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-algebra associated with Y₂.  The I-cross product of Y₂ with itself  Y₂  Y₂  ...   Y₂ =  is the popula­tion from which the trader's beliefs   are `drawn.'

    Let B₂  ...  B₂ =  denote the product σ-algebra (see Parthasarathy 1967; Definition 1.2 and Theorem 1.10, p.6).  Although each trader does not know the other traders' beliefs, he assigns a sec­ond probability measure  over the possibilities.  This measure represents trader i's second-order belief over the first-order beliefs of the other traders (and for convenience himself as well).³

    Continuing by induction for n  2, let Y_n be the space of all probability measures on  en­dowed with the topology of weak convergence.  Again, this metric space is complete and separable.  Let B_n be the Borel σ-algebra associated with Y_n.  The I-cross product  denotes the popu­lation of the traders' (n-1)-order beliefs.  Define  as the I-cross product σ-algebra.  The n^th-order beliefs of trader i, , are defined over this set of events.  Note that  = 1 since  is a probability measure and 80 represents the entire population of the n^th-order `sample' points.

    Traders thus experience uncertainty not only over the initial probability space Y₁, repre­sent­ing extrinsic uncertainty, but over  as well.  This latter uncertainty is over the beliefs (1st and higher-order) of other traders, and it is called intrinsic uncertainty.  To capture this notion of an `ex­panded' state space and to transform this model into the form of the canonical arbitrage pricing mod­el, we construct the following space.  This construction comes from Parthasarathy (1967;, pp. 135-139).

    Fix a trader i ε {1, 2, ..., I}.  Define  and   Next, define for n  2, ,  , and   , where  is the n-fold cross product measure (see Breiman 1968; p. 399).  By construction, since  is a complete, separable, metric space for all n  1, the set  will be a complete, separable, metric space under the topology.  Hence,  is a separable and standard Borel space for all n  1.

    Define γ_n:   as the projection map, i.e., γ_n (y₁, y₂, ..., y_n, y_n+1) = (y₁, ..., y_n) where y₁ ε Y₁ and y_j ε   for j  2.  The expanded state space (Borel space) (Ω,ö) is defined to be the inverse limit of the Borel spaces  for n = 1, 2, 3,....  That is,   , and ö is the smallest σ-algebra of subsets of  such that  defined by  is measurable for all n.

    It is easy to see that the n^th-order cross-product measure  satisfies   Indeed,

                            =

                                                =

                                                = .

This consistency condition states that the modified probability beliefs at the n^th step, , restrict the joint distribution of the beliefs at step n+1, .  Hence, by Parthasarathy (1967; p. 139, Theorem 3.2), there exists a  unique probability  measure  P ⁱ defined  on (Ω,ö)  such that

P ⁱ() =  for all  and each n = 1, 2, 3, ....  This probability measure P ⁱ, defined on the expanded Borel space (Ω, ö), is the unique extension of the n^th-order cross product measures  for all n.  (This measure omits the sub­script denoting the order of beliefs over extrinsic uncertainty.)

    The σ-algebra ö includes events with both intrinsic and extrinsic uncertainty.  It is useful for the subsequent analysis to decompose ö  into those events which are either only extrinsic or intrin­sic.

    We claim that , the smallest σ-algebra generated by the class of events  represents those events in ö associated with extrinsic uncertainty alone.  Indeed, by standard arguments,   Recall that  is the projection of the universal beliefs space Ω into the extrinsic uncertainty space Y₁.  Hence, the inverse mapping   identifies all those events A  B₁ with events in the corresponding universal beliefs space Ω.  Since  is measurable, ö.  This proves the first claim.  Denote 132 for extrinsic events.

    Secondly, we claim that  133 are those events in ö associated only with intrinsic uncertainty.  In­deed,  =  = ö since each  is measurable.  In this set of events, no information is revealed about extrinsic uncertainty since the events consid­ered in the inverse mapping  always take as given the first set Y₁.  All that remains is information about higher beliefs.  Define

 for intrinsic events.  It follows easily then that ö.

    Each trader i = 1, 2, ..., I is endowed with an information set F ⁱ  ö.  To be consistent with the interpretation of the expanded state space, we add the following assumption.

 

Assumption 3:  (Traders Know Their Own Beliefs)

    144 and i = 1,..., I.

