6.1  Introduction¹

Financial theorists' recent interest in common knowledge stems from the need to eliminate arbitrage opportunities in models of financial markets.  Without implicit or explicit common knowledge assumptions, many so-called `riskless arbitrage' trades in reality entail speculative exposure to risk from changing market beliefs and asset valuations.

    Aumann (1976) introduced a mathematical definition of common knowledge using information partitions (σ-algebras) and coarsenings.  Milgrom (1981) and Brandenberger and Dekel (1987) further developed the mathematical structure for representing common knowledge.  The necessity of common priors has been studied for repeated communication by Geanakoplos and Polemarchakis (1982), Nielsen (1984), Aumann (1987), and more recently in imputed rationality to game players by Nau and McCardle (1990) and Nau (1992).  This chapter examines the necessity for common knowledge priors in a contingent claims economy.  The `equi­librium' solution concept utilized in this chapter is the absence of arbitrage.  This particular structure underpins much of contingent claims analysis, option pricing, and arbitrage pricing theory (see Babbs and Selby (1992) in this regard).

    Our work also touches upon results found in Milgrom and Stokey (1982), Milgro­m (1981), Tirole (1983) and Sebenius and Geanakoplos (1983).  These papers relate common knowledge (of trader preferences and market structures) to no trading theorems in pure exchange economies with asymmetric information.  Conversely, our paper investigates what the assumption of no trading of a particular type (arbitrage opportunities) implies about common knowledge (priors).

    The no-trading proposition of Milgrom and Stokey (1982), Tirole (1983), and others asserts that if agents have common priors regarding the value of some assets, and receive private information about future values, then one agent's mere willingness to trade means no trading partner should com­plete the trade.  The problem with this logic is that individuals with common priors have no way of knowing a priori that the rest of the market agrees with them.  In fact, it would be just as reason­able for them to be­lieve the rest of the market, or some fraction thereof, disagrees (at least in part) with their own priors.  When priors are common but not common knowledge, the mere willingness to trade could indicate someone has (1) superior information or (2) apparently a different prior.  No trader would be able to distinguish between these two cas­es a priori, and Chapter 4 demonstrated the formal proof of this lack of consensus.²  Therefore, we draw an important distinction be­tween priors being common (homogeneous) and common knowledge (leaving no uncertainty about how others hold their priors).

    The condition for no arbitrage opportunities applied in this chapter leads to a series of no-price-exists results, which are analogous to the no-trading result of the common knowledge litera­ture.  For example, we show that when any trader knows with certainty that an asset poses  limited lia­bility, no positive price will exist for the asset.

    Our work is also related to the more fundamental analysis of Nau and McCardle (1990).  They formulate a new axiomatic foundation for rational behavior in a finite non-cooperative game, which takes as one of its basic axioms the absence of arbitrage.  From this foundation, they deduce an equiv­alence theorem between `correlated' equilibria (Aumann 1987) and strategies, which they call `jointly co­herent' (which require the absence of arbitrage).  Further, they show that with an additional complete­ness hypothesis the identical common knowledge priors assumption is a consequence of their theory.  Nau (1992) further explores the relaxation of this completeness as­sumption in a subsequent, related paper.

    Our work uses similar concepts to Nau and McCardle, yet the focus differs significantly. Rather than exploring foundational issues, we take a state space, information partition (σ-algebra), and individual prior beliefs as given.  From these, in a market context, we deduce what the absence of arbitrage (no trading) implies about the common knowledge of prior beliefs.

    Our line of argument requires the construction of a universal beliefs space with consistent beliefs along the lines of Mertens and Zamir (1985) or Tan and Werlang (1988).  In our construction, succes­sively high­er-order beliefs are independent of both lower order beliefs and the first-order beliefs over states of the world.  Thus no individual is restricted into believing everyone else must assign the same probability as his own (subjective) priors.  The construction uses well-known results from probability theory.

