Complete and competitive financial markets in a complex world
aa r X i v : . [ q -f i n . M F ] M a r COMPLETE AND COMPETITIVE FINANCIAL MARKETS IN A COMPLEX WORLD
GIANLUCA CASSESEAbstract. We investigate the possibility of completing financial markets in a model with no ex-ogenous probability measure and market imperfections. A necessary and sufficient condition isobtained for such extension to be possible.
1. Introduction.Since the seminal contributions of Arrow [4] and of Radner [24], market completeness and the noarbitrage principle have played a prominent rˆole in financial economics. Market completeness, asfirst noted by Arrow, is a crucial property as it permits the optimal allocation of risk bearing amongrisk averse agents. In fact the equilibria of an economy under conditions of uncertainty but withcompetitive and complete financial markets are equivalent to those of an ordinary static economyso that classical welfare theorems apply. The equilibrium analysis on which this conclusion restsrequires that financial markets are free of arbitrage opportunities.General equilibrium theory with financial markets, however, is traditionally cast in the frameworkof a finite state space (or at least of an infinite sequence economy with finitely many states at eachdate) in which an appropriate justification of market incompleteness is more difficult. Our modelwill assume a completely arbitrary set Ω as the sample space – a situation to which we shall referas complexity. We believe that, despite the fast pace of financial innovation, the complexity ofmodern economic systems seems to be growing as fast which makes market completion an ongoingprocess. A natural consequence of this analysis is the assumption that in a complex world financialmarkets are incomplete. Given this general premise, the main questions we address in the paperare: (a) can an incomplete set of financial markets be extended to a complete one while preservingthe basic economic principle of absence of arbitrage opportunities? (b) if so, can such an extensionbe supported by a competitive market mechanism?Our answer is that this need not be the case. Competition on financial markets may in principleproduce two distinct outcomes. On the first hand it lowers margins on currently traded assets andresults thus in lower prices. On the other hand, competition involves the design and issuance ofnew securities. We argue that lower prices on the existing securities may destroy the possibility toobtain complete markets free of arbitrage opportunities. In principle the net effect of competitionon collective welfare may be unclear. Second, we argue that the completion of financial markets
Date: March 3, 2020.2010 Mathematics Subject Classification. Primary: , Secondary: .Key words and phrases. Absolute continuity, Market completeness. in respect of the no arbitrage principle may not be possible under linear pricing (which we takeas synonymous of perfect competition). We actually provide an explicit example. On the otherhand we show that if such an extension is possible with a limited degree of market power, it is thenpossible under perfect competition 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, we do not model thestrategic behaviour of intermediaries, as done, e.g., by Allen and Gale [2] or Bisin [7]. We ratherstudy the properties of pricing functions described as a sublinear functional on the space of tradedassets’ payoffs. The non linearity of prices captures the non competitive nature of financial marketsas well as the role of other market imperfections.In addition to market power, our model departs from traditional financial literature inasmuch asit lacks of any particular mathematical structure, topological or measure theoretic. In particular,following the thread of our previous papers [10] and [13], we do not assume the existence of anyexogenously given probability measure. Although this choice implies giving up the powerful artilleryof stochastic analysis, particularly in continuous time, it permits, we believe, a better understandingof how financial markets work in a context of unrestricted complexity. A thorough discussion ofthe reasons supporting this choice may be found in [10].In recent years there have been several papers in which the assumption of a given referenceprobability is relaxed, if not abandoned. Riedel [25] (and more recently Burzoni, Riedel and Soner[9]) suggests that an alternative approach to finance should be based on the concept of Knightianuncertainty. A typical implication of this approach is that a multiplicity of probability priorsis given – rather than a single one. Some authors, including Bouchard and Nutz [8], interpretthis multiplicity as an indication of model uncertainty, a situation in which each prior probabilitycorresponds to a different model that possesses all the traditional properties but in which it isunknown which of the models should be considered the correct one. An exemplification is the paperby Epstein and Ji [20] in which model uncertainty simply translates into ambiguity concerning thevolatility parameter. Other papers, among which the ones by Davis and Hobson [16] and by Acciaioet al [1], take the sample space to consist of all of the trajectories of some underlying asset andstudy the prices of options written thereon based on a path by path or model-free definition ofarbitrage.In our model, and similarly to Arrow’s setting, contingent claims are described simply as functionsof the sample space Ω. Differently from the papers mentioned above, this need not be a space oftrajectories (and thus a Polish space) and the functions describing securities payoffs need not becontinuous in any possible sense. Moreover, we do not adopt the pointwise definition of arbitragesuggested in [1], as this would implicitly correspond to assuming a form of rationality on economicagents even more extreme than probabilistic sophistication. Our starting point is rather a criterionof economic rationality embodied in a partial order which describes on what all agents agree when
ARKET COMPLETENESS 3 saying that “ f is more than g ”. This modeling of economic rationality, first introduced in [13], isreferred to as common order in [9].In section 2 we describe the model in all details, we introduce the notion of arbitrage and provesome properties of prices. In section 3 we characterize the set of pricing measures and in thefollowing section 4 we prove one of our main results, Theorem 1, in which the existence of marketextensions is fully characterized. Then, in section 5 we establish a second fundamental result,Theorem 2, in which we give exact conditions for such an extension to be competitive. Severaladditional implications are proved. Given its importance in the reference literature, in section6 we examine the question of countable additivity and eventually, in section 7 we return on theinterpretation of the common order as a probabilistic ranking.2. The Economy.We model the market as a triple, ( X , ≥ ∗ , π ), in which X describes the set of payoffs generatedby the traded assets, ≥ ∗ the criterion of collective rationality used in the evaluation of investmentprojects and π is the price of each asset as a function of its payoff. Each of these elements will nowbe described in detail.Before getting to the model we introduce some useful notation. Throughout Ω will be an arbi-trary, non empty set that we interpret as the sample space so that the family F (Ω) of real valuedfunctions on Ω will be our ambient space. If A ⊂ F (Ω), we write A u to denote its closure in thetopology of uniform distance. A class of special importance in F (Ω) is the family B (Ω) of boundedfunctions. The symbol P (Ω) designates the collection of finitely additive, probability functionsdefined on the power set of Ω. All probabilities in this paper will be considered to be just finitelyadditive, unless explicitly indicated, in which case the symbol P is replaced with P ca . Generalreferences for the theory of finitely additive set functions and integrals are [18] and [5].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 naturalorder.Natural as it may appear, pointwise order is not an adequate description of how economic agentsrank random quantities according to their magnitude, save when the underlying sample space isparticularly simple, such as a finite set. For example, it is well documented that investors basetheir decisions on a rather incomplete assessment of the potential losses arising from the selectedportfolios, exhibiting a sort of asymmetric attention that leads them to neglect some scenarios,in contrast with a pointwise ranking of investment projects . In a complex world, in which theattempt to formulate a detailed description of Ω is out of reach, rational inattention is just onepossible approach to deal with complexity. A different approach is the one followed in probabilitytheory to reduce complexity by restricting to measurable quantities. See [10] for a short discussion of some inattention phenomena relevant for financial decisions.
