω , F )+ Z − ) is closed under pointwise convergencewhere L (Ω , F ) denotes the class of pointwise nonnegative random variables.Once this result is shown, by observing that X − c n − D ( X ) = W n ∈ C convergespointwise to X − D ( X ) we obtain the attainment property.We proceed by induction on the number of time steps. Suppose first T = 1.Let W n = ℓ n − K n − Z n → W, (C.1)where ℓ n ∈ I , K n ≥ Ω Z n ∈ Z − . We need to show W ∈ C . Note that any ℓ n can be represented as ℓ n = H n · ∆ S with H n ∈ L (Ω , F ).Let Ω := { ω ∈ Ω | lim inf | H n | < ∞} . From Lemma 2 in Kabanov and Stricker(2001) there exist a sequence { ˜ H k } such that { ˜ H k ( ω ) } is a convergent subse-quence of { H k ( ω ) } for every ω ∈ Ω . Let H := lim inf H n χ Ω and ℓ := H · ∆ S .Note now that Z n ≤ Ω
0, hence, if lim inf | Z n | = ∞ we have lim inf Z n = −∞ .We show that we can choose ˜ Z n ∈ Z − , ˜ K n ≥ Ω W n := ℓ n − ˜ K n − ˜ Z n → W and lim inf ˜ Z n is finite on Ω . On { ℓ n ≥ Ω W } set ˜ Z n = 0 and ˜ K n = ℓ n − W .On { ℓ n < Ω W } set˜ Z n = Z n ∨ ( ℓ n − W ) , ˜ K n = K n χ { Z n = ˜ Z n } .
31t is clear that Z n ≤ Ω ˜ Z n ≤ Ω
0. From Lemma E.1 we have ˜ Z n ∈ Z . Moreover,it is easily checked that ˜ W n := ℓ n − ˜ K n − ˜ Z n → W . Nevertheless, from the con-vergence of ℓ n on Ω and ˜ Z n ≥ Ω − ( W − ℓ n ) + , we obtain { ω ∈ Ω | lim inf ˜ Z n > −∞} = Ω . As a consequence also lim inf ˜ K n is finite on Ω , otherwise we couldnot have that ˜ W n → W . Thus, by setting ˜ Z := lim inf ˜ Z n and ˜ K := lim inf ˜ K n ,we have W = ℓ − ˜ K − ˜ Z ∈ C .On Ω C we may take G n := H n / | H n | and let G := lim inf G n χ Ω C . Define, ℓ G := G · ∆ S . We now observe that, { ω ∈ Ω C | ℓ G ( ω ) ≤ } ⊆ { ω ∈ Ω C | lim inf Z n ( ω ) = −∞} . Indeed, if ω ∈ Ω C is such that lim inf Z n ( ω ) > −∞ , applying again Lemma 2 inKabanov and Stricker (2001), we have thatlim inf n →∞ X ( ω ) + Z n ( ω ) | H n ( ω ) | = 0 , implying ℓ G ( ω ) is nonnegative. Set now˜ Z n := Z n ∨ − ( ℓ G ) − . From Z n ≤ Ω ˜ Z n ≤ Ω
0, again by Lemma E.1, ˜ Z n ∈ Z . By taking the limit for n → ∞ we obtain ( ℓ G ) − ∈ Z and thus, ℓ G ∈ P . Since the financial market hasno arbitrages G · ∆ S = Z ∈ Z and hence one asset is redundant. Considera partition Ω i of Ω C on which G i = 0. Since Z is stable under multiplication(Lemma E.2), for any ℓ ∗ ∈ I , there exists Z ∗ ∈ Z and H ∗ ∈ L (Ω i , F ) with( H ∗ ) i = 0, such that ℓ ∗ = H ∗ · ∆ S + Z ∗ on Ω i . Therefore, the term ℓ n in(C.1) is composed of trading strategies involving only d − d -steps we have the conclusion.Assuming now that C.1 holds for markets with T − T periods. Set againΩ := { ω ∈ Ω | lim inf | H n | < ∞} . Since on Ω we have that, W n − H n · ∆ S = T X t =2 H nt · ∆ S t − K n − Z n → W − H · ∆ S . The induction hypothesis allows to conclude that W − H · S ∈ C and therefore W ∈ C . On Ω C we may take G n := H n / | H n | and let G := lim inf G n χ Ω C .Note that W n / | H n | → T X t =2 H nt | H n | · ∆ S t − K n | H n | − Z n | H n | → − G · ∆ S . Since Z is stable under multiplication Z n | H n | ∈ Z and hence, by inductive hy-pothesis, there exists ˜ H t for t = 2 , . . . , T and ˜ Z ∈ Z such that˜ ℓ := G · ∆ S + T X t =2 ˜ H t · ∆ S t ≥ Ω ˜ Z ∈ Z . ℓ ∈ Z . Once again, this means that oneasset is redundant and, by considering a partition Ω i of Ω C on which G i = 0,we can rewrite the term ℓ n in (C.1) with d − d -steps we have the conclusion.The above result is consistent with the fact that in classical “probabilistic”model for finite discrete-time markets only the no-arbitrage condition and notthe no-free lunch condition has been utilized. D Countably Additive Measures