    Consider a pure exchange economy where the universal beliefs space, its σ-algebra, individuals' information sets, and prices are all common knowledge, but where traders can have private information.  We prove that the absence of arbitrage implies that prices partially reflect private infor­mation involving higher-order beliefs used in trading, and this partial information be­comes common knowledge.  (In another minor contribution, this chapter formulates a new and generalized definition of the no arbitrage condition when traders have private infor­mation.)  Those events which traded securities can isolate as pure bets become common knowledge.  Furthermore, the richer the set of traded securities, the more information about beliefs must be common knowl­edge.  In the limiting case of complete markets (see Green and Jarrow (1987) for the relevant defini­tion), common knowledge priors are a necessary condition of the analysis.  Our definition of complete markets differs from the standard definition used in general equilibrium theory.  Complete markets span not only the possible states of the world but also the possible distributions for higher-order beliefs over the states.

    Market completeness plays a crucial role in the con­tinu­ous time stochastic market models used to price contingent claims.  These models require completeness in the strong sense used above and exclude speculation based on incomplete information about other traders' beliefs.  Our analysis sheds considerable light on the restrictive na­ture (and perhaps inap­propriateness) of this completeness assumption in many models in the options pricing literature.

    The outline for the chapter is as follows.  Section 6.2 provides the canonical arbitrage pricing model without imposing com­mon knowledge priors.  Section 6.3 generalizes the range of contingent claims to distinguish intrin­sic un­certainty (a term coined in Guth 1989) from uncertainty over extrinsic events.  Section 6.3 also constructs a hierarchy of higher-order beliefs into a universal beliefs space.  Section 6.4 briefly applies the work in Section 6.2 and 6.3 to option pricing theory, and Section 6.5 concludes the chapter.

6.2.  The Canonical Arbitrage Pricing Model

The canonical arbitrage pricing model extant consists of a single time period, starting at time 0 and ending at time 1.  The state of the economy at time 1 is uncertain.  The state space and possible events are represented by a mea­sure space (Ω,ö) with Ω an arbitrary set and ö the associated -algebra.

    There are a finite number of traders {1, 2, ..., I }, indexed by (a superscript) i, each with an information set F ⁱ.  The information set F ⁱ is a σ-algebra contained in ö, the set of events.  Let  denote the largest σ-algebra of sets containing the in­formed or true state across all the F ⁱ.   These information sets allow for infinite state spaces and, therefore, generalize the information partition concept applicable to finite state spaces used in the common knowledge litera­ture.  This added generality is needed to incorporate the universal beliefs space (an infinite set) as a possible set for Ω.

    The fact that some set of events is known to individual i after he has received his private signal means that it is F ⁱ-measurable.  For that set to be common knowledge across all I individuals means that it must be -measurable.

    The term `common knowledge' can be used as a noun or an adjective.  Thus, later when we speak of `common knowledge price functionals', π, we mean the π are -measurable.  Just as we use the expressions `homoge­neous beliefs' or `common priors,' so can we use the term `common knowledge beliefs' or `common knowledge priors' to express the notion that each individual's prior is -measurable. 

    We assume that the state space Ω, the set of all possible events, ö, and the sets F ⁱ for i = 1, ..., I are common knowledge.  Although the collection of sets F ⁱ and ö are common knowl­edge, those events (elements of these sets) containing the true state ω₀ ε Ω are generally not common knowledge.

    We say that individual i's information set, F ⁱ, is common knowledge, when everyone knows everyone knows .... everyone else's information set.  Mathematically, we could define a measure space for a random variable over possible values of the original information sets, F ⁱ, for each indi­vidual.  Each individual i would have another information set, say , in this new measure space.  We would then say an information set F ⁱ is common knowledge when F ⁱ is -measurable.  Once it is recognized how to write mathe­matically that the information sets are common knowledge, we do not need to intro­duce the additional notation and carry it throughout the chapter.  Alternatively, we could redefine our original event space to include not only states of the world but also a specification of each individual's information set.

