Complete and competitive financial markets in a complex world

We investigate the possibility of completing financial markets in a model with no exogenous probability measure and market imperfections. A necessary and sufficient condition is obtained for such extension to be possible.

Since the seminal contributions of Arrow [4] and of Radner [32], market completeness and the no arbitrage principle have played a prominent rôle in financial economics. Market completeness, as first noted by Arrow, is a crucial property as it permits the optimal allocation of risk bearing among risk averse agents.
In fact the equilibria of an economy under conditions of uncertainty but with competitive and complete financial markets are equivalent to those of an ordinary static economy so that classical welfare theorems apply. The equilibrium analysis on which this conclusion rests requires that financial markets are free of arbitrage opportunities. In the present paper we consider the validity of these classical results in a complex world, that is in the context of an economic model in which uncertainty is treated as a completely general and unrestricted phenomenon. As a general conclusion we find that, in a complex world the interplay between uncertainty and asset prices is richer and more interesting than expected.
Indeed the model of Arrow and of Radner, and of much of the following literature on general equilibrium theory with financial markets, focuses on an economy with a finite state space. This modelling choice is instrumental to and has its main advantage in the description of financial assets as contingent contracts, i.e. in purely economic terms. On the other hand, it has the drawback that this simplified representation of uncertainty makes it more challenging, in the absence of further elements, to justify market incompleteness. To the other extreme, in infinite dimensional economic theory (see e.g. the review of Mas-Colell and Zame [31]) commodities and assets need to be defined as elements of some given space, the choice of which is most o en a first step of crucial importance.
In the approach we propose herea er we retain Arrow's original idea that assets should be described solely in terms of the rights and obligations of the two counterparties on the occurrence of each future contingency and yet allow for an arbitrary sample space Ω. In particular, Ω will not be endowed with any special structure and the real valued functions X defined thereon, which describe assets payoffs, need not be continuous nor measurable in any specific sense. In fact, following the thread of [11] and [13], we do not assume the existence of any exogenously given probability measure, a major difference with a large part of the financial literature as well as many important papers in equilibrium theory, such as Bewley [5].
Our starting point is rather a criterion of economic rationality which describes what all agents agree upon when saying that "f is more than g" (this modelling of rationality, first introduced in [13], is referred to as common order in [10]).
We believe that our general framework is indeed the natural setting for assuming incomplete markets and to investigate the main aspects of a process of gradual completion of markets. In particular, we address the following questions: (a) can an incomplete set of financial markets be extended to a complete one while preserving the basic economic principle of absence of arbitrage opportunities? (b) if so, can such an extension be supported by a competitive market mechanism?
Our finding is that neither question need have a positive answer. Concerning the former, we observe that competition on financial markets may in principle produce two distinct outcomes. On the one hand it lowers margins on currently traded assets and results thus in lower prices. On the other hand, competition involves the design and issuance of new securities. We argue that lower prices on the existing securities may destroy the possibility to obtain complete markets free of arbitrage opportunities in much the same way as predatory pricing in a monopolistic setting may prevent the entry of new, potential competitors.
In principle the net effect of competition on collective welfare may be unclear. Relatively to the latter question, we show that the completion of financial markets in respect of the no arbitrage principle may not be possible under linear pricing (which we take as synonymous of perfect competition). We actually provide an explicit example from which it clearly emerges that this negative conclusion has to do with the complexity of the economic environment described in the model. Indeed, most economic models treat economic complexity under probabilistic assumptions which do not permit a clear comprehension of this phenomenon. On the other hand, we show that if a no arbitrage extension of markets with a limited degree of market power is possible, then markets admit a perfectly competitive extension as well.
We should make clear that, although it is indeed natural and appropriate on a general ground, to interpret the extension of markets as the effect of financial innovation, an explicit model of the strategic behaviour of intermediaries, such as Allen and Gale [2] or Bisin [8], is beyond the scope of the present work. We rather study the properties of pricing functions described as a sublinear functional on the space of traded assets' payoffs. The non linearity of prices captures the non competitive nature of financial markets as well as the role of other market imperfections.
In recent years there have been several papers in which the assumption of a given reference probability is relaxed, if not abandoned. Riedel [33] (and more recently Burzoni, Riedel and Soner [10]) suggests that an alternative approach to finance should be based on the concept of Knightian uncertainty. A typical implication of this approach is that a multiplicity of probability priors is given -rather than a single one. Some authors, including Bouchard and Nutz [9], interpret this multiplicity as an indication of model uncertainty, a situation in which each prior probability corresponds to a different model that possesses all the traditional properties but in which it is unknown which of the models should be considered the correct one. An exemplification is the paper by Epstein and Ji [22] in which model uncertainty translates into ambiguity concerning the volatility parameter. Other papers, among which the ones by Davis and Hobson [17] and by Acciaio et al [1], take the sample space to consist of all of the trajectories of some underlying asset and study the prices of options written thereon based on a path by path or model-free definition of arbitrage.
In section 2 we describe the model in all details, we introduce the notion of arbitrage and prove some properties of prices. In section 3 we characterize the set of pricing probability charges which is of crucial importance in our construction. In the following section 4 we prove the first of our main results, Theorem 2, in which the possibility of completing markets is fully characterized. We show that in a complete financial market, although prices may in principle contain bubbles, there cannot be assets with a positive price but no intrinsic value. Theorem 2 provides an answer to question (a) above. In section 5 we examine viability of financial prices which, in our setting, turns out to be a stronger property than the extension property.