 

This  assumption  asserts that  trader i  has included  in  his  information

set all those events which contain his n^th-order beliefs (for all n).  Finally, we assume that for each i, P ⁱ(A*F ⁱ)(ω) is a regular conditional probabili­ty measure, which is posterior complete and proper.

    Thus, the probability space (Ω, ö, P ⁱ) is a universal beliefs space in the sense of Mertens and Zamir (1985) which has consistent probability beliefs.  To separate the influence of information about intrinsic uncertainty versus extrinsic uncertainty, we impose the following assumption.

 

Assumption 4:  (Information Only Over Intrinsic Uncertainty)

    For every trader i, , and F ⁱ are independent σ-algebras with respect to P ⁱ. 

 

    To understand Assumption 4 note that two σ-algebras are independent with respect to P ⁱ if P ⁱ(A₁  A₂) = P ⁱ(A₁)P ⁱ(A₂) for all A₁  and all A₂  F ⁱ (see Loeve 1977; p. 235).  Hence, knowledge of the events in F ⁱ provides no information about the extrinsic events in .  The individual could still receive a private signal about the extrinsic events   in this model, although it would be necessary to introduce new notation to distinguish this information set from F ⁱ.  In this section, we focus on information events about the other market participants.  Assumption 4, therefore, indicates that F ⁱ provides only information about intrinsic uncertainty , i.e., higher-order beliefs.

    We imposed Assumption 4 to highlight a fallacy in many `private information' models, name­ly, that an individual's priors over other traders' higher-order beliefs depend on that individual's priors over the extrinsic events of the economy.  Let's consider a simple two-state example of `rain' or `shine'.  Just because one trader personally feels that `shine' is three times more probable than `rain,'  he has no reason to assume everyone else shares his sunny disposition.  This trader may have bad priors that need updating, or he may represent the market's view.  Either case or any convex combination of the two is possible. 

    Assuming people agree with your own priors is no more rational/ir­rational than assuming they disagree with your priors.  Assump­tion 4 allows traders the flexibility to formulate higher-order beliefs that may not correspond to their own priors over the extrinsic events.  How do arbitrage opportunities relate to higher-order beliefs?  The next proposition answers this question.

 

Proposition 3 (Common Knowledge Higher-Order Beliefs):

Given    for  some n  2.  If    (Y₁ x A₂)

is -measurable, then

,  a.e. .

 

Proof:  First,   Indeed,

  and  are trivial.  Conversely, if ω = , then     and   Thus,   i.e.,   Second, take , then

    =

                       =

                       =  by definition of

                       =  by Assumption 4.

                       =  by the previous claim.  QED.

 

    For clarity, let us write  as `Y<sub>1</sub>  A<sub>2</sub>' and  as `A₁  A₂'.  To understand the meaning of this proposition, consider the event `A<sub>1</sub>  A<sub>2</sub>'.  It exhibits both extrinsic uncertainty  and intrinsic uncertainty   Now we replace A<sub>1</sub> with Y<sub>1</sub>, so that `Y₁  A₂' = `A<sub>2</sub>' reflects only the intrinsic uncertainty.  If the event `Y₁  A₂' is common knowledge, then only uncertainty over the event `A<sub>1</sub>' remains.  The proposition states that if `A₂' is common knowledge then the individual assigns probability to the event `A₁  A₂' based on his first-order probability beliefs (over A₁) alone.

    We say that first-order beliefs  are common knowledge if each  is -measurable.  A direct application of Proposition 3 yields the following corollary that only extrinsic uncertainty matters:⁵

 

Corollary 3.1:  If first-order beliefs  for all i  {1, ..., I} are common knowledge, then  is common knowledge for all i.

 

    Corollary 3.1 indicates that if first-order beliefs are common knowledge, then all n-order beliefs can be shown to be common knowledge by induction.  If first-order beliefs are common knowledge, then no intrinsic uncertainty remains over them.  This certainty implies the second-order beliefs are fixed, known, and common knowledge, which in turn means third-order beliefs are fixed and common knowledge, etc.

    When beliefs are homogeneous or identi­cal, traders agree in the sense that  their first-order beliefs  happen to  be equal.  However,  they need

not know that everyone else knows that everyone knows ... they agree unless these beliefs are common knowledge.  Corollary 3.1 also applies when traders' beliefs are unequal and yet com­mon knowledge, as, for example,  when  the market  knows  that 2/3  of its members are bullish

on future earnings and 1/3 of its members are bears.

    Proposition 2 of Section 6.2 applied to the extrinsic/intrinsic state space gives the major result of this section.