GIANLUCA CASSESE
In this paper, following the thread of [13], we treat monotonicity as a primitive economic notionrepresented by a further transitive, reflexive binary relation on F (Ω). To distinguish it from thepointwise order ≥ , we us the symbol ≥ ∗ .Let (cid:23) α represent the preference system of agent α over all acts F (Ω) and assume that(1) ( i ) . ≻ α ii ) . f ≥ g implies f (cid:23) α g i.e. that (cid:23) α is non trivial and pointwise monotonic. Then an implicit, subjective criterion ofmonotonicity (or rationality), ≥ α , may be deduced by letting(2) f ≥ α g if and only if b ( f − g ) + h (cid:23) a h b, h ∈ F (Ω) , b ≥ . A mathematical criterion ≥ ∗ describing collective rationality may then be defined as the meet ofall such individual rankings, i.e. as(3) f ≥ ∗ g if and only if f ≥ α g for each agent α the asymmetric part of ≥ ∗ will be written as > ∗ . One easily deduces the following, useful properties:(4a) ( i ) . > ∗ ii ) . f ≥ g implies f ≥ ∗ g, (4b) if f ≥ ∗ g then bf + h ≥ ∗ bg + h, b ∈ B (Ω) + , h ∈ F (Ω)(4c) if f > ∗ f ∧ > ∗ .It will be useful to remark that if f ≥ ∗ f { f ≤ } ≥ ∗ f ≥ ∗ f − f { f ≤ } = | f | . Associated with ≥ ∗ is the collection of negligible sets(5) N ∗ = { A ⊂ Ω : 0 ≥ ∗ A } together with the subset P (Ω , N ∗ ) ⊂ P (Ω) which consists of probability measures which vanish on N ∗ . Every subset of Ω not included in N ∗ will be called non negligible.It is immediate to note that any exogenously given probability measure P (countably or finitelyadditive) induces a corresponding ranking defined as(6) f ≥ P h if and only if inf ε> P ( f > h − ε ) = 1 f, h ∈ F (Ω) , which satisfies the above axioms (4). The same would be true if P were replaced with a family P ⊂ P (Ω) and if we defined accordingly(7) f ≥ P h if and only if f ≥ P h for all P ∈ P . The first paper to treat monotonicity in an axiomatic way was, of course, Kreps [22]. In a recent paper, Burzoniet al [9] adopt an approach quite similar to the present one. In [13, Theorem 1] we show that ≥ ∗ may arise from acash sub additive risk measure. ARKET COMPLETENESS 5
The ranking ≥ P defined in (7) arises in connection with the model uncertainty approach mentionedin the Introduction and exemplified by the paper by Bouchard and Nutz [8]. In this approach eachelement P of the given collection P is a model .A more interesting question concerns the conditions under which the ranking ≥ ∗ coincides withthe ranking ≥ P for some endogenous probability measure P . In this case we shall say that ≥ ∗ isrepresented by P . We shall address this question in the last section of this paper.2.2. Assets. We posit the existence of an asset whose final payoff and current price are used asnum´eraire of the payoff and of the price of all other assets, respectively. Each asset is identifiedwith its payoff expressed in units of the num´eraire and is modelled as an element of F (Ω). Themarket is then a convex set X ⊂ F (Ω) containing the origin as well as the function identicallyequal to 1 (that will be simply indicated by 1). Notice that we do not assume that investmentsmay be replicated on any arbitrary scale, i.e. that X is a convex cone, as is customary in thisliterature.We assume in addition that (i) each f ∈ X satisfies f ≥ ∗ a for some a ∈ R and (ii) that(8) f + λ ∈ X f ∈ X , λ ≥ . The first of these assumptions constraints the assets traded on the market to bear a limited risk oflosses and may be interpreted as a restriction imposed by some regulator; the second one permitsagents, which in principle may only form convex portfolios, to invest into the num´eraire asset anunlimited amount of capital. Notice that, since the num´eraire cannot be shorted, the constructionof zero cost portfolios – or self-financing strategies – is not possible. We are not assuming thatthe market prohibits short positions but rather that, in the presence of credit risk, long and shortpositions even if permitted should be regarded as two different investments as they bear potentiallydifferent levels of risk. In other words, when taking short positions, investors affect the implicitcounterparty risk and modify de facto the final payoff of the asset shortened.The issuance of new securities may result in the extension of the set X of traded assets. We mayconsider to this end several possibilities, varying from one another by the degree of completeness.A minimal extension is obtained when, along with each asset in X , investors are permitted totake a short position in the corresponding call option (a strategy very common on the market andknown as call overwriting). The resulting set of assets is(9) X = (cid:8) X ∧ k : X ∈ X , k ∈ R + ∪ { + ∞} (cid:9) . To the other extreme, we have the case of complete markets. In a model with poor mathematicalstructure such as the one considered here, the definition of market completeness is not entirelyobvious. The idea to define completeness as a situation in which all functions f ∈ F (Ω) are traded It should be noted that the choice of Bouchard and Nutz to take P to be a set of countably additive probabilitieshas considerable implications on N ∗ which needs e.g. be closed with respect to countable unions. GIANLUCA CASSESE is indeed too ambitious, as it would be difficult to define a price function on such a large domain.We rather identify market completeness with the set(10) L ( X ) = (cid:8) g ∈ F (Ω) : λX ≥ ∗ | g | for some λ > , X ∈ X (cid:9) which may be loosely interpreted as the set of superhedgeable claims. It is easily seen that L ( X )is a vector lattice containing X as well as B (Ω).2.3. Prices. In financial markets with frictions and limitations to trade, normalized prices are bestmodelled as positively homogeneous, subadditive functionals of the asset payoff, π : X → R ,satisfying π (1) = 1 and the monotonicity condition(11) X, Y ∈ X , X ≥ ∗ Y imply π ( X ) ≥ π ( Y ) . We also require that prices be free of arbitrage opportunities, a property which we define as (12) X ∈ X , X > ∗ π ( X ) > . Of course, (12) implies that π ( X ) ≥ X ≥ ∗ X > ∗ π ( X ) >