In section 6 we establish a second fundamental result, Theorem 5, which answers question (b). It offers a complete characterization of the existence of a fully competitive completion of financial markets. In particular we prove that a competitive completion is possible if only one can obtain a completion in which the market power of intermediaries is limited. This result can also be read as a theoretical justification of the microstructure formula stating that asset prices are obtained by applying a spread on the asset fundamental value. Several additional implications are proved. We provide an explicit example of a financial market that admits no competitive completions as a consequence of a high degree of complexity. Given its importance in the reference literature, in section 7 we examine the question of countable additivity and give exact necessary and sufficient conditions for the existence of a countably additive pricing probability.

T E .
We model the market as a triple, (X , ≥ * , π), in which X describes the set of payoffs generated by the traded assets, ≥ * the criterion of collective rationality used in the evaluation of investment projects and π is the price of each asset as a function of its payoff. Each of these elements will now be described in detail.
Before getting to the model we introduce some useful notation. Throughout, Ω will be an arbitrary, non empty set that we interpret as the sample space so that the family F(Ω) of real valued functions on Ω will be our ambient space; B(Ω) will represent bounded functions. If A ⊂ F(Ω), we write A u (resp. A τ ) to denote its closure in the topology of uniform distance (resp. in the topology τ ). If A is a σ algebra of subsets of Ω, F(Ω, A ) denotes the family of A measurable functions and we set B(Ω, A ) = F(Ω, A ) ∩ B(Ω). The symbol ba(A ) denotes the set of bounded, finitely additive set functions on A , to which we refer as charges, while P ba (A ) designates the collection of probability charges. We reserve the word probability with no further qualification and the symbol P(A ) to classical (i.e. countably additive) probabilities. General references for the theory of charges and finitely additive integrals are [20] and [6].
2.1. Economic Rationality. A natural order to assign to F(Ω) is pointwise order, to wit f (ω) ≥ g(ω) for all ω ∈ Ω, also written as f ≥ g. The lattice symbols |f | or f + will always refer to such order.
Natural as it may appear, pointwise order is not an adequate description of how economic agents rank random quantities save when the underlying sample space is particularly simple, such as a finite set. For example, it is well documented that investors base their decisions on a rather incomplete assessment of the potential losses arising from the selected portfolios, exhibiting a sort of asymmetric attention that leads them to neglect some scenarios, in contrast with a pointwise ranking of investment projects 1 . In a complex world, in which the attempt to formulate a detailed description of Ω is out of reach, rational inattention is just one possible approach to deal with complexity, possibly not too different from the restriction to measurable quantities adopted by classical probability.
In this paper, following the thread of [13], we treat monotonicity as a primitive economic notion represented by a further transitive, reflexive binary relation on F(Ω). To distinguish it from the pointwise order ≥, we use the symbol ≥ * (the asymmetric part of ≥ * will be written as > * ).
We assume the following properties: which will be the basis for what follows 2 .
Of course, the symmetric part of ≥ * induces a corresponding equivalence relation, ∼ * . It will be useful |f | ∼ * f . Based on ∼ * we can also construct the collection of negligible sets (2) N * = {A ⊂ Ω : 0 ∼ * 1 A } and, if A is an arbitrary σ algebra of subsets of Ω which contains N * , the subset P ba (A , N * ) consisting of probability charges on A which vanish on N * . Every subset of Ω not included in N * will be called non negligible. It will be useful to define the space It is immediate to note that any exogenously given probability charge P ∈ P ba (A ) on some σ algebra A (countably or finitely additive) induces a corresponding ranking defined as which satisfies the above axioms (1). In this case a set is negligible if and only if it is P null. Clearly, if P is a probability the ranking ≥ P is just the P -a.s. ranking. The same construction may be extended by replacing P with a family P ⊂ P ba (A ) of probability charges and defining accordingly (5) f ≥ P h if and only if f ≥ P h for all P ∈ P.
The ranking ≥ P defined in (5) arises in connection with the model uncertainty approach mentioned in the Introduction and exemplified by the paper by Bouchard and Nutz [9]. In this approach each element P of the given collection P is a model 3 .
We provide some concrete examples of the partial order ≥ * that arise from decision theory.
Example 1. Let α represent the preference system of agent α defined over the whole of F(Ω). Assume that α satisfies the following monotonicity properties valid for all f, g ∈ F(Ω) 4 : We may deduce an implicit, subjective criterion ≥ α of monotonicity (or rationality), by letting A mathematical criterion ≥ * describing collective rationality may then be defined as the meet of all such individual rankings, i.e. as (8) f ≥ * g if and only if f ≥ α g for each agent α ∈ A.
Notice that in this case f > * g means that all agents agree on f ≥ α g and there exists at least one agent α 0 ∈ A according to which f > α 0 g.
If we aggregate the ≥ α rankings via the unanimity rule (8), we obtain thus the ranking ≥ P defined in (5), W shall return on Example 2 in Section 6, Example 6.
which is sometimes referred to as null-additivity. A strongly subadditive (or submodular) capacity [19,III.30] clearly possesses all the above properties while a supermodular one need not satisfy (9).
3 It should be noted that the choice of Bouchard and Nutz to take P to be a set of probabilities has considerable implications on N * which coincides with the collection of sets which are P null for every P ∈ P and is therefore closed with respect to countable unions. We shall return on the ranking ≥ P in Example 6. 4 These assumptions are indeed minimal in economic models with an infinite dimensional commodity space, see e.g. Bewley Again, one easily concludes that ≥ F possesses the above properties (1) if and only if F is a filter of subsets of Ω. In accordance with [20, I.7.11] let F * be some ultrafilter refining F , and define P ∈ P ba (2 Ω ) by letting true.