 

Proposition 4 (Information About Prior Beliefs Revealed by Prices):

Given  for some n  2.  If  for some i and  for some   then  is common knowledge.

 

Proof:  If    for some i,  then

                      .

By Proposition 2,  is common knowledge.  QED.

 

    This proposition states that if individual i has private knowledge about higher-order beliefs that he can `bet' upon with traded assets ,  then this information about higher-order beliefs is common knowledge.<sup>7</sup>  The following two corollaries further illustrate this proposition's usefulness.  If some individual i's priors can be `bet' upon through traded assets, then these priors must be common knowledge. 

 

Corollary 4.1:  If  for some  and some i, then  is common knowledge.

 

Corollary 4.2:  If ö, then first-order beliefs are common knowledge.

 

Proof:  This corollary follows since if markets are complete,  for all , all n, and all j.  Hence, by Corol­lary 4.1,  are -measurable for all , all n, all j.  This result implies  is -measurable for all   , for all n, i.e.,  is common knowledge.  QED.

 

    Corollary 4.2 states that complete markets imply first-order beliefs are common knowledge.  Completeness guarantees that traded securities isolate the person specific knowledge in pure bets, i.e.,  where  for all .  Common knowledge prices for  across all events , therefore, will reveal

    If markets are incomplete (i.e., D   (M) =ö), first-order beliefs over extrinsic uncer­tainty need not be common knowledge.  In fact in the extreme case, prices reveal no infor­mation about intrinsic uncertainty at all.  The following class of mod­els, which is often used in financial economics, see Jarrow (1988; Chapters 9, 15), illustrates this case.

    We say that only extrinsic uncertainty governs cash flows if m(ω)  for all m  M.  Cash flows would depend only on extrinsic factors if time 1 cash flows had no capital gains component or capital gains were independent of investor be­liefs.  Under either of these conditions, , because the other market participants effectively have been shut out of determing prices.  Since the events σ(M) govern arbitrage pricing, intrinsic uncertainty has no influence in this (unrealistic) theory.  Indeed, the next proposition restates this point more formally.

 

Proposition 5 (Extrinsicly-Determined Cash Flows):

    If , then given A₁  σ(M), P ⁱ(A₁*F ⁱ)(ω₀) =  with P ⁱ probability one.

 

Proof:  Fix a C  F ⁱ, then

= ,  since   implies ,

                                      =

 

                                      =

 

                                      =  by Assumption 4.  QED.

 

    This proposition shows that in models where intrinsic uncertainty does­n't influence future prices, information about higher-order beliefs has no influence on arbitrage pricing.  Generating probability statements requires only first-order beliefs.  Indeed, D contains no event in the intrinsic uncertainty set .  Hence, prices reveal no infor­mation about higher-order beliefs.  To recap, only first-order beliefs are required to generate all probability statements and these probability statements alone, through Assumption 2, generate the arbitrage pricing theo­ry.

 

 

6.4  Application to Option Pricing Theory

 

This section applies the previous analysis to standard option pricing theory as analyzed in Harrison and Pliska (1981) and Duffie and Huang (1985).  Option pricing theory takes as given a continuous trading economy and a stochastic process for asset price movements.  Although option pricing theory takes place in a dynamic setting, the use of self-financing trading strategies enables one to transform the multiper­iod model into the single period economy.  Under this transforma­tion, the terminal value of all self-financing trading strategies represent the traded assets .  In this setting, to obtain prefer­ence-free valuation formulas (like the Black-Scholes model), one needs to assume that the asset mar­kets are complete in the sense of Green and Jarrow (1987).  This as­sumption is required to invoke the martingale representation theorem, which provides `valuation for­mu­las' for any arbitrarily determined ran­dom variable ().  In this setting, ö because trading strategies yielding $1 un­der any event ö and zero otherwise, are needed to construct all contingent claims.

    There are two ways to interpret these option pricing models.  First, the probabili­ty space underlying the asset price dynamics is based solely on extrinsic uncertainty.  By construction, this broad class of option pricing models precludes speculative trading based on anticipated gains from changing investor beliefs.  Second, let the probability space underlying the asset price process represent both ex­trinsic and intrinsic uncertainty, as in Section 6.3.  However, to develop preference-free valuation formulas, one imposes market completeness  (ö).  Corollary 4.1 implies that first-order beliefs are common knowledge, and then Corollary 3.1 implies that only extrin­sic uncertainty matters.  Here again, by implication rather than by construction, these option pricing models exclude speculation based on higher-order beliefs.  This im­plication reveals a strong assumption, which is not likely to be satisfied in practice, of models attempt­ing to provide valuation formulas for any arbitrarily determined random variable.