0, exceptional asit appears, does not represent in our model an arbitrage opportunity because of the infeasibilityof short positions in the num´eraire asset. A firm experiencing difficulties in raising funds for itsprojects and competing with other firms in a similar position may offer abnormally high returns tothose who accept to purchase its debt.A functional satisfying all the preceding properties – including (12) – will be called a pricefunction and the corresponding set will be indicated with the symbol Π( X ). We thus agree thatmarket 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 the noarbitrage property (12) may fail. These will be denoted by the symbol Π ( X ).The non linearity of financial prices is a well known empirical feature documented in the mi-crostructure literature (see e.g. the exhaustive survey by Biais et al. [6]) and essentially accountsfor the auxiliary services that are purchased when investing in an asset, such as liquidity provisionand inventory services. Subadditivity captures the idea that these services are imperfectly divisible.Another important property of price functions is cash additivity, defined as (13) π ( X + a ) = π ( X ) + a X ∈ X , a ∈ R such that X + a ∈ X . (the collection of cash additive price functions will be denoted by Π a ( X )). Although π ∈ Π( X )may fail to be cash additive, it always has a cash additive part π a , i.e. the functional(14) π a ( X ) = inf { t ∈ R : X + t ∈ X } π ( X + t ) − t X ∈ X . See [13] for a short discussion of alternative definitions of arbitrage in an imperfect market. This property, defined in slightly different terms, is discussed at length relatively to risk measures in [19]. In thecontext of non linear pricing cash additivity is virtually always assumed in a much stronger version, namely for all X ∈ X and all a ∈ R , see e.g. [3, Definition 1]. ARKET COMPLETENESS 7
It is routine to show that π a is the greatest element of Π a ( X ) dominated by π .Notice that the same quantity defined in (14) may be computed for each element of the setˆ X = X − R . If we denote by ˆ π a the corresponding extension, ˆ π a ∈ Π( ˆ X ) if and only if thereexists no ˆ X ∈ ˆ X such that ˆ X > ∗ ˆ π a ( ˆ X ). This corresponds to the classical definition of absence ofarbitrage, as given in the literature. This remark further clarifies the differences with our definition.3. Pricing MeasuresAssociated with each price π ∈ Π( X ) is the space (15) C ( π ) = (cid:8) g ∈ F (Ω) : λ [ X − π ( X )] ≥ ∗ g for some λ > X ∈ X (cid:9) . and, more importantly, the collection of pricing measures (16) M ( π ) = n m ∈ P (Ω) : L ( X ) ⊂ L ( m ) and π ( X ) ≥ Z Xdm for every X ∈ X o . The following is a very basic result illustrating the role of cash additivity and of the set M ( π ).Lemma 1. For given π ∈ Π ( X ) the set M ( π ) is non empty and each m ∈ M ( π ) satisfies(17) Z f dm ≥ Z gdm f, g ∈ L ( X ) , f ≥ ∗ g. Moreover, M ( π ) = M ( π a ) and(18) π a ( X ) = sup m ∈ M ( π ) Z Xdm X ∈ X ∩ B (Ω) . Eventually, the set M ( π ) is convex and compact in the topology induced by L ( X ).Proof. We simply use Hahn-Banach and the representation(19) φ ( f ) = φ ⊥ ( f ) + Z f dm φ f ∈ L ( X )established in [14, Theorem 3.3] and valid for positive linear functionals on a vector lattice ofreal valued functions. In (19), φ ⊥ is a positive linear functional on L ( X ) with the property that φ ⊥ ( f ) = 0 whenever f ∈ B (Ω) while m φ is a positive, finitely additive measure on the power setof Ω such that L ( X ) ⊂ L ( m φ ). Then, m φ ∈ P (Ω) if and only if φ (1) = 1.We easily realize that the functional defined by(20) π ( g ) = inf (cid:8) λπ ( X ) : λ > , X ∈ X , λX ≥ ∗ g (cid:9) g ∈ L ( X )is an element of Π ( L ( X )) extending π . By Hahn-Banach, we can find a linear functional φ on L ( X ) such that φ ≤ π and φ (1) = 1. Necessarily, f ≥ ∗ φ ( f ) ≥ φ is ≥ ∗ -monotone and thus m φ ∈ P (Ω). To show that m φ ∈ M ( π ) observe that, by assumption, each X ∈ X admits a ∈ R such that X ≥ ∗ a so that φ ⊥ ( X ) ≥ φ ⊥ ( a ) = 0 and thus π ( X ) = π ( X ) ≥ It is easily seen that C ( π ) is a convex cone containing X . In [13] a pricing measure was defined to be a positive, finitely additive measures dominated by π withoutrestricting it to be a probability. The focus on probabilities will be clear after Theorem 1 GIANLUCA CASSESE φ ( X ) ≥ R Xdm φ . Suppose now that m ∈ M ( π ) and that f, g ∈ L ( X ) and f ≥ ∗ g . Then, by (4b), { f − g ≤ − ε } ∈ N ∗ for all ε > Z ( f − g ) dm = Z { f − g> − ε } ( f − g ) dm ≥ − ε. We deduce (17) from f, g ∈ L ( m ).Concerning the claim M ( π ) = M ( π a ), it is clear that the inequality π a ≤ π induces theinclusion M ( π a ) ⊂ M ( π ). However, if X ∈ X , t ∈ R and m ∈ M ( π ) then X + t ∈ X implies π ( X + t ) − t ≥ Z ( X + t ) dm − t = Z Xdm so that m ∈ M ( π a ).The cash additive part π a of π , obtained as in (14), is easily seen to be an extension of π a to L ( X ). Of course, π a is the pointwise supremum of the linear functionals φ that it dominates sothat (18) follows if we show that m φ ∈ P (Ω) for all such φ . But this is clear since π a ≥ φ impliesthat φ is positive on L ( X ). Moreover, π a ( f ) = π a ( f + t ) − t ≥ φ ( f + t ) − t = φ ( f ) − t (1 − k m φ k ) t ∈ R which contradicts the inequality π a ≥ φ unless m φ ∈ P (Ω).The last claim is an obvious implication of Tychonoff theorem [18, I.8.5]. It is enough to notethat L ( X ) contains B (Ω) so that a cluster point of M ( π ) in the topology induced by L ( X ) isnecessarily represented by a finitely additive probability. (cid:3) Pricing measures closely correspond to the risk-neutral measures which are ubiquitous in thetraditional financial literature since the seminal paper of Harrison and Kreps [21]. We only highlightthat the existence of pricing measures and their properties are entirely endogenous here and donot depend on any special mathematical assumption – and actually not even on the absence ofarbitrage. In traditional models, the condition M ( π ) = ∅ is obtained via Riesz representationtheorem (here replaced with (19)) and requires an appropriate topological structure. Also noticethat (18) may be considered as our version of the superhedging Theorem. Upon re reading thepreceding proof one deduces that a version of (18) may be obtained for π (rather than π a ) byreplacing M ( π ) with the collection of finitely additive, positive set functions satisfying (16).It is customary to interpret the integral(21) Z Xdm as the asset fundamental value, although the values obtained for each m ∈ M ( π ) chosen maydiffer significantly from one another. We cannot at present exclude the extreme situation of anasset X ∈ X such that X > ∗ π a ( X ) > m ∈ M ( π ) Z Xdm = 0 . ARKET COMPLETENESS 9
This case, which, in view of (18), requires X to be unbounded, describes an asset with no intrinsicvalue (no matter how computed) which still receives a positive market price. For this reason itwould be natural to interpret such price as a pure bubble. In Theorem 1 we shall provide necessaryand sufficient conditions which exclude pure bubbles.Notice that bubbles need not always be pure. We define a bubble as the quantity:(23) β π ( X ) ≡ π a ( X ) − lim k →∞ π a ( X ∧ k ) = π a ( X ) − sup m ∈ M ( π ) Z ( X ∧ k ) dm where we used (18) and the fact thatlim k →∞ sup m ∈ M ( π ) Z ( X ∧ k ) dm = sup m ∈ M ( π ) Z ( X ∧ k ) dm. Our definition of a bubble is thus quite conservative as it amounts to the minimum spread of theprice over the fundamental value of the asset, no matter how computed.Another possible failure of pricing measures under the current assumptions is that it may not bepossible to support the view expressed in many microstructure models and according to which theask price π ( X ) of an asset is obtained by applying some mark-up to its fundamental value, such as(24) π ( X ) = [1 + α ( X )] Z Xdm with α ( X ) ≥ α (1) X ∈ X . In fact (24) not only requires the absence of pure bubbles, but also the existence of a pricing measurewith the property that
X > ∗ R Xdm >
0. This further property will be discussed at lengthin section 5. 4. Market Completeness.Competition among financial intermediaries may involve existing assets and/or the launch ofnew financial claims. As a consequence it may produce two different effects: (a) a reduction ofintermediation margins, and thus lower asset prices, and (b) an enlargement of the set X of tradedassets, thus contributing to complete the markets. This short discussion justifies our interest forthe set (25) Ext( π ) = (cid:8) π ′ ∈ Π ( L ( X )) : π ′ | X ≤ π (cid:9) . In this section we want to address the following question: under what conditions is it possible toextend the actual markets to obtain an economy with complete financial markets without violatingthe no arbitrage principle? This translates into the mathematical condition Ext( π ) = ∅ and if π ′ ∈ Ext( π ) we speak of ( L ( X ) , ≥ ∗ , π ′ ) as a completion of ( X , ≥ ∗ , π ).We obtain the following complete characterisation for the case of cash additive completions.Theorem 1. For a market ( X , ≥ ∗ , π ) the following properties are mutually equivalent:(a). π satisfies(26) C ( π ) u ∩ { f ∈ F (Ω) : f > ∗ } = ∅ , Dropping the index 0 or adding the superscript a to Π will result in a similar transformation of Ext. (b). the market ( X , ≥ ∗ , π ) admits a cash-additive completion,(c). the set M ( π ) is such that(27) sup m ∈ M ( π ) Z ( f ∧ dm > f > ∗ . Proof. Assume that (26) holds and define the functional(28) ρ ( f ) = inf (cid:8) λπ ( X ) − a : a ∈ R , λ > , X ∈ X such that λX ≥ ∗ f + a (cid:9) f ∈ L ( X ) . Property (4b) and ≥ ∗ monotonicity of π imply that ρ ∈ Ext ( π a ). Moreover, it is easily seen that(29) ρ ( f + x ) = ρ ( f ) + x f ∈ L ( X ) , x ∈ R i.e. that ρ is cash additive. To prove that ρ ∈ Π( L ( X )), fix f ∈ L ( X ) and k ∈ N such that f k = f ∧ k > ∗