An interesting question raised by the preceding examples concerns the conditions under which a given ranking ≥ * coincides with the ranking ≥ P for some endogenous probability charge P . Notice that in Example 4 we only showed that ≥ P is compatible with ≥ F but we may still have that f ≥ P h while f ≥ F h. We shall obtain an indirect answer to this question in Theorem 7.
2.2. Assets. We posit the existence of an asset whose final payoff and current price are used as numéraire of the payoff and of the price of all other assets, respectively. Each asset is identified with its payoff expressed in units of the numéraire and is modelled as an element of F(Ω). The market is then a convex set X ⊂ F(Ω) containing the origin as well as the function identically equal to 1 (that will be simply indicated by 1). Notice that, similarly to Jouini and Kallal [26], we do not assume that investments may be replicated on any arbitrary scale, i.e. that X is a convex cone, as is customary in this literature. Restrictions to the investment scale are commonly met on the market whenever investors are subject to margin requirements or the provision of a collateral.
We assume in addition that (i) each X ∈ X satisfies X ≥ * a for some a ∈ R and (ii) that The first of these assumptions constraints the assets traded on the market to bear a limited risk of losses and may be interpreted as a restriction imposed by some regulator; the second one permits agents, which in principle may only form convex portfolios, to invest into the numéraire asset an unlimited amount of capital. Notice that, since the numéraire cannot be shorted, the construction of zero cost portfolios -or In order to have an appropriate notion of completeness of markets, we find it convenient to introduce a σ algebra A of subsets of Ω satisfying which may be loosely interpreted as the set of (A measurable) superhedgeable claims. We then define markets to be complete relatively to A (or A complete, for brevity) whenever each element of L(X , A ) is attained by a corresponding asset. In other words, we define market completeness conditionally on the σ algebra A and of course a whole hierarchy of completions is possible as A ranges between the two extrema of the minimal σ algebra A 0 generated by the elements equivalent to some X ∈ X and the power set of Ω. The introduction of a measurable structure, which implies no restriction in our analysis, serves two distinct purposes: on the one hand, it makes the comparison with the traditional literature fully transparent; on the other hand, it permits to put the extension problem in a clearer relation with the notion of complexity, that we identify with the fineness of A . A fully complex model corresponds then to the case in which A is the power set of Ω, a case perfectly possible in our setting.
2.3. Prices. In financial markets with frictions and limitations to trade, normalized prices are best modelled as positively homogeneous, subadditive functionals of the asset payoff, π : X → R, satisfying π(1) = 1, the monotonicity condition and cash additivity 5 The non linearity of financial prices is a well known empirical feature documented in the microstructure literature (see e.g. the exhaustive survey by Biais et al. [7]) and essentially accounts for the auxiliary services that are purchased when investing in an asset, such as liquidity provision and inventory services.
Subadditivity captures the idea that these services are imperfectly divisible and that they are supplied in conditions of limited competition. The degree of non linearity of prices, computed as will play a major role in section 6 and it measures the lack of competitiveness on financial markets. Although deviations from the competitive paradigm may originate from several sources, we find it convenient to interpret the polar cases m(π) = 0 and m(π) = 1 as indication of perfect competition and of full monopoly power, respectively.
Cash additivity implicitly assumes that the numéraire is traded and priced on a separate market -as is normally the case with treasury bonds, but not necessarily so with other assets. Concerning positive homogeneity, this property is consistent with market imperfections such as the bid ask spread, but contrasts in general with fixed costs. In [26] prices are described by a convex functional, thus not necessarily positively homogeneous 6 . Indeed a number of our results still hold true even for convex price functions although the lack of positive homogeneity cannot be reconciled with superhedging duality established in Theorem 1.
We also require that prices be free of arbitrage opportunities, a property which we define as 7 5 This property, defined in slightly different terms, is discussed at length relatively to risk measures in [21]. In the context of non linear pricing cash additivity is virtually always assumed in a much stronger version, namely for all X ∈ X and all a ∈ R, Of course, (17) implies that π(X) ≥ 0 whenever X ≥ * 0 while (14) need not follow from (17) if short selling is not permitted. We notice that the situation X > * π(X) > 0, though clearly exceptional, does not represent in our model an arbitrage opportunity because of the potential infeasibility of short positions in the numéraire asset. A firm experiencing difficulties in raising funds for its projects and competing with other firms in a similar position may offer abnormally high returns to induce investors to purchase its debt.
A functional satisfying all the preceding properties -including (17) -will be called a price function and the corresponding set will be indicated with the symbol Π(X ). We thus agree that market prices are free of arbitrage by definition and we shall avoid recalling this crucial property. At times, though, it will be mathematically useful to consider pricing functionals for which cash additivity and/or the no arbitrage property (17) may fail. These will be denoted by the symbol Π 0 (X ). To this end we note that even if π 0 ∈ Π 0 (X ) fails to be cash additive, it always has a cash additive part π a 0 , i.e. the functional It is routine to show that π a 0 is the greatest cash additive element dominated by π 0 .