    But what if option pricing models yield fair values for, not any arbitrarily determined ran­dom variable but, a proper subset of random variables?  The work of Harrison and Pliska (1981) concerns finite-variance random variables, yet this section's analysis still applies.  Other option pricing models may not rely on market completeness or martingale representation theorems, in which case our results may need to be qualified accordingly.  However, few works, if any, in the option pricing literature extant have even acknowledged the existence of intrinsic uncertainty in financial markets, let alone in­corporate it as a trading motive.

    One of the more interesting questions in contingent claims models with asymmetric infor­ma­tion is how to transform multiperiod models with heterogeneous beliefs into single period mod­els à la Duffie and Huang (1985).  Prices in these intertemporal models with information events do more than just clear markets; prices convey information.  Changes in (heterogeneous) beliefs, speculative consid­erations, and the capital gains motive should drive trading in these models.  Thus prices should not be exoge­nously specified as part of the state of the world.  However, if prices are not specified as part of the state of the world, then traders might rationally speculate over what prices will prevail in future spot markets.

    These complex issues and related ones about mechanisms for revealing prior belief infor­ma­tion can create multiple equilibria for prices, rather than a unique solution. In an intertemporal investment model with no information events, only one round of trading is required to act on the information available to traders.  Without the arrival of new information and changes in beliefs, these multiperiod models can be transformed into single period models with little difficulty.

    Differences in construction between the various models in the contingent claims literature prevent a full discussion of how to transform multiperiod models here.  We can comment on  these issues with­in the context of our model.  The model in Section 6.2 avoids the issue of intertemporal equilibrium with changing investor beliefs by assuming prices -- for whatever unspecified reason -- are determined by forces outside the control of the market traders.  No information events trigger belief updating and intertemporal trading.  The model contains only one trading round by assumption, but it could just as easily depict a transformation of a multiperiod model.

    In Section 6.3, we adopt a framework in which both extrinsic and intrinsic uncertainty can affect prices.  Obviously, prices in this subsequent framework are going to reveal something about both external factors and the market's internal perception of these factors.  We do not address how this more complex economy, where traders may have heterogeneous beliefs about the rest of the market, can be transformed into a single period.

    For the model in Section 6.3, it turns out that if a trader has some private information about the beliefs of others, his purchase of contingent claims that payoff according these higher-order beliefs communicates information to the market.  His purchase makes com­mon knowl­edge what this rational individual believes about higher-order beliefs.  Interestingly, Proposition 4 indicates this information on higher-order beliefs will be revealed if the individual chooses to bet; if he fails to transact, his private information will not be revealed.  Our second find­ing (Proposition 5) states that if you can identify an asset -- even in an intertemporal framework -- that does not depend on in­trinsic uncertainty, then prices will reveal no information about higher-order beliefs for that asset.  In fact, the intrinsic uncertainty is irrelevant.

    Thus, our model sheds some light on particular transactions in which common knowledge beliefs must not be assumed for the absence of arbitrage.  We are aware that not all multiperiod, heterogeneous beliefs models can be transformed into a single period model.  The transformation relies on a deeper understanding of the relationship between multiperiod economies with hetero­ge­neous beliefs and those where beliefs may be homogeneous but not common knowledge.

 

 

6.5  Summary

 

The structure of an asset's cash flows determines the necessity for imposing common knowledge priors in arbitrage pricing models.  This structure (represented by M) is exogenously specified for arbitrage pricing theory.  If market participants determine prices endogenously so that price changes and capital gains depend on higher-order beliefs, then those events that generate poten­tial arbitrage oppor­tunities, (the set D), reveal information about higher-order beliefs and imply necessary conditions on common knowledge priors (Proposition 4).  If markets are com­plete (the set D is large), then priors are common knowledge.  Conversely, if the cash flows and capital gains do not depend on higher-order beliefs, then prices reveal no information about high­er-order beliefs.  Applying these insights, we have shown that standard option pricing theory im­plicitly as­sumes common knowledge priors.  If one relax­es this restriction, then the intrinsic uncer­tainty in the model can form the basis for securities trading apart from uncertainty over extrinsic events.