0. Observe that f k ∈ L ( X ) and, in search of a contradiction, suppose that ρ ( f k ) ≤
0. Then for each n ∈ N there exist a n ∈ R , λ n > X n ∈ X such that λ n X n ≥ ∗ f k + a n but λ n π ( X n ) < − n + a n . This clearly implies(30) λ n [ X n − π ( X n )] ≥ ∗ f k − − n and f k ∈ C ( π ) u , contradicting (26). It follows that ρ ( f k ) > ⇒ (b).Choose ρ ∈ Ext a ( π ) and let f k be as above. Consider the linear functional(31) ˆ φ ( x + bf k ) = x + bρ ( f k ) x, b ∈ R defined on the linear subspace L ⊂ L ( X ) spanned by { , f k } . Given that ρ satisfies (29), ˆ φ isdominated by ρ on L so that we can find an extension φ of ˆ φ to the whole of L ( X ) still dominatedby ρ . As in Lemma 1, given that φ is a positive linear functional on a vector lattice, we obtain therepresentation (19) with m φ ∈ M ( ρ ) ⊂ M ( π ). Moreover, φ ( f ++ ) ≤ ρ ( f + k ) = ρ ( f k ) = φ ( f k ) so that φ ( f k ) = φ ( f + k ) by positivity. Eventually observe that, again by Lemma 1,0 < ρ ( f k ) = ρ ( f + k ) = Z f + k dm φ = Z f k dm φ . Thus (c) follows from (b).Assume now (c). Let f ∈ C ( π ) u be such that f ≥ ∗ n ∈ N arbitrarily. Then thereexist λ n > X n ∈ X such that 2 − n + λ n [ X n − π ( X n )] ≥ ∗ f . Notice that this implies theinclusion C ( π ) u ∩ { f ∈ F (Ω) : f ≥ ∗ } ⊂ L ( X ). But then, for every m ∈ M ( π ), Lemma 1 and(16) imply Z f dm ≤ − n + λ n h Z X n dm − π ( X n ) i ≤ − n so that (a) follows. (cid:3) It is immediate to recognize a very close relationship between (26) and the No-Free-Lunch-with-Vanishing-Risk (NFLVR) notion formulated long ago by Delbaen and Schachermayer [17] in ahighly influential paper. This similarity is quite surprising in view of the deep differences in the
ARKET COMPLETENESS 11 starting assumptions of the present model with theirs . The main point is that, in our setting theelements of the form λ [ X − π ( X )] with λ > X ∈ X cannot be interpreted as net payoffs of acorresponding trading strategy since the possibility of borrowing funds by shorting the num´eraireis precluded as well as the strategy of replicating a given investment on an arbitrary scale. Noticealso that in [17] the NFLVR condition was formulated in purely mathematical terms (and withreference to an exogenously given probability measure) while its economic content has remainedlargely unexplained.Upon relating condition (26) with the existence of a strictly positive, cash additive extensionof the pricing functional, Theorem 1 characterizes the economic role of NFLVR as a conditionnecessary and sufficient for financial markets to admit an extension that, while completing thefamily of assets traded, preserves the absence of arbitrage opportunities. The focus on the extensionproperty of financial prices was clear in the papers by Harrison and Kreps [21] and Kreps [22] (seealso [10, Theorem 8.1]) but has been somehow neglected in the following literature. Notice that astrictly positive extension may still exist even when (26) fails. In this case, however, it cannot becash additive.Notice that, in the light of the discussion following (22), the above condition (27) correspondsto a No-Pure-Bubble (NPB) condition while it does not exclude more general bubbles defined asin (23).Incidentally we remark that, from the equality M ( π ) = M ( π a ), it follows that π satisfiescondition (26) if and only if so does π a . More precisely,Lemma 2. Let π ∈ Π ( X ). Then: (a) C ( π ) ⊂ C ( π a ) ⊂ C ( π ) u and (b) for every X ∈ X , π a ( X ) ≤ X ∈ C ( π ) u . Therefore, π a ∈ Π( X ) if and only if(32) C ( π ) u ∩ { X ∈ X : X > ∗ } . Proof. (a). For each X ∈ X it is obvious that X − π ( X ) ≤ X − π a ( X ). However, X − π a ( X ) isthe limit, uniformly as t → + ∞ , of X + t − π ( X + t ) ∈ C ( π ). (b). X ∈ X and π a ( X ) ≤ n ∈ N and for t n > X ≤ X − π a ( X ) ≤ − n + [ X + t n − π ( X + t n )]so that X ∈ C ( π ) u . Viceversa, if X ≤ − n + λ n [ X n − π ( X n )] for some X n ∈ X and λ n ≥ λ n π ( X n ) to the left hand side if positive and using cash additivity, we conclude π a ( X ) ≤ − n + λ n [ π a ( X n ) − π ( X n )] ≤ − n . (cid:3) To highlight the role of competition in financial markets, consider two pricing functions π, π ′ ∈ Π( X ). If π ≤ π ′ then C ( π ′ ) ⊂ C ( π ). Thus, lower financial prices are less likely to satisfy (26) andthus to admit an extension to a complete financial market free of arbitrage. Competition amongmarket makers, producing lower spreads, may thus have two contrasting effects on economic welfare. For example, one may remark in mathematical terms that although (26) makes use of the uniform topology(perhaps the one closest to the L ∞ ( P ) one adopted in [17]) the set C ( π ) does not consist of bounded functions. 