Example 5. In classical, continuous-time financial models, the discounted, final payoff of each investment takes the form in which w is the initial wealth invested, S, the discounted price process, is a semimartingale on some filtered probability space and θ an element of a suitably defined set Θ of admissible trading strategies. Then, π(X w,θ ) = w. Given the possibility of shorting the numéraire, each investment is naturally associated with a corresponding self financing strategy with final payoff X 0,θ = θdS. The no arbitrage condition (17), restricted to self financing strategies, implies then that no such strategy may produce a strictly positive final payoff, i.e. that

P C
Associated with each price π ∈ Π(X ) is the convex cone and, more importantly, the collection of pricing probability charges 8 M 0 (π, A ) = m ∈ P ba (A ) : L(X , A ) ⊂ L 1 (Ω, A , m) and π(X) ≥ Xdm for every X ∈ X . 8 In [13] we used the term pricing measure to define a positive charge dominated by π without restricting it to be a probability charge. The focus on probability charges will be clear a er Theorem 2. A definition of the finitely additive integral and its properties may be found in [20, III.2.17].
The following, basic result illustrates the role of cash additivity and of the set M 0 (π, A ). In particular, property (24) may be considered as our version of the classical superhedging duality Theorem 9 .
Theorem 1. For given π 0 ∈ Π 0 (X ) the set M 0 (π, A ) is non empty and each m ∈ M 0 (π, A ) satisfies Eventually, the set M 0 (π 0 , A ) is convex and compact in the weak * topology of ba(A ) (i.e. the topology induced by B(Ω, A ) on ba(A )).
so that m ∈ M 0 (π a 0 , A ). The cash additive part π a 0 of π 0 , obtained as in (18), is easily seen to be an extension of π a 0 to L(X , A ). Of course, π a 0 is the pointwise supremum of the linear functionals φ that it dominates so that (24) follows 9 The first part of this Theorem is mainly a consequence of the Hahn-Banach Theorem and it is true even if the price function π0 is just convex. The superhedging formula (24), on the other hand, necessarily requires positive homogeneity.
if we show that m φ ∈ P ba (A ) for all such φ. But this is clear since π a 0 ≥ φ implies that φ is positive on L(X , A ). Moreover, which contradicts the inequality π a 0 ≥ φ unless φ(1) = 1 and thus m φ ∈ P ba (A ). The last claim is an obvious implication of Tychonoff theorem [20, I.8.5].
Pricing probability charges closely correspond to the risk-neutral measures which are ubiquitous in the traditional financial literature since the seminal paper of Harrison and Kreps [24]. We only highlight that the existence of pricing charges and their properties are entirely endogenous here and do not depend on any special mathematical assumption -and actually not even on the absence of arbitrage 10 . In traditional models, the condition M 0 (π, A ) = ∅ is obtained via Riesz representation theorem (here replaced with Theorem 10) and requires an appropriate topological structure. Finite additivity is a direct consequence of our minimal approach. The view expressed by Bewley [5, p. 516] that charges have "no economic interpretation" and the elements he offers in favour of the choice of the Mackey topology only make sense if a reference, countably additive measure is assumed to be given exogenously. It is, in other words, a somewhat circular argument. We shall argue in section 7 that in a model treating economic rationality as a primitive concept, the economic role of countable additivity is far from clear.
It is customary to interpret the expected value Xdm of the asset payoff as its fundamental value, although the value so obtained may vary significantly as m ranges across the set of pricing charges. The intrinsic value of the asset, computed as (27) sup Xdm, corresponds to the most optimistic among such evaluations. A bubble is customarily defined as the spread between the price of an asset and its intrinsic value, that is

Xdm
According to Theorem 1, assets with bounded payoff admit no bubbles. At present, however, we cannot exclude the extreme situation of a pure bubble, i.e. of an asset X ∈ X such that X ≥ * 0, π(X) > 0 but sup m∈M 0 (π,A ) Xdm = 0.
In Theorem 2 we obtain, among other things, a full characterisation of pure bubbles.
At this level of generality we cannot conclude in favour of the pricing formula popular in microstructure models and according to which asset prices are obtained by applying some spread to its fundamental value, such as (30) π(X) = [1 + α(X)] Xdm with |α(X)| < 1 X ∈ X .
We easily see that (30) is a stronger condition than absence of pure bubbles. This formula will be discussed again in section 6.
We close remarking that each m ∈ M 0 (π, A ) induces a linear functional on L(X , A ). The locally convex linear topology induced by such functionals and denoted by τ (π) is weaker than the classical weak topology.
Competition among financial intermediaries may involve existing assets and/or the launch of new financial claims. As a consequence it may produce two different effects: (a) a reduction of intermediation margins, and thus lower asset prices, and (b) an enlargement of the set X of traded assets, thus contributing to complete the markets. This short discussion justifies our interest for the set 11 In this section we want to address the following question: under what conditions is it possible to extend the actual markets to obtain an economy with complete financial markets without violating the no arbitrage principle? This translates into the mathematical condition Ext(π, A ) = ∅ and, if π ′ ∈ Ext(π, A ), we speak of (L(X , A ), ≥ * , π ′ ) as an A -completion of (X , ≥ * , π).
The first papers to focus on the extension property of financial prices were those by Harrison and Kreps [24] and Kreps [27] (see also [11,Theorem 8.1]), although this aspect has later been somehow neglected in the following literature. In their approach the completion property is related to (and in fact in most cases equivalent to) the concept of viability that we discuss later. Recalling the discussion in subsection 2.2.1, we remark that our definition of A -completion does not preclude that markets may be in principle further extended, e.g. by passing to a larger σ algebra than A . However, given that A is quite arbitrary, our results carry over very simply.