    Our results offer then both good news and bad news to financial theorists.  Those financial theorists, who suspected arbitrage pricing models implicitly incorporate common knowledge belief assumptions, will be relieved to learn that a large class of these models requires only the weaker homogeneous or common priors assumption.  Problems associated with asymmetric information do not crop up in the canonical arbitrage pricing model, even though individual traders do not know how other market participants assign probabilities.  These findings are the good news.

    The bad news is that if cash flows depend only on extrinsic factors, which seems reasonable for real earnings but not for price movements, then any capital gains component of cash flows must not exist or must be independent of investor beliefs.  It is hardly surprising that information about other market participants then has no impact on arbitrage pricing (Proposition 5).  However, the ease with which this assumption has worked its way into, and now permeates, arbitrage pricing models is surprising.  Most outside reviewers would find this assumption unrealistic and severely limiting the robustness of these models.  We recommend that a more plausible treatment of the capital gains motive, speculative considerations, and the impact of uncertain or changing beliefs feature prominently in new theories on arbitrage pricing.

Notes

 

1. This chapter was co-authored with Robert A. Jarrow, S.C. Johnson Graduate School of Management, Cornell University.

 

2. Even with heterogeneous and common knowledge priors, no trades will execute, because each individual would know with certainty whether someone's will­ingness to trade was motivated by his having superior information.

 

3. Lemma.  If there exists  such that  then Assumption 2(a) implies Assumption 2(b).

 

        Proof:   Suppose m^*  (M⁰) ^j(ω₀) for some trader j.  Let π(m^*) > 0 (without loss of generality).  Consider the .  By Assumption  1(a),  π(m) > 0.  But,  -π(m)m^*/π(m^*) + m  M   where P ^j[-π(m)m^*(ω)/π(m^*) + m(ω)  0 *F ^j](ω₀) = 1  since

        P ^j[m^*(ω) = 0*F ^j](ω₀) = 1 and P ^j[m(ω)  0 *F ^j](ω) = 1.  Now

        P ^j[-π^*m)m^*(ω)/π(­m^*) + m(ω) > 0*F ^j](ω₀) > 0  since

        P ^j[m(ω) > 0*F ^j](ω₀) > 0; and π(-π(m)m^*/π(m^*) + m) = 0.  This result shows π(m)m^*/π(m^*) + m  (M+) ^j(ω₀) with zero price, contra­dicting Assumption 2(a).  Hence, π(m^*) = 0.  QED.

 

4. Common knowledge beliefs are defined later in Section 6.3, immediately prior to Proposition 4.

 

5. Mertens and Zamir (1985), by contrast, define trader i's second-order belief as .  They always define higher-order beliefs over the cross product with the original space.

 

6. Assuming first-order beliefs are common knowledge is identically equivalent to assuming all higher-order beliefs are common knowledge.

 

7. Market traders do not engage in strategic trading to `hide' their beliefs.  This honest revelation of beliefs comes from arbitrage pricing's assumption of price taking behavior, so the trader does not believe his trades influence prices.

References

 

Aumann, R., (1976), `Agreeing to Disagree,' The Annals of Statistics, Vol. 4, pp. 1236-1239.

 

Aumann, R., (1987), `Correlated Equilibrium as an Expression of Bayesian Rationality,' Econometrica, Vol. 55:1, pp. 1-18.

 

Babbs, S. and M. Selby, (1992), `Contingent Claims Analysis,' in The Palgrave Dictionary of Finance, London.

 

Brandenburger, A. and E. Dekel, (1987), `Common Knowledge with Probability 1,' Journal of Mathematical Eco­nomics, Vol. 16, pp. 237-245.

 

Breiman, L., (1968), Probability, Addison-Wesley:  Reading, Massachusetts.

 

Cass, D. and K. Shell, (1983), `Do Sunspots Matter?,' Journal of Political Economy, Vol. 91:2, pp. 193-227.

 

Duffie, D. and C. Huang, (1985), `Implementing Arrow-Debreu Equilibria by Continuous Trading of Few Long-Lived Securities,' Econometrica, Vol. 54, pp. 1161-1184.

 

Geanakoplos, J. and H. Polemarchakis, (1982), `We Can't Disagree Forever,' Journal of Economic Theory, Vol. 28, pp. 192-200.

 

Green, R. and R. Jarrow, (1987), `Spanning and Completeness in Markets with Contingent Claims,' Journal of Econom­ic Theory, Vol. 41:1, pp. 202-210.