    Each trader is endowed with a probability measure pⁱ: ö ® [0, 1] representing the trader's sub­jective prior beliefs.  These probability measures pⁱ can differ across traders and need not be common knowledge.  The necessity of this restriction, or the lack thereof, is the topic of this chapter.

    In forming their time 0 trading decisions, individuals will condition on their information sets.  To facilitate the analysis, we assume for all i  14 {1, 2, ..., I },  and A Î ö,  pⁱ(A*F ⁱ)(ω) is a regu­lar condi­tional probability of ö given F ⁱ, i.e., for fixed ω₀    Ω, pⁱ(*F ⁱ)(ω₀) is a probability measure of ö, and for fixed A  ö, pⁱ(A*F ⁱ)() is F ⁱ-measurable.  We also assume pⁱ is posterior complete, i.e., F ⁱ contains the pⁱ(*F ⁱ)(ω) zero probability events for all ω  Ω.  Finally, we assume that pⁱ is proper (i.e., pⁱ(A*F ⁱ)(ω) = 1_A(ω) for every A   F ⁱ ).  These last two hypotheses ensure that our definition of common knowledge matches Aumann (1976) when beliefs are common knowledge (Brandenburger and Dekel 1987, Prop. 2.1).

    In the economy, traders exchange financial securities entitling the owner to time 1 cash flows and the resale value of the paper claim (the capital gain).  Let M represent a linear sub­space of the ö-measurable functions mapping Ω into .  The linear subspace condition implies that if the assets m₁ and m₂ trade, then all portfolios with  shares of m₁ and   shares of m₂ trade as well (for all ).  The set M must be common knowledge; otherwise, the traders do not know they are playing the same game.  Consequently, the cash flows and prices that will prevail in each of the states are common knowledge.

    The time zero value of the traded assets m  M are represented by a price functional π:  M , i.e., π(m) denotes the time 0 price of asset m  M.  We require that the asset markets are fric­tionless in that π is linear, i.e., given m₁, m₂  M and  then  (m₁+m₂) = π(m₁) + βπ(m₂).  The price func­tional π is common knowledge.

    For easy reference, we collect in a first assumption those quantities which are common knowl­edge.

Assumption 1:  (Common Knowledge Parameters)

    (Ω,ö), {F ⁱ: i = 1, ..., I}, M, and π are all common knowledge.

    Let ω₀ ε Ω represent the unknown, `true' realization of the state space.  Define the  sets    and  and  

   These sets represent special collections of traded assets, as seen by trader i.  The first, (M⁺)ⁱ(ω₀), represents those traded assets which (under ω₀ ε Ω) the trader believes are nonzero and of limited liability since they always have non-negative payoffs at time 1.  These assets will typically have zero payoffs for some states and positive payoffs in others.  The set (M⁰)ⁱ(ω₀) represents those traded assets that trader i believes (under ω₀ ε Ω) have zero payoffs with probability one.  The set (M⁰)ⁱ(ω₀) repre­sents an equivalence class, under the i ^th investor's beliefs about assets with zero payoffs, i.e., it is the `zero' element in the space of traded assets.  Arbitrage opportunities are trader- and state-specific.

Definition (An Arbitrage Opportunity):  m ε M is called an arbitrage opportunity for individual i  under state  ω₀  ε  Ω  if  (1.a) m ε (M⁺)ⁱ(ω₀)

and (1.b) π(m)  0, or  (2.a) m ε (M⁰)ⁱ(ω₀) and (2.b) π(m)  0.

    This definition of arbitrage is a straightforward extension to asymmetrically informed trad­ers of that contained in the literature, see Jarrow (1987).  An arbitrage opportunity is a traded asset which satisfies one of two conditions:  either it is of limited liability with a non-positive price, or it is the `zero^th' asset, but with a non-zero price.  A well-functioning market (one that is in equi­librium) would not contain any of these opportunities, hence, we assume:

Assumption 2:  (No Arbitrage Opportunities)  Fix ω₀ ε Ω,  (a) if m ε  then π(m) > 0 and (b) if m ε  then π(m) = 0.