We obtain the following complete characterisation for the case of cash additive A -completions.
Theorem 2. For a market (X , ≥ * , π) the following properties are mutually equivalent: (a). π satisfies the condition (b). π satisfies the condition (c). the market (X , ≥ * , π) admits an A -completion; (d). the set M 0 (π, A ) of pricing probability charges satisfies the condition Proof. The implication (a)⇒(b) is immediate in view of the fact that τ (π) is a topology weaker than the topology of uniform distance. Assume that (33) holds and define the functional Property (1b) and ≥ * monotonicity of π imply that ρ ∈ Ext 0 (π, A ). Moreover, it is easily seen that and thus that ρ is cash additive. To prove that ρ ∈ Π L(X , A ) , fix f ∈ L(X , A ) such that f 1 = f ∧ 1 > * 0. In search of a contradiction, suppose that ρ(f 1 ) ≤ 0. Then for each n ∈ N there exist a n ∈ R, λ n > 0 and X n ∈ X such that λ n X n ≥ * f 1 + a n but λ n π(X n ) < 2 −n + a n . This clearly implies (33). It follows that ρ(f 1 ) > 0 and that (b)⇒(c).
Choose ρ ∈ Ext(π, A ) and let f 1 be defined as above. Consider the linear functional Given that ρ satisfies (36),φ is dominated by ρ on L 0 so that we can find an extension φ ofφ to the whole of L(X , A ) still dominated by ρ.
Let now f ∈ C (π, A ) τ (π) . For each m ∈ M 0 (π, A ) and n ∈ N there exist λ n > 0, X n ∈ X and h n ∈ L(X , A ) such that λ n (X n − π(X n )) ≥ * h n and (f − h n )dm ≤ 2 −n . But then, Under (d) this excludes that f > * 0 and proves the implication (d)⇒(a).

It is immediate to recognize the close relationship linking the condition (33) to the No-Free-Lunch-with-
Vanishing-Risk principle formulated long ago by Delbaen and Schachermayer [18] in a highly influential paper. Since then, this condition, despite its unclear economic interpretation, has been unanimously accepted as the most convenient mathematical restatement of the no arbitrage principle. Leaving aside the dissimilarity in the set-up adopted, the major difference between the two conditions lies in the interpretation. In fact, due to the restrictions to trade considered here, the elements of the form X − π(X), with X ∈ X , need not correspond to the payoff of any feasible trading strategy so that (33) cannot be interpreted as a mathematical reformulation of the no arbitrage principle. Rather, the set C (π, A ) u represents those potential claims that cannot be assigned a strictly positive price by any extension of the actual price function. Thus, Theorem 2 characterizes (33) as a condition necessary and sufficient for financial markets to admit a no arbitrage A -completion. Notice that a strictly positive extension may still exist even when (33) fails. In this case, however, it cannot be cash additive.
In the light of the discussion following (29), condition (34) corresponds to a No-Pure-Bubble (NPB) condition, although it does not exclude more general bubbles defined as in (28). In other terms, with complete financial markets there cannot exist pure bubbles, which are instead possible with incomplete markets 12 .
Given the arbitrariness of the σ algebra A , one may consider the possibility of further extending the market from L(X , A ) to L(X ′ , A ′ ) where X ′ = L(X , A ) and A ⊂ A ′ . If π ′ ∈ Ext(π, A ), it is then immediate from Theorem 2 that (X ′ , ≥ * , π ′ ) admits an A ′ -completion if and only if Notice that each π ′′ ∈ Ext(π ′ , A ′ ) satisfies the supermartingale-like inequality π ′′ |X ′ ≤ π ′ Incidentally we remark that, from the equality M 0 (π 0 , A ) = M 0 (π a 0 , A ), it follows that π 0 ∈ Π 0 (X ) satisfies (33) if and only if so does π a 0 . In fact, Proof. (a). For each X ∈ X it is obvious that X − π(X) ≤ X − π a (X). However, X − π a (X) is the limit in the uniform topology as t → +∞, of X + t − π(X + t) ∈ C (π, A ). (b). X ∈ X and π a (X) ≤ 0 imply that for each n ∈ N and for t n > 0 sufficiently large so that X ∈ C (π, A ) u . Viceversa, if X ≤ 2 −n + λ n [X n − π(X n )] for some X n ∈ X and λ n ≥ 0, then, moving λ n π(X n ) to the le hand side if positive and using cash additivity, we conclude π a (X) ≤ To highlight the role of competition in financial markets, consider two pricing functions π, π ′ ∈ Π(X ).
If π ≤ π ′ then C (π ′ , A ) ⊂ C (π, A ). Thus, lower financial prices are less likely to satisfy (33) and thus to admit an extension to a complete financial market free of arbitrage. Competition among market makers, producing lower spreads, may thus have two contrasting effects on economic welfare. On the one side it reduces the well known deadweight loss implicit in monopolistic pricing while, on the other, it imposes a limitation to financial innovation and its benefits in terms of the optimal allocation of risk. It may be conjectured that fully competitive pricing, i.e. the pricing of assets by their fundamental value, may not be compatible with the extension property discussed here. We investigate this issue in the following Section 6. 12 The fact that completeness of financial markets may change the structure of asset bubbles has already been noted by Jarrow, Protter and Shimbo [25].