 

Guth, M.A.S., (1989), `Intrinsic Uncertainty and Common Knowledge Priors in Financial Economics,' Journal of Finan­cial Research, Vol. 22:4, pp. 269-283.

 

Harrison, J.M. and S. Pliska, (1981), `Martingales and Stochastic Integrals in the Theory of Continuous Trading,' Stochastic Processes and Their Applications, Vol. 11, pp. 215-260.

 

Jarrow, R., (1987), `Beliefs and Arbitrage Pricing,' Economics Letters, Vol. 24:2, pp. 165-169.

 

Jarrow, R., (1988), Finance Theory, Prentice-Hall, Inc.:  Englewood Cliffs, New Jersey.

 

Loeve, M., (1977), Probability Theory 1, 4th Edition, Springer-Verlag Inc.:  New York.

 

Mertens, J.F. and S. Zamir, (1985), `Formulation of Bayesian Analysis for Games with Incomplete Information,' International Journal of Game Theory, Vol. 14:1, pp. 1-29.

 

Milgrom, P., (1981), `An Axiomatic Characterization of Common Knowledge,' Econometrica, Vol. 49:1, pp. 219-222.

 

Milgrom, P. and N. Stokey, (1982), `Information, Trade, and Common Knowledge,' Journal of Economic Theory, Vol. 26, pp. 17-27.

 

Nau, R., (1992), `Joint Coherence in Games of Incomplete Information,' Management Science, Vol. 38:3, pp. 374-387.

 

Nau, R. and K. McCardle, (1990), `Cooperative Behavior in Noncooperative Games,' Journal of Economic Theory, Vol. 50, pp. 424-444.

 

Nielsen, L., (1984), `Common Knowledge, Communication, and Convergence of Beliefs,' Mathematical Social Sciences, Vol. 8, pp. 1-14.

 

Parthasarathy, K., (1967), Probability Measures on Metric Spaces, Academic Press:  New York.

 

Royden, H., (1968), Real Analysis, 2nd Edition, MacMillan Publishing Co.:  New York.

 

Sebenius, J. and J. Geanakoplos, (1983), `Don't Bet On It:  Contingent Agreements with Asymmetric Informa­tion,' Journal of the American Statistical Association, Vol. 78, pp. 424-426.

 

Tan, T.C. and S.R.C. Werlang, (1988), `The Bayesian Foundations of Solution Concepts of Games, Journal of Economic Theory, Vol. 45, pp. 370-391.

 

 

Tirole, J., (1983), `On the Possibility of Speculation Under Rational Expectations,' Econometrica, Vol. 50, pp. 1163-1181.

 

Comment

 

This chapter derives important results concerning the role of common knowledge and prior probabilities in asset pricing models:  it develops a framework for studying the relevance of higher-order beliefs and identifies conditions under which no-arbitrage requires common prior beliefs to be common knowledge.  The agents in this analysis are assumed to be risk neutral, as in much of the asset pricing literature and also in the cited papers by Nau and McCardle (1990) and Nau (1992), which use no-arbitrage arguments to derive the existence of a common prior in game theoretic models.

    In commenting on this chapter, I would merely like to suggest a direction for generalization ! namely, heterogeneous risk preference ! which leads to a rather different perspective on common knowledge and common priors.  If agents are risk averse, it becomes appropriate to assume that probabilities are neither common property nor common knowledge, and the various agreeing-to-disagree and no-trade theorems and game-theoretic solution concepts acquire new interpretations.

    An arbitrage opportunity is basically a publicly observed inconsistency in prices.  Under risk neutrality, competitive security prices reflect the agents' true expectations for payoffs, and in particular their (normalized) prices for Arrow-Debreu securities must be their true state probabilities.  Hence, the true probabilities of risk neutral agents would naturally be common knowledge in a complete, competitive market, and these probabilities would then have to be identical in order to avoid arbitrage.  (Conversely, as the chapter shows, if the market is complete and there are no arbitrage opportunities and priors are commonly held, then the latter must also be common knowledge.)  However, it is hard to tell a convincing story about how exact agreement on prior probabilities would arise in nature, and the model has an uncomfortable knife-edge quality:  small departures from common beliefs would produce unlimited volumes of trade in contingent claims.