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 optimalallocation of risk. It may be conjectured that fully competitive pricing, interpreted as the pricingof assets by their fundamental value, may not be compatible with the extension property discussedhere. We investigate this issue in the following section.Eventually, we give a mathematical reformulation of Theorem 1.Corollary 1. For a market ( X , ≥ ∗ , π ) the condition (26) is equivalent to the following:(33) \ m ∈ M ( π ) C ( π ) L ( m ) ∩ { f ∈ F (Ω) : f > ∗ } = ∅ . Proof. Of course, the topology of uniform distance is stronger than the one induced by the L ( m )distance, for any m ∈ ba (Ω). Thus, (33) implies (26). On the other hand, if m ∈ M ( π ) thennecessarily(34) Z f dm ≤ f ∈ C ( π ) L ( m ) . By Theorem 1, if (26) holds then the inequality (34) excludes the existence of f ∈ T m ∈ M ( π ) C ( π ) L ( m ) such that f > ∗ (cid:3) Corollary 1 suggests the choice of a topology, τ ( π ), weaker than the one induced by the uniformdistance and generated by the family of open sets (35) O εm ( h ) = n f ∈ F (Ω) : f − h ∈ L ( m ) , Z | f − h | dm < ε o h ∈ F (Ω) , m ∈ M ( π ) , ε > . This topology has the advantage of being endogenously generated by market prices.To close this section, we observe that a strictly related problem is whether markets my beextended, even if remaining incomplete. Much of what precedes remains true and we thus only givesome hints for the case X defined in (9).Corollary 2. The market ( X , ≥ ∗ , π ) admits a cash-additive extension ( X , ≥ ∗ , π ) if and only if(36) C ( π ) u ∩ { f ∈ X : f > ∗ } = ∅ , or, equivalently,(37) sup m ∈ M ( π ) Z ( f ∧ dm > f ∈ X , f > ∗ . It is in fact obvious that T m ∈ M ( π ) A L ( m ) = A τ ( π ) for all A ⊂ T m ∈ M ( π ) L ( m ). ARKET COMPLETENESS 13
5. Competitive Complete Markets.In this section we investigate the conditions under which the set M ( π ) contains a strictly positiveelement, i.e. some m such that(38) Z ( f ∧ dm > f > ∗ . If such m ∈ M ( π ) may be found it is then clear that pricing each payoff in L ( X ) by its fundamentalvalue results in an arbitrage free price function. Given our preceding discussion this condition maybe rightfully interpreted as the possibility of a fully competitive market completion. The subsetof those m ∈ M ( π ) which satisfy (38) will be indicated by M ( π ) . The set M ( π ; X ) can bedefined by restricting (38) to elements f ∈ X .Notice that if m satisfies (38) it is then strictly positive on any non negligible set while theconverse is, in general, not true. In fact if f > ∗ { f > } isnecessarily non negligible, but, since N ∗ need not be closed with respect to countable unions, theyare not sufficient to exclude that { f > ε } ∈ N ∗ for all ε > π as an indica-tion of market imperfections, e.g. the market power of market makers. If ρ ∈ Π ( L ( X )) a possiblemeasure of market power is defined as follows (with the convention 0 / m ( ρ ; f , . . . , f N ) = P i ≤ N ρ ( f i ) − ρ (cid:0) P i ≤ N f i (cid:1)P i ≤ N ρ ( f i ) and m ( ρ ) = sup m ( ρ ; f , . . . , f N )the supremum in (39) being over all finite sequences of positive and bounded functions f , . . . , f N ∈ B (Ω) + . If ρ ∈ Π( X ) we can define the quantity m ( ρ ; X ) as in (39) but with f , . . . , f N ∈ X ∩ B (Ω) + .Clearly, 0 ≤ m ( ρ ) ≤ m ( ρ )which is then strictly less than unity. On the other side, if prices include fixed costs, then m ( ρ )may well reach 1. As we shall see, the case m ( ρ ) = 1 is an extreme case of special importance.The question we want to address next is: given a market, is it possible to find a completion thatpermits some degree of competitiveness? In symbols, this translates into the question of whetherthere exists ρ ∈ Ext( π ) such that m ( ρ ) <
1. This condition has in fact far reaching implications.Theorem 2. A market ( X , ≥ ∗ , π ) satisfies M ( π ) = ∅ if and only if admits a cash additive comple-tion ( L ( X ) , ≥ ∗ , ρ ) with m ( ρ ) < m ∈ M ( π ), define ρ : L ( X ) → R by(40) ρ ( f ) = Z f dm f ∈ L ( X ) . In [25] a set function satisfying (38) was said to have full support and the emergence of measures of full supportfollows in that paper from the assumption that Ω is a complete, separable metric space and X consists of continuousfunctions defined thereon. See also [9]. Then, ρ is cash additive and m ( ρ ) = 0, by linearity. Conversely, assume that ρ ∈ Ext( π ) is cashadditive and that m ( ρ ) <
1. Define for each n ∈ N the set(41) B n = (cid:8) b ∈ L ( X ) : 1 ≥ b > ∗ ρ ( b ) > /n (cid:9) and notice that { f ∈ L ( X ) : 1 ≥ f > ∗ } = S n B n , because ρ ∈ Ext( π ). Denote by co( B n ) theconvex hull of B n . If f = P Ni =1 w i b i ∈ co( B n ) then, ρ ( f ) ≥ (cid:0) − m ( ρ ) (cid:1) N X i =1 w i ρ ( b i ) ≥ (cid:0) − m ( ρ ) (cid:1) /n. In view of the properties of M ( π ) proved in Lemma 1, we can then apply Sion minimax Theorem[26, Corollary 3.3] and obtain from (18)0 < inf f ∈ co( B n ) ρ ( f ) = inf f ∈ co( B n ) sup µ ∈ M ( ρ ) Z f dµ = sup m ∈ M ( ρ ) inf f ∈ co( B n ) Z f dµ. Therefore, for each n ∈ N there exists µ n ∈ M ( ρ ) such that inf f ∈ co( B n ) R f dµ n ≥ (1 − m ( ρ )) / n > m = P n − n µ n . Then, m ∈ M ( ρ ) ⊂ M ( π ) and, as a consequence, L ( X ) ⊂ L ( m ). Butthen, if f > ∗ f ∧ ∈ L ( X ), that f ∧ > ∗ R ( f ∧ dm > m ∈ M ( π ). (cid:3) What the preceding Theorem 2 asserts in words is that if a complete, arbitrage free marketis possible under limited market power, it is then possible under perfect competition – i.e. withassets priced according to their fundamental value. This does not exclude, however, the somewhatparadoxical situation in which the only possibility to complete the markets is by admitting unlimitedmarket power by financial intermediaries. As noted above, this describes the terms of a potentialconflict between the effort of regulating the market power of intermediaries and the support toa process of financial innovation that does not disrupt market stability by introducing arbitrageopportunities.Let us remark that the condition m ( ρ ) <