V
Following Harrison and Kreps [24] and Kreps [27], several authors in financial economics have characterized the notion of viability, i.e. the property that price functions support optimal choice by individuals with monotone, convex and continuous preferences. Continuity of preferences is a key property in proving the existence of economic equilibrium (see the survey by Mas-Colell and Zame [31]). In addition, in most models this property provides the necessary justification for formulating the separating condition (33) in terms of the closure of C (π, A ).
In our model, viability induces a conclusion much stronger than condition (33) which is in fact equivalent to a local version of it. Moreover, the restrictions to trade considered, particularly the constraint that prevents shorting the numéraire, drive a wedge between our approach and other papers in this literature (but one should refer to Jouini and Kallal [26] for a comparison).
Let us consider the space X = R × F(Ω, A ), endowed with the topology of uniform convergence, as a description of agents consumption space at two different instants of time. Agents with no initial endowment are described by a strict preference ≻ (i.e. a transitive and non reflexive binary relation) which induce family of preferred sets and by the budget set, defined as (42) B(π) = (c, h) ∈ X : c + λπ(X) ≤ 0 and λX ≥ * h for some λ > 0 and X ∈ X .
We can then define viability in formal terms: Definition 1. The price function π is viable relatively to the strict preference relation ≻ if Viability has special importance when preferences are monotonic, convex and continuous. In order to define these properties, we extend > * to X by writing and define the convex cone (with the origin deleted) K plays here the same role as in [27]. We define now the class of admissible preferences.

Definition 2. A strict preference ≻ on R × F(Ω, A ) is of class A if it satisfies the following properties:
Condition (46a) is a monotonicity property while (46c) is a weak form of semicontinuity. We can now define two distinct notions of viability as follows: The literature has rarely considered the need to reinforce the notion of viability as we do here introducing * -viability. The reason is that the two definitions coincide whenever either the numéraire is traded freely or preferences are defined over net trades. And a combination of either one of these two features appears in virtually all contributions. It is thus of some importance to remark that outside of the narrow, traditional approach viability may need to be reformulated.
Another new feature of our model is the focus on a local version of the above properties. If k ∈ K, a strict preference is of class A k if it satisfies the above properties (46) in restriction to the convex cone {ak + (b, 0) : a, b ∈ R + } (rather than the whole of K). If for any k ∈ K the price function π is viable for some preference of class A k , we speak of π as locally viable (resp. locally * -viable).
Eventually, if X ∈ X , there exists a ∈ R such that X ∼ * X ∨ a so that, for any n ∈ N we have the This, being true for all X ∈ X and n ∈ N, implies via [20,III.3.6] that X ⊂ L 1 (Ω, A , m) and, eventually, a condition equivalent to (33) by Theorem 2.
Viceversa, assume (33), Properties (46) and (47) are easily seen to hold true so that ≻ is of class A k 0 . This, being true for all k 0 ∈ K, proves that π is locally * -viable.
(ii). If π is * -viable, it is necessarily locally * -viable. The proof is identical to that of claim (i) with the only difference that Int(W ≻ (0, 0)) contains now appropriate multiples of each k ∈ K. Thus, if Φ is the continuous linear functional introduced in (49) we conclude that Φ(k) > 0 for all k ∈ K. Proceeding exactly as above, we obtain m ∈ M 0 (π, A) that satisfies (48). Conversely, if m ∈ M 0 (π, A) satisfies such properties, the strict preference defined in (53) is necessarily * -viable.
As is clear from the proof, * -viability is required to conclude that the set function m obtained by normalizing Φ is a probability charge. Assuming simple viability, we would in fact obtain an element of ba(A ) + which is dominated by π on X and this is weaker than (33).
The main finding of Theorem 3 is that full viability of financial prices is a much more restrictive condition than (33) and is in fact equivalent to the existence of m ∈ M 0 (π, A ) that satisfies (48). This point will be discussed at length in the following section 6 in which we provide examples in which such set functions may not exist 15 . In these cases, the general notion of viability as defined above is not useful. Although this may appear at first as a limitation, the next result suggests that in fact it may not be so provided we enlarge the set of preferences that we consider as admissible 16 .
14 The existence of such m is trivial if c0 > 0 or else follows from (34 Proof. Let ≻ meet the conditions of the claim and choose X ∈ X . If π(X) ≤ 0 then (0, X−π(X)) ∈ B(π); if on the other hand π(X) > 0 then the same conclusion follows from either It is thus clear that if π is ≻-viable then necessarily {X − π(X) : X ∈ X } and H = {f ∈ F(Ω, A ) : (0, f ) ∈ W ≻ (0, 0)} are disjoint sets. By (46a) the same conclusion remains valid upon replacing the first set with C (π, A ). For each f ∈ F(Ω, A ) such that f > * 0 (and thus f ∈ H) we deduce from (46c) that f / ∈ C (π, A ) u . This shows that (33) holds. If, conversely, π satisfies (33), then one may define (c, h) ≻ (c ′ , h ′ ) simply by letting It is immediate that this is a strict preference and that it satisfies (46a), (46c) as well as (54a) while it is obvious that π is ≻-viable.
Thus, (33) i.e. local * -viability, are indeed perfectly sensible conditions from the point of view of market equilibrium, at least under either (54a) or (54b) and as long as we do not insist on convexity of preferences.