This assumption implies that any individual, acting independently of the other traders, can force prices to be positive or zero, depending upon his beliefs.  Any individual can act as a price dictator.

    It is easy to show that if there exists an m ε M such that m ε , then Assumption 2(a) implies Assumption 2­(b).³  This condition is satisfied, for example, if a riskless asset trades (1_Ω ε M where 1_Ω(ω) = 1  for all ω ε Ω).

    Assumptions 1 and 2 impose these necessary conditions on common knowledge:

(a)    if  m ε (M⁺)ⁱ(ω₀) for some i, then

       -m  (M⁺) ^j(ω₀) and ε (M⁰) ^j(ω₀) for all j.

(b)    if  m ε (M⁰)ⁱ(ω₀) for some i, then

       -m  (M⁺) ^j(ω₀) and ε (M⁺) ^j(ω₀) for all j.

    If some person believes with certainty an asset is of limited liability (condition a) or the `zero<sup>th</sup>' asset (condition b), then the absence of arbitrage implies that the other traders' beliefs must be minimally consistent.  Indeed, if one trader believes with certainty an asset is of limited liability, then no one else can believe that shorting the asset is of limited liability or that it is a `ze­ro' cash flow security.  Similarly, if one trader sees an asset as having `zero' cash flows, then no one else can believe with certainty that it or its negative is of limited liability.

    Consider the set of traded assets that everyone agrees has non-negative time 1 cash flows, i.e.,   These assets, by Assumption 2, have non-negative prices.  This set could be empty.

    Define σ(M) to be the smallest σ-algebra generated by all m ε M.  These are the events in 54 relevant to asset payoffs.  Next, define the set of events that generate potential arbitrage opportunities as D, where

 .  The set D could possibly contain only the empty set.  If the I individuals trade, however, the set  of traded assets D contains those assets that pay off $1 in state A and 0 otherwise, denoted 1_A(ω).  These are analogous to `Arrow-Debreu' securities.

Proposition 1  (Absolute Continuity of Beliefs):  Given Assumptions 1 and 2 and ω₀ ε Ω, if A ε D, then pⁱ(A*F ⁱ)(ω₀) = 0 if and only if p ^j(A*   F ^j)(ω₀) = 0, .

Proof:  If D = {φ} this is trivially true.  If D  {φ}, choose A ε D such that  A  =  {m > 0}  for  some  m  ε     If

pⁱ(A*F ⁱ)(ω₀) = 0  for some i, then  m  ε  (M⁰)ⁱ(ω₀).  By definition, m

(M⁺) ^j(ω₀) for any j.  Hence, m ε (M⁰) ^j(ω₀) for all j, i.e., p ^j(A*F ^j)(ω₀) = 0.  QED.

    Proposition 1 states that traders agree on zero probability events in the set of events D.  This proposition generalizes a similar proposition contained in Jarrow (1987).  Jarrow requires stronger assumptions on preferences and the market structure than that required in Proposition 1 above.  As a special case of this proposition, if D = σ(M), then traders agree on all the zero proba­bility events relevant to asset payoffs.  This is called a complete market (see Green and Jarrow 1987).  Thus in complete markets, equivalent probability beliefs are a necessary and sufficient condition for the ab­sence of arbitrage opportunities.

    Our use of equivalent probability beliefs differs from the `homogeneous or identical be­liefs' assumption of mean-variance theory and the arbitrage pricing literature.  Proposition 1 only implies that given an event A ε D, p<sup>i</sup>(A*F <sup>i</sup>)(ω<sub>0</sub>) = 0 if and only if p <sup>j</sup>(A*F <sup>j</sup>)(ω<sub>0</sub>) = 0 for all i, j.  This assumption is also known as `unconditional beliefs' in the literature.  In contrast, homogeneous or identical beliefs are defined as pⁱ(A) = p ^j(A) for all i, j and A ε ö.  With homogeneous beliefs, market traders objectively agree.  However, the traders do not know they agree unless their beliefs are both equal and common knowledge.⁴

Proposition 2  (Common Knowledge Information Revealed by Prices):

Given Assumptions 1 and 2 and ω₀ ε Ω, if A ε D and pⁱ(A*F ⁱ)(ω₀) = 0 then A is common knowledge.