In this section, we show that in general finite discrete time markets, it is possibleto characterize viability through countably additive functionals. We prove thisresult by combining some results from Burzoni, Frittelli, Hou, Maggis, and Ob l´oj(2017) which we collect in Appendix E.2. We refer to that paper for the precisetechnical requirements for (Ω , F , S ), we only point out that, in addition to theprevious setting, Ω needs to be a Polish space.We let Q ca be the set of countably additive positive probability measures Q ,with finite support, such that S is a Q -martingale and Z − := {− Z − | Z ∈ Z} .For X ∈ H , set Z ( X ) := (cid:8) Z ∈ Z − : ∃ ℓ ∈ I such that D ( X ) + ℓ ≥ Ω X + Z (cid:9) , which is always non-empty when D ( X ), e.g. ∀ X ∈ B b . By the lattice propertyof Z , if D ( X ) + ℓ ≥ Ω X + Z the same is true if we take Z = Z − . FromTheorem C.5 we know that, under no arbitrage, the attainment property holdsand, hence, Z ( X ) is non-empty for every X ∈ H . For A ∈ F , we define D A ( X ) := inf { c ∈ R : ∃ ℓ ∈ I such that c + ℓ ( ω ) ≥ X ( ω ) , ∀ ω ∈ A }Q caA := { Q ∈ Q ca : Q ( A ) = 1 } . We need the following technical result in the proof of the main Theorem.
Proposition D.1.
Suppose Assumption C.3 holds and the financial market hasno arbitrages. Then, for every X ∈ H and Z ∈ Z ( X ) , there exists A X,Z suchthat A X,Z ⊂ { ω ∈ Ω : Z ( ω ) = 0 } , (D.1) and D ( X ) = D A X,Z ( X ) = sup Q ∈Q caAX,Z E Q [ X ] . Before proving this result, we state the main result of this section.
Theorem D.2.
Suppose Assumption C.3 holds. Then, the financial market hasno arbitrages if and only if for every ( Z, R ) ∈ Z − × P + there exists Q Z,R ∈ Q ca satisfying E Q Z,R [ R ] > and E Q Z,R [ Z ] = 0 . (D.2)33 roof. Suppose that the financial market has no arbitrages. Fix (
Z, R ) ∈ Z − ×P + and Z R ∈ Z ( R ). Set Z ∗ := Z R + Z ∈ Z ( R ). By Proposition D.1, thereexists A ∗ := A R,Z ∗ satisfying the properties listed there. In particular,0 < D ( R ) = sup Q ∈Q caA ∗ E Q [ R ] . Hence, there is Q ∗ ∈ Q caA ∗ so that E Q ∗ [ R ] >
0. Moreover, since Z R , Z ∈ Z − , A ∗ ⊂ { Z ∗ = 0 } = { Z R = 0 } ∩ { Z = 0 } . In particular, E Q ∗ [ Z ] = 0.To prove the opposite implication, suppose that there exists R ∈ P + , ℓ ∈ I and Z ∈ Z such that ℓ ≥ Ω R + Z . Then, it is clear that ℓ ≥ Ω R − Z − . Let Q ∗ := Q − Z − ,R ∈ Q ca satisfying (D.2). By integrating both sides against Q ∗ ,we obtain 0 = E Q ∗ [ ℓ ] ≥ E Q ∗ [ R − Z − ] = E Q ∗ [ R ] > . which is a contradiction. Thus, there are no arbitrages.We continue with the proof of Proposition D.1. Proof of Proposition D.1.
Since there are no arbitrages, by Theorem C.5 wehave the attainment property. Hence, for a given X ∈ H , the set Z ( X ) is non-empty. Step 1.