In this section we investigate the conditions under which the set M 0 (π, A ) contains a strictly positive element, i.e. some m satisfying (48). We denote the corresponding set with the symbol M (π, A ) 18 . By Theorem 3, the condition M (π, A ) = ∅ is equivalent to the price function π being viable. More importantly, if m ∈ M (π, A ) then each asset with payoff in L(X , A ) may be priced by its fundamental value and this price rule ρ would be free of arbitrage and linear. In other words, ρ would provide an A -extension of π which is fully competitive, in symbols ρ ∈ Ext(π, A ) with m(ρ) = 0 (recall the definition (16) of the market power index m).
This is of course, an extreme situation. The question we want to address is rather: given a market, (X , ≥ * , π), is it possible to find an A -completion that permits some degree of competitiveness?. In other words, we look for ρ ∈ Ext(π, A ) satisfying the No-Full-Monopoly (NFM) condition (56) m(ρ) < 1.
17 Properties (46a) and (46c) are unduly restrictive and a local version may be used instead. We may, in other words, require that for each k ∈ K there exists a strict preference which is of class A k (save for convexity) and such that π is ≻-viable. This more general formulation would however require a quite involved statement. 18 In [33] a set function satisfying (48) is said to have full support and the emergence of measures with full support follows easily from the assumption that Ω is a complete, separable metric space and X consists of continuous functions defined thereon.
Notice that for a pricing probability charge to meet (48) it is necessary that it be strictly positive on each non negligible set. This condition is, however, not sufficient. In fact, although f > * 0 implies {f > 0} / ∈ N * , the fact that N * is not closed with respect to countable unions does not rule out that {f > ε} ∈ N * for all ε > 0 which contradicts (48).
As we shall see, this condition has in fact far reaching implications.
For each n ∈ N define the sets and let co(B n ) be the convex hull of B n . Notice that B 0 = n B n , because ρ ∈ Ext(π, A ).
What the preceding Theorem 5 asserts in words is that if an A -complete, arbitrage free market is possible under limited market power, it is then possible under perfect competition -i.e. with assets priced by their fundamental value. This conclusion does not exclude, however, the somewhat paradoxical situation in which the only possibility to complete the markets is by admitting unlimited market power by financial intermediaries. As noted in the introduction, this situation describes the terms of a potential conflict between the effort of regulating the market power of intermediaries and the support to a process of financial innovation that does not disrupt market stability by introducing arbitrage opportunities.
We stress that the conclusions of Theorem 5 do crucially depend on the intervening σ algebra A and that the NFM condition is the less likely to hold the finer is A . This remark suggests that a limited or null market power may not be possible as the degree of complexity of the financial market, as embodied in the width of A , ranks high. Actually, given the requirement N * ⊂ A , even the smallest possible extension of markets as we defined it may not admit any extension satisfying (56).
A stronger form of the preceding example can be proved.

Lemma 4. Let the σ algebra A admit an uncountable collection of non negligible sets, the intersection of any
two of which is negligible. Then, any extension ρ ∈ Ext(π, A ) is necessarily such that m(ρ) = 1.
Proof. Using exactly the same notation of the example above, we obtain that ρ(f αn ) > δ as before while f αn ∧ f αm ∼ * 0 when n = m. We obtain Lemma 4 shows that the complete and competitive financial market imagined by Arrow [4] may not be feasible in a complex world, that is with a large enough A . We have thus an instance in which complexity acts as a restriction to perfect competition 20 . The Lemma also provides a negative answer to the question of the existence of viable price systems, as defined in section 5. Essentially this occurs because in the situation considered the elements of the positive cone K defined in (45) are too many and too diverse from one another.
Example 6. Let us return to Example 2. Let each agent α in the economy be endowed with a probability prior P α , forming the collection P ⊂ P(A ) (thus countably additive). Define ≥ P as in (5). If P is undominated then, by [12,Theorem 3], there exists an uncountable, pairwise disjoint family of sets in A each of which is 19 In the theory of Boolean algebras the condition that no uncountable, pairwise disjoint collection of non empty sets may be given, is known as the countable chain (CC) condition and was first formulated by Maharam [29]. See the comments in [15]. 20 The only theoretical contribution to the study of the link between competitive equilibria and complexity of which I am aware is the paper of Gale and Sabourian [23] -and the references therein. The authors prove that in a game theoretic, matching model rational agents with aversion to complexity end up playing a subgame perfect equilibrium which is perfectly competitive despite the finite number of players. Complexity refers here to strategies, and one strategy is considered to be more complex than another whenever the two coincide save on a set of states on which the former is constant. i.e. relatively to P α . In other words, no agent respects the condition (46a) (and a fortiori (46c)) relatively to K. Nevertheless, for each α ∈ A there is a preference system of class A fα and thus it is certainly true that an equilibrium supporting price π must be locally * -viable although it cannot be linear (i.e. competitive). This example complements the results obtained by Bouchard and Nutz [9].
Those extensions that satisfy the NFM condition have further, special mathematical properties.