    If agents are assumed to be risk averse and heterogeneous in their risk tolerances as well as probabilities, we can tell a more plausible story:  they should trade contingent claims and exploit arbitrage opportunities until they reach a competitive equilibrium in which differences in probabilities are compensated by differences in marginal utilities for money.  In such an equilibrium, the state prices are still commonly known and commonly held, but they are no longer the agents' true probabilities.  Rather, they are products of true probabilities and relative marginal utilities (Drèze 1970).  This product is, of course, the `risk neutral probability distribution' of the representative agent in theories of asset pricing by arbitrage (e.g., Cox and Ross 1976), but it represents the probabilities of real agents only in the aggregate.

    This story explains how, in equilibrium, agent can fail to agree on probabilities:  they never credibly observe each others' true probabilities.  Instead, they observe only prices, in which probabilities and utilities are combined differently for every agent.  This illustrates a phenomenon which has been studied in the literature of statistical decision theory, namely that observations of material gambling or trading behavior may be insufficient to separate an agent's probabilities from her utilities, particularly when her utilities are state-dependent and/or her prior endowment of state-contingent wealth is unknown (Kadane and Winkler 1988; Schervish, Seidenfeld, and Kadane 1990; Nau 1994a).

    Thus, it can be argued that risk neutral probabilities, rather than true probabilities, are the natural objects of common knowledge and common prior assumptions in information economics.  In this manner, a great deal of literature on game theory, decision theory, and finance can be unified under the umbrella of no-arbitrage (Nau and McCardle 1991).  For example, if the common priors in Aumann's (1976) and Sebenius and Geanakoplos' (1983) agreeing-to-disagree models are reinterpreted as risk neutral distributions, the results take on a simple no-arbitrage characterization (Nau 1994b).  The same reinterpretation of the common prior in game theoretic models leads to a refined form of subjective correlated equilibrium, rather than Nash or objective correlated equilibrium, as the natural solution concept, and the outcome-contingent allocation of wealth is guaranteed to be efficient.  (A little-noted property of Nash and objective correlated equilibria is that they can produce inefficient ex ante wealth allocations in strictly competitive games among risk averse players:  the players may be left with incentives to share risks by making side gambles.)  Milgrom and Stokey's (1982) no-trade model already assumes common prior risk neutral probabilities on the entire state space, through the assumptions of an ex ante Pareto optimal wealth distribution over payoff-relevant events and concordant conditional probabilities for informational events.  The conclusion that there will be no subsequent trade contingent on common knowledge events is therefore tautological:  the assumed initial conditions are precisely that all incentives to trade contingent claims have already been exhausted.

    Of course, this more realistic view of common knowledge priors still refers only to a static characterization of equilibrium.  It does not per se address issues of market dynamics such as speculative behavior of the effects of price history on beliefs.  The authors have rightly called attention to the importance and difficulty of incorporating these concerns into future extensions of arbitrage pricing models.  In a world of heterogeneous beliefs and risk preferences, it would of course be even more difficult to model higher-order beliefs explicitly.  As a simpler alternative, we could seek to directly model beliefs about future prices and how these beliefs respond to price trajectories, perhaps through simulation as well as analysis.

 

 

Robert F. Nau

Fuqua School of Business

Duke University

 

References

 

Cox, J. and S. Ross, (1976), `The Valuation of Options for Alternative Stochastic Processes,' Journal of Financial Economics, Vol. 3, pp. 145-166.

 

Drèze, Jacques, (1970), `Market Allocation Under Uncertainty,' European Economic Review, Vol. 2, pp. 133-165.

 

Kadane, J. B., and R. L. Winkler, (1988), `Separating Probability Elicitation from Utilities,' Journal of the American Statistical Association, Vol. 83, pp. 357-363.

 

Nau, Robert F., (1992), `Joint Coherence in Games of Incomplete

    Information,' Management Science, Vol. 38:3, pp. 374-387.

 

Nau, Robert F., (1994a), `Coherent Decision Analysis With Inseparable Probabilities and Utilities,' forthcoming in Journal of Risk and Uncertainty.

 

Nau, Robert F., (1994b), `The Incoherence of Agreeing to Disagree,'   forthcoming in Theory and Decision.

 

Nau, Robert F. and Kevin F. McCardle, (1991), `Arbitrage, Rationality,         and Equilibrium,' Theory and Decision, Vol. 31, pp. 199-240.

 

Schervish, M. J., T. Seidenfeld, and J. B. Kadane, (1990), `State-Dependent Utilities,' Journal of the American Statistical Association, Vol. 85:411, pp. 840-847.

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