1, although economically sound, is not a trivial one, atleast when the structure of non negligible sets is sufficiently rich i.e. when uncertainty is a complexphenomenon. Consider, e.g., the case in which an uncountable family of possible, alternativescenarios is given. In mathematical terms we can model this situation via an uncountable, pairwisedisjoint collection { A α : α ∈ A } of non negligible subsets of Ω. Then, if ρ ∈ Ext( π ) and f α = A α it must be that ρ ( f α ) > α ∈ A and thus, for some appropriately chosen δ > α , α , . . . ∈ A ,(42) inf n ρ ( f α n ) > δ. But then, P ≤ n ≤ N (1 /N ) ρ ( f α n ) > δ while ρ (cid:0) P ≤ n ≤ N (1 /N ) f α n (cid:1) ≤ /N so that m ( ρ ) = 1. Thus,in the case under consideration m ( ρ ) = 1. Let us also remark that in the probabilistic approach, in ARKET COMPLETENESS 15 which N ∗ coincides with the collection of null sets of some a priori given probability, the existenceof the collection { A α : α ∈ A } above is not possible .Lemma 3. Assume the existence of uncountably many, pairwise disjoint non negligible sets. Thenfor each market ( X , ≥ ∗ , π ) (and with inf ∅ = 1),(43) inf ρ ∈ Ext( π ) m ( ρ ) = 1The cash additive extensions ρ of π that satisfy the condition m ( ρ ) < ρ ∈ Π ( L ( X )) be cash additive and such that m ( ρ ) <
1. Then there exists µ ∈ M ( ρ ) such that(44) lim µ ( | f | ) → ρ ( | f | ∧
1) = 0 . Proof. For each α in a given set A , let h A αn i n ∈ N be a decreasing sequence of subsets of Ω satisfyingthe following properties: (i) for each α, β ∈ A there exists n ( α, β ) ∈ N such that(45) A αn ∩ A βn = ∅ n > n ( α, β )and (ii) for each α ∈ A there exists m α ∈ M ( ρ ) such that lim n m α ( A αn ) >
0. If the set A is uncountable, then, as in the preceding Lemma 3, we can fix δ > α , α , . . . ∈ A such that(46) inf i ∈ N lim n → + ∞ ρ (cid:0) A αin (cid:1) > δ. For each k ∈ N define n ( k ) = 1 + sup { i,j ≤ k : i = j } n ( α i , α j ) and f ki = A αin ( k ) for i = 1 , . . . , k . Then f k , . . . , f kk ∈ B (Ω) are pairwise disjoint functions with values in [0 ,
1] and such that(47) inf ≤ i ≤ k ρ ( f ki ) > δ. But then, taking w i = 1 /k , we obtain(48) k X i =1 ρ ( w i f ki ) > δ while ρ (cid:16) k X i =1 w i f ki (cid:17) = 1 k ρ (cid:16) k X i =1 f ki (cid:17) ≤ k so that m ( ρ ) = 1. We thus reach the conclusion that A must be countable and deduce from thisand from [15, Theorem 2] that M ( ρ ) is dominated by some of its elements, µ . In addition, M ( ρ )is weak ∗ compact as a subset of ba (Ω), as proved in Lemma 1. It follows from [28, Theorem1.3] that M ( ρ ) is weakly compact. If µ does not dominate M ( ρ ) uniformly, we can then find asequence h E n i n ∈ N of subsets of Ω, a sequence h m n i n ∈ N in M ( ρ ) and some constant d > µ ( E n ) → m n ( E n ) > d . Passing to a subsequence, we can assume that h m n i n ∈ N is In mathematics 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 [23]. See the comments in [15].It is clear that in the following statement the collection { A α : α ∈ A } may be chosen to meet a weaker condition,namely that the pairwise intersections are negligible sets. weakly convergent and so, by the finitely additive version of the Theorem of Vitali, Hahn and Saks(see e.g. [5, Theorem 8.7.4]), that the set { m n : n ∈ N } is uniformly absolutely continuous withrespect to m = P n − n m n and, since µ ≫ m , with respect to µ as well which is contradictory.We conclude that(49) lim µ ( A ) → sup m ∈ M ( ρ ) m ( A ) = 0 . Let h f n i n ∈ N be a sequence in L ( X ) that converges to 0 in L ( µ ) in therefore in µ measure. Then,by (18) lim n ρ ( | f n | ∧ ≤ lim n ρ ( {| f n | >c } ) + c = lim n sup m ∈ M ( ρ ) m ( | f n | > c ) + c so that the claim follows. (cid:3) Notice that Theorem 3 does not require the no arbitrage property and may thus be adaptedto the case in which L ( X ) is a generic vector lattice of functions on Ω containing the boundedfunctions and ρ a monotonic, subadditive and cash additive function, such as the Choquet integralwith respect to a sub modular capacity.Another characterization of the condition M ( π ) = ∅ may be obtained as follows:Theorem 4. A market ( X , ≥ ∗ , π ) satisfies the condition M ( π ) = ∅ if and only if there exists µ ∈ P (Ω) such that(50) X ⊂ L ( µ ) and C ( π ) L ( µ ) ∩ { f ∈ F (Ω) : f > ∗ } = ∅ . In this case one may choose µ ∈ M ( π ).Proof. If µ ∈ M ( π ) then, by definition, X ⊂ L ( µ ) and R f dµ ≤ f ∈ C ( π ) L ( µ ) whichrules out f > ∗
0. Conversely, if µ ∈ P (Ω) satisfies (50) and h > ∗
0, then h ∧ ∈ L ( µ ) and h ∧ > ∗
0. There exists then a positive and continuous linear functional φ h on L ( µ ) such that(51) sup n φ h ( f ) : f ∈ C ( π ) L ( µ ) o ≤ < φ h ( h ∧ . Given that necessarily φ h (1) >
0, (51) remains unchanged if we replace φ h by its normalization sothat we can assume φ h (1) = 1. This implies that φ h ∈ Ext ( π ) and, by [11, Theorem 2], that φ h admits the representation(52) φ h ( f ) = Z f dm h f ∈ L ( µ )for some m h ∈ M ( π ) such that L ( µ ) ⊂ L ( m h ) and m h ≪ µ . Moreover, by exploiting thefinitely additive version of Halmos and Savage theorem, [12, Theorem 1], we obtain that the set { m h : h > ∗ } is dominated by some m ∈ M ( π ). It is then clear that m ( f ∧ > f > ∗ m ∈ M ( π ). (cid:3) ARKET COMPLETENESS 17