Theorem 6. Let ρ ∈ Π(L(X , A )) satisfy (56). Then there exists µ ∈ M (ρ, A ) such that Proof. Of course |f | ∧ 1 ∈ B(Ω, A , N * ) for each f ∈ F(Ω, A ) so that (62) is well defined. For each α in a given set A, let A α n n∈N be a decreasing sequence of sets in A satisfying the following properties: (i) for each distinct pair α, β ∈ A there exists n(α, β) ∈ N such that and (ii) for each α ∈ A there exists m α ∈ M 0 (ρ, A ) such that lim n m α (A α n ) > 0. If the set A is uncountable, then, as in the preceding Lemma 4, we can fix δ > 0 and extract a sequence α 1 , α 2 , . . . ∈ A such that, For each k ∈ N define n(k) = 1 + sup {i,j≤k:i =j} n(α i , α j ). Then f 1 n(k) , . . . , f k n(k) ∈ B(Ω, A ) are pairwise disjoint functions with values in [0, 1] and such that But then, taking w i = 1/k, we obtain so that m(ρ) = 1, contradicting our initial assumption. We thus reach the conclusion that A must be countable and deduce from this and from [15, Theorem 2] that M 0 (ρ, A ) is dominated by some of its elements: let this be µ. In addition, M 0 (ρ, A ) is weak * compact as a subset of ba(A ), as proved in Theorem 1. It follows from [37, Theorem 1.3] that M 0 (ρ, A ) is weakly compact.
Take a sequence E n n∈N in A such that µ(E n ) ≤ 2 −n and for each n ∈ N choose m n ∈ M 0 (ρ, A ) such that with respect to m 0 = n 2 −n m n . However, given that m 0 ∈ M 0 (π, A ), we also have µ ≫ m 0 so that {m n : n ∈ N} is uniformly absolutely continuous with respect to µ as well. Thus, Let f n n∈N be a sequence in L(X , A ) that converges to 0 in L 1 (Ω, A , µ), and therefore in µ measure.
Then, by (24) and for arbitrary 0 < c < 1, so that the claim follows.
To highlight the importance of this last claim, we note that the absolute continuity property for charges, even if defined on a σ algebra cannot be simply stated in terms of null sets. As a consequence, the existence of a strictly positive element of M 0 (π, A ) established in Theorem 5, is not sufficient to imply that the set M 0 (π, A ) is dominated, i.e. that each of its elements is absolutely continuous with respect to a given one.
It rather induces the weaker conclusion that there is a given pricing measure m 0 such that m 0 (A) = 0 implies m(A) = 0 for all m ∈ M 0 (π, A ).
On the other hand, if such a dominating element exists then, by weak compactness, it dominates M 0 (π, A ) uniformly. A similar conclusion is not true in the countably additive case treated in the traditional approach. In that approach, the fact that risk neutral measures are dominated is an immediate consequence of the assumption of a given, reference probability measure but the set of such measures is not weakly * compact when regarded as a subset of ba(A ). This special feature illustrates a possible advantage of the finitely additive approach over the countably additive one.
Eventually, notice that Theorem 6 does not require the no arbitrage property and may thus be adapted to the case in which L(X , A ) is a generic vector lattice of functions on Ω containing the bounded functions and ρ a monotonic, subadditive and cash additive function, such as the Choquet integral with respect to a sub modular capacity.
Another characterization of the condition M (π, A ) = ∅ may be obtained as follows: If, conversely, P ∈ P ba (A , N * ) satisfies (68) then N * coincides with the collection N P of P null sets.
Implicitly, Theorem 7 provides an answer to the question raised in subsection 2.1 relative to the conditions under which the ranking ≥ * takes the form ≥ P for some P ∈ P ba (A ).

C A M
Given the emphasis on countable additivity which dominates the traditional financial literature, it is natural to ask if it possible to characterise those markets in which the set M 0 (π, A ) contains a countably additive element. A more ambitious question is whether such measure is strictly positive, i.e. an element on M (π, A ).
Not surprisingly, an exact characterisation may be obtained by considering the fairly unnatural possibility of forming portfolios which invest simultaneously in countably many different assets. This induces to modify the definition (16) into the following (again with the convention 0/0 = 0): for all sequences f 1 , f 2 , . . . ∈ B(Ω, A ) + such that n f n ∈ B(Ω, A ).
It may at first appear obvious that, upon buying separately each component of a given portfolio, the investment cost results higher, but considered more carefully, this is indeed correct only if the infinite sum n ρ(f n ) corresponds to an actual cost, i.e only if such a strategy of buying separately infinitely many assets is feasible on the market. implies that, whenever f n n∈N is a uniformly bounded sequence of negligible functions, then necessarily sup n f n has to be negligible as well. An obvious implication is that N * has to be closed with respect to countable unions, a property which requires a rather deep reformulation of the axioms (1) that characterize economic rationality, as embodied into ≥ * . There may well be cases in which such additional condition is simply contradictory and which cast doubts on the economic adequacy of the countably additive paradigm.
A model in which the sample space Ω is a separable metric space is a good case in point. In fact, if we take If L is such that f ∧ 1 ∈ L for each f ∈ L, then φ ⊥ = 0 if and only if Proof. The proof of the existence of the representation (76) for some λ ∈ ba(Ω) + follows immediately from [14,Theorem 3.3]: given that φ is assumed to be positive, the identity map on L is clearly φ conglomerative and, since L is a lattice, directed as well. Given that L consists of A measurable functions, the representing charge λ may be restricted to A . Denote such restriction again by λ. If f ∧ 1 ∈ L for each f ∈ L, then Since f ∈ L 1 (Ω, A , λ) we obtain from ordinary results on finitely additive integrals, [20,III.3.6], that lim k |f − f k |dλ = 0. Thus if φ ⊥ = 0 then (77) holds; conversely, (77) implies [2] A , F., G , D. Arbitrage, short sales, and financial innovation. Econometrica 59, 4 (1991), 1041-1068. [4] A , K. J. The rôle of securities in the optimal allocation of risk bearing. Rev. Econ. Stud. 31 (1964), 91-96.