Proof:  pⁱ(A*F ⁱ)(ω₀) = 0 implies A ε F ⁱ.  But, by Proposition 1, A ε D and pⁱ(A*F ⁱ)(ω₀) = 0 implies p ^j(A*F ^j)(ω₀) = 0 for all j.  Hence, A is -measurable.  QED.

    Although the set of possible assets to trade, D, is common knowledge, individuals do not know a priori which contingent claims will actually be traded.  This key proposition demonstrates that those events that can be bet upon (associated with traded assets) and that everyone agrees have non-negative cash flows (A ε D) become common knowledge through prices.  The reason is that if someone assigns 0 probability to such an asset having positive cash flow, (pⁱ(A*F­ ⁱ)(ω₀) = 0 for some i), then this asset's price must be zero, or there is an arbitrage opportu­nity.  But, the price  being  zero  is  com­mon  knowledge.   If  anyone  else  believes

p ^j(A*F ^j)(ω₀) > 0, then they would see an arbitrage opportuni­ty, and these do not exist.  Hence, as previously noted, everyone agrees that A can never occur, so A is common knowledge.

    An important case of this proposition is for complete markets.  In this case, D = σ(M), since all `Arrow-Debreu' securities trade (1_A ε M for all A ε σ(M)).  Here, all zero probability asset payoff relevant events will be common knowledge.  This occurs, for example, in standard option pricing theo­ry, see Section 6.4 below.

6.3  Extrinsic and Intrinsic Uncertainty

This section generalizes the canonical arbitrage pricing model to include uncertainty over investor beliefs in asset pricing.  We call the initial state space governing the actual state of the world extrinsic uncertainty and the remaining part pertaining to the beliefs of other market traders intrinsic uncertainty.  This distinction will clarify whether the absence of arbitrage opportunities requires com­mon knowledge priors over extrinsic uncertainty.

    Mertens and Zamir (1985) provide a formal model of Harsanyi's infinite hierarchies of be­liefs by constructing a universal beliefs space generated by a compact state space Ω.  They also examine consistency of a set of beliefs P over this universal beliefs space.  The following develop­ment gives an alternative construction of such a universal beliefs space Ω and a consistent set of beliefs, P, for the case where Ω is a complete, separable, metric space.  The results then follow from standard probability theory theorems (see Mertens and Zamir 1985, remark 2.18, p. 14).  This construction, although not the analysis of this section, also resembles the work of Tan and Werlang (1988).  Our higher-order beliefs are not artificially constrained to match first-order beliefs.

    Let (Y₁, B₁) be a measurable space where Y₁ is a complete, separable, metric space (Part­hasarathy 1967; p. 1) and B₁ is the Borel σ-algebra.  The set Y₁ represents the extrinsic uncer­tainty in the economy.  Extrinsic uncertainty is randomness due to exogenously-determined phe­nome­na, independent of traders' beliefs, which has a real effect on earnings.  Examples of ex­trinsic uncer­tainty are returns on firm investments, merger and acquisition policies, technological dis­coveries, etc.  This definition is distinct from that of extrinsic uncertainty used by Cass and Shell (1983).  `Sunspot equilib­ria' can occur in our model through intrinsic uncertainty when traders believe others believe sunspots are relevant.  This, however, is not the topic of the remaining analy­sis.

    Each trader (i = 1, 2, ..., I) has a subjective probability measure  defined over the extrinsic uncertainty.  The traders may not know the other traders' probability beliefs; however, they know the population from which these beliefs are `drawn.'

    Let Y₂ be the space of all probability measures on (Y₁, B₁) endowed with the topology of weak convergence.  Under this topology, Y₂ is a complete, separable, metric space (see Parthasa­rathy 1967; Theorem 6.2, page 43 and Theorem 6.5, page 46).  Let B₂ be the Borel σ