Let us mention that under finite additivity the existence of a strictly positive element of M ( π )established in Theorem 2, is not sufficient to imply that the set M ( π ) is dominated, i.e. that eachof its elements is absolutely continuous with respect to a given one. It rather induces the weakerconclusion that there is a given pricing measure m such that m ( A ) = 0 implies m ( A ) = 0 for all m ∈ M ( π ).On the other hand, if such a dominating element exists then, by weak compactness, it domi-nates M ( π ) uniformly. A similar conclusion is not true in the countably additive case treated inthe traditional approach. In that approach, the set of risk neutral measures is dominated as animmediate consequence of the assumption of a given, reference probability measure but such set isnot weakly ∗ compact when regarded as a subset of the space of finitely additive measures. Thisspecial feature illustrates a possible advantage of the finitely additive approach over the countablyadditive one.As above, we can formulate a version of the preceding results valid for partial extensions. Again,the proofs remain essentially unchanged. Let X be defined as in (9). Specializing definitions (39)and (38) by replacing F (Ω) with X we obtain the definitions of m ( ρ ; X ) replacing m ( ρ ) when ρ ∈ Π( X ) and of M ( π ; X ) replacing M ( π ).Corollary 3. The market ( X , ≥ ∗ , π ) satisfies M ( π ; X ) = ∅ if and only if it admits a cash additiveextension ( X , ≥ ∗ , ρ ) with m ( ρ ; X ) < M ( π ) contains acountably additive element. A more ambitious question is whether such measure is strictly positive,i.e. an element on M ( π ).Not surprisingly, an exact characterisation may be obtained by considering the fairly unnaturalpossibility of forming portfolios which invest in countably many different assets. This induces tomodify the quantity appearing in (39) into the following (again with the convention 0 / m c ( ρ ; f , f , . . . ) = lim k → + ∞ P n ≤ k ρ ( f n ) − ρ (cid:0) P n f n (cid:1)P n ≤ k ρ ( f n ) ρ ∈ Π( L ( X ))for all sequences h f n i n ∈ N in B (Ω) + such that P n f n ∈ B (Ω). Notice that in principle, the inequality m c ( ρ ; f , f , . . . ) ≥ m c ( ρ ; f , f , . . . ) ≤ P n ρ ( f n ) corresponds to an actual cost, i.e only if such a strategy of buyingseparately infinitely many assets is feasible on the market.Define then the functionals(54) m c ( ρ ) = sup m c ( ρ ; f , f , . . . ) and n c ( ρ ) = − inf m c ( ρ ; f , f , . . . ) where both the supremum and the infimum are computed with respect to all sequences in B (Ω) + with bounded sum. Notice that m c ( ρ ) = m ( ρ ).Theorem 5. Let π ∈ Π ( X ). Then:(a). M ( π ) ∩ P ca (Ω) = ∅ if and only if there exists ρ ∈ Ext ( π ) such that n c ( ρ ) < + ∞ and that(55) X n ρ ( f n ) < ∞ for all f , f , . . . ∈ B (Ω) + with X n f n ∈ B (Ω);(b). M ( π ) ∩ P ca (Ω) = ∅ if and only if there exists ρ ∈ Ext( π ) such that n c ( ρ ) < + ∞ and m c ( ρ ) < m ∈ M ( π ) ∩ P ca (Ω) when considered as a pricing function is anelement of Ext ( π ) such that n c ( m ) = m c ( m ) = 0. This proves necessity for both claims. To provesufficiency, let ρ ∈ Π ( L ( X )) and choose a sequence h f n i n ∈ N in B (Ω) + with P n f n ∈ B (Ω). If m c ( ρ ) <
1, then(56) X n ρ ( f n ) ≤ − m c ( ρ ) ρ (cid:16) X n f n (cid:17) < ∞ so that (55) is satisfied. It is therefore enough to show that if ρ meets the conditions listed under (a)then M ( ρ ) ⊂ P ca (Ω). Assume to this end that ρ ∈ Ext ( π ) is such a function, choose m ∈ M ( ρ )and let h A n i n ∈ N be a disjoint sequence of subsets of Ω. Then, upon setting f n = A n , we get P n ρ ( f n ) < ∞ and therefore m (cid:16) [ n A n (cid:17) = X n m ( A n ) + lim k m (cid:16) [ n>k A n (cid:17) ≤ X n m ( A n ) + lim k ρ (cid:16) X n>k f n (cid:17) ≤ X n m ( A n ) + [1 + n c ( ρ )] lim k X n>k ρ ( f n )= X n m ( A n ) . Thus, M ( ρ ) ⊂ P ca (Ω). If, m c ( ρ ) < m ( ρ ) < (cid:3) The conditions for the existence of a countably additive pricing measure listed under (a) and(b) are perhaps deceptively simple. In fact the inequality n c ( ρ ) > −∞ implies that, if h f n i n ∈ N isa uniformly bounded sequence of negligible functions, then necessarily sup n f n has to be negligibleas well. Thus, e.g., N ∗ has to be closed with respect to countable unions. This property requiresa rather deep reformulation of the axioms (4) that characterize economic rationality, ≥ ∗ , andthere may well be cases in which such additional conditions are simply contradictory. If, e.g.,Ω is a separable metric space and N ∗ consists of sets of first category then, as is well known, P ca (Ω , N ∗ ) = ∅ , see [27, Th´eor`eme 1].In the classical setting of continuous time finance, the order ≥ ∗ is generated by some countablyadditive probability and each X ∈ X is of the form X = W X + R θ X dS where S is, e.g., a locally ARKET COMPLETENESS 19 bounded semimartingale with respect to the given probability measure, θ is a predictable processintegrable with respect to S and its price, in the absence of frictions, is ρ ( X ) = W X . Although thecondition m c ( ρ ) = 0 is immediate the inequality n c ( ρ ) > −∞ is not at all obvious. In fact, evenwhen 0 ≤ X n ≤ X n +1 and X = lim n X n ∈ B (Ω) it is not easy to show that there is θ such that(57) X n Z θ X n dS = Z θ dS.