We show that, for any Z ∈ Z ( X ), D ( X ) = D { Z =0 } ( X ).Note that, since D ( X )+ ℓ ≥ Ω X + Z , for some ℓ ∈ I , the inequality D { Z =0 } ( X ) ≤D ( X ) is always true. Towards a contradiction, suppose that the inequality isstrict, namely, there exist c < D ( X ) and ˜ ℓ ∈ I such that c + ˜ ℓ ( ω ) ≥ X ( ω ) forany ω ∈ { Z = 0 } . We show that˜ Z := ( c + ˜ ℓ − X ) − χ { Z< } ∈ Z . This together with c + ˜ ℓ ≥ Ω X + ˜ Z yields a contradiction. Recall that Z is alinear space so that nZ ∈ Z for any n ∈ N . From nZ ≤ Ω ˜ Z ∨ ( nZ ) ≤ Ω
0, we alsohave ˜ Z n := ˜ Z ∨ ( nZ ) ∈ Z , by Lemma E.1. By noting that { ˜ Z < } ⊂ { Z < } we have that ˜ Z n ( ω ) → ˜ Z ( ω ) for every ω ∈ Ω. From the closure of Z underpointwise convergence, we conclude that ˜ Z ∈ Z . Step 2.
For a given set A ∈ F T , we let A ∗ ⊂ A be the set of scenarios visitedby martingale measures (see (E.2) in the Appendix for more details). We showthat, for any Z ∈ Z ( X ), D ( X ) = D { Z =0 } ∗ ( X ).Suppose that { Z = 0 } ∗ is a proper subset of { Z = 0 } otherwise, from Step1, there is nothing to show. From Lemma E.6 there is a strategy ˜ ℓ ∈ I suchthat ˜ ℓ ≥ { Z = 0 } . Lemma E.5 (and in particular (E.4)) yield a finite Note that restricted to { Z = 0 } this strategy yields no risk and possibly positive gains,in other words, this is a good candidate for being an arbitrage. ℓ t , . . . ℓ tβ t with t = 1 , . . . T , such that { ˆ Z = 0 } = { Z = 0 } ∗ where ˆ Z := Z − T X t =1 β t X i =1 χ { Z =0 } ( ℓ ti ) + . (D.3)Moreover, for any ω ∈ { Z = 0 }\{ Z = 0 } ∗ , there exists ( i, t ) such that ℓ ti ( ω ) > ℓ ti ∈ Z for any i = 1 , . . . β t , t = 1 , . . . T . In particular, from the lattice property of the linearspace Z , we have ˆ Z ∈ Z .We illustrate the reason for t = T , by repeating the same argument upto t = 1 we have the thesis. We proceed by induction on i . Start with i = 1. From Lemma E.5 we have that ℓ Ti ≥ { Z = 0 } and, therefore, { ℓ T < } ⊆ { Z < } . Define ˜ Z := − ( ℓ T ) − ≤ Ω
0. By using the same argumentas in Step 1, we observe that nZ ≤ Ω ˜ Z ∨ ( nZ ) ≤ Ω nZ ∈ Z for any n ∈ N .From { ℓ T < } ⊆ { Z < } and the closure of Z under pointwise convergence,we conclude that ˜ Z ∈ Z . From no arbitrage, we must have ℓ T ∈ Z .Suppose now that ℓ Tj ∈ Z for every 1 ≤ j ≤ i −
1. From Lemma E.5, we have that ℓ Ti ≥ { Z − P i − j =1 ℓ Ti = 0 } and, therefore, { ℓ Ti < } ⊆ { Z − P i − j =1 ℓ Ti < } .The argument of Step 1 allows to conclude that ℓ Ti ∈ Z .We are now able to show the claim. The inequality D { Z =0 } ∗ ( X ) ≤ D { Z =0 } ( X ) = D ( X ) is always true. Towards a contradiction, suppose that the inequality isstrict, namely, there exist c < D ( X ) and ˜ ℓ ∈ I such that c + ˜ ℓ ( ω ) ≥ X ( ω ) forany ω ∈ { Z = 0 } ∗ . We show that˜ Z := ( c + ˜ ℓ − X ) − χ Ω \{ Z =0 } ∗ ∈ Z . This together with c + ˜ ℓ ≥ Ω X + ˜ Z , yields a contradiction. To see this recallthat, from the above argument, ˆ Z ∈ Z with ˆ Z as in (D.3). Moreover, again by(D.3), we have { ˜ Z < } ⊂ { ˆ Z < } . The argument of Step 1 allows to concludethat ˜ Z ∈ Z . Step 3.
We are now able to conclude the proof. Fix Z ∈ Z ( X ) and set A X,Z := { Z = 0 } ∗ . Then, D ( X ) = D { Z =0 } ( X ) = D ( A X,Z ) ∗ ( X ) = sup Q ∈Q caAX,Z E Q [ X ] , where the first two equalities follow from Step 1 and Step 2 and the last equalityfollows from Proposition E.7. E Some technical tools
E.1 Preferences
We start with a simple but a useful condition for negligibility.35 emma E.1.
Consider two negligible contracts ˆ Z, ˜ Z ∈ Z . Then, any contract Z ∈ H satisfying ˆ Z ≤ Z ≤ ˜ Z is negligible as well.Proof. By definitions, we have, X ≤ X + ˆ Z ≤ X + Z ≤ X + ˜ Z ≤ X ⇒ X ∼ X + Z. Thus, Z ∈ Z . Lemma E.2.
Suppose that Z is closed under pointwise convergence. Then, Z is stable under multiplication, i.e., ZH ∈ Z for any H ∈ H .Proof. Note first that Z n := Z (( H ∧ n ) ∨ − n ) ∈ Z . This follows from by LemmaE.1 and the fact that Z is a cone. By taking the limit for n → ∞ , the resultfollows.We next prove that E ( Z ) = 0 for every Z ∈ Z . Lemma E.3.
Let E be a sublinear expectation. Then, E ( c + λ [ X + Y ]) = c + E ( λ [ X + Y ]) = c + λ E ( X + Y ) (E.1) ≤ c + λ [ − ( −E ( X ) − E ( Y ))] , for every c ∈ , λ ≥ , X, Y ∈ H . In particular, E ( Z ) = 0 , ∀ Z ∈ Z . Proof.
Let
X, Y ∈ H . The sub-additivity of U E implies that U E ( X ′ ) + U E ( Y ′ ) ≤ U E ( X ′ + Y ′ ) , ∀ X ′ , Y ′ ∈ H , even when they take values ±∞ . The definition of U E now yields, E ( X + Y ) = − U E ( − X − Y ) ≤ − [ U E ( − X ) + U E ( − Y )] = − ( −E ( X ) − E ( Y )) . Then, (E.1) follows directly from the definitions.Let Z ∈ Z . Then, − Z, Z ∈ P and E ( Z ) , E ( − Z ) ≥
0. Since − Z ∈ P , themonotonicity of E implies that E ( X − Z ) ≥ E ( X ) for any X ∈ H . Choose X = Z to arrive at 0 = E (0) = E ( Z − Z ) ≥ E ( Z ) ≥ . Hence, E ( Z ) is equal to zero. E.2 Finite Time Markets
We here recall some results from Burzoni, Frittelli, Hou, Maggis, and Ob l´oj (2017)(see Section 2 therein for the precise specification of the framework). We aregiven a filtered space (Ω , F , F ) with Ω a Polish space and F containing thenatural filtration of a Borel-measurable process S . We denote by Q the setof martingale measures for the process S , whose support is a finite number of36oints. For a given set A ∈ F , Q A = { Q ∈ Q | Q ( A ) = 1 } . We define the set ofscenarios charged by martingale measures as A ∗ := { ω ∈ Ω | ∃ Q ∈ Q A s.t. Q ( ω ) > } = [ Q ∈Q A supp ( Q ) . (E.2) Definition E.4.
We say that ℓ ∈ I is a one-step strategy if ℓ = H t · ( S t − S t − )with H t ∈ L ( X, F t − ) for some t ∈ { , . . . , T } . We say that a ∈ I is a one-pointArbitrage on A iff a ( ω ) ≥ ∀ ω ∈ A and a ( ω ) > ω ∈ A .The following Lemma is crucial for the characterization of the set A ∗ interms of arbitrage considerations. Lemma E.5.
Fix any t ∈ { , . . . , T } and Γ ∈ F . There exist an index β ∈{ , . . . , d } , one-step strategies ℓ , . . . , ℓ β ∈ I and B , ..., B β , a partition of Γ ,satisfying:1. if β = 0 then B = Γ and there are No one-point Arbitrages, i.e., ℓ ( ω ) ≥ ∀ ω ∈ B ⇒ ℓ ( ω ) = 0 ∀ ω ∈ B .
2. if β > and i = 1 , . . . , β then: ⊲ B i = ∅ ; ⊲ ℓ i ( ω ) > for all ω ∈ B i ⊲ ℓ i ( ω ) ≥ for all ω ∈ ∪ βj = i B j ∪ B . We are now using the previous result, which is for some fixed t , to identify A ∗ . Define A T := AA t − := A t \ β t [ i =1 B it , t ∈ { , . . . , T } , (E.3)where B it := B i, Γ t , β t := β Γ t are the sets and index constructed in Lemma E.5with Γ = A t , for 1 ≤ t ≤ T . Note that, for the corresponding strategies ℓ ti wehave A = T \ t =1 β t \ i =1 { ℓ ti = 0 } . (E.4) Lemma E.6. A as constructed in (E.3) satisfies A = A ∗ . Moreover, Noone-point Arbitrage on A ⇔ A ∗ = A . Proposition E.7.
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