In a financial economy with asymmetric information and incomplete markets, we study how agents, having no model of how equilibrium prices are determined, may still refine their information by eliminating sequentially "arbitrage state(s)", namely, the state(s) which would grant the agent an arbitrage, if realizable.Arbitrage, Incomplete markets, Asymmetric information, Information revealed by prices

Bernard Cornet

Lionel D Boisdeffre

English

Research Papers in Economics

Bernard Cornet1  Lionel De Boisdere2
Elimination of Arbitrage States
in Asymmetric Information Models
Abstract In a nancial economy with asymmetric information and incomplete
markets, we study how agents, having no model of how equilibrium prices are
determined, may still rene their information by eliminating sequentially \arbi-
trage state(s)", namely, the state(s) which would grant the agent an arbitrage,
if realizable.
Keywords: Arbitrage  Incomplete markets  Asymmetric information 
Information revealed by prices
JEL Classication Number: D52.
1University of Kansas, USA and Universit e de Paris 1, France.
E-mail: cornet@ku.edu, cornet@univ-paris1.fr
2INSEE, Paris and Universit e de Paris 1, France.
E-mail: lionel.de.boisdere@wanadoo.fr
11 Introduction
In a nancial economy with asymmetric information and incomplete markets,
will agents be able to learn from prices about their partners' private information
when they have no prior \model" or \expectations" of how equilibrium prices are
determined? This paper, which complements an earlier one on arbitrage and
price revelation under asymmetric information [Cornet-De Boisdere (2002)],
answers positively and introduces a sequential and decentralized process of in-
ferences, where agents learn from prices by analyzing arbitrage opportunities on
nancial markets. Renement of information is then achieved in a decentralized
manner by each agent eliminating sequentially her \arbitrage state(s)", namely,
the state(s) which would grant the agent an arbitrage, if realizable. The paper
also shows that a coarse renement of information, which precludes arbitrage,
may always be attained, alternatively, in a decentralized way through prices, or
by a similar sequential elimination process when no price is given.
Section 2 presents the framework and recalls the basic concepts of informa-
tion structures, renements and arbitrage with asymmetric information. Section
3 introduces the \no-arbitrage principle", by which agents who only know their
own characteristics may rene their information in successive steps, by eliminat-
ing \arbitrage state(s)". Section 4 denes a concept of no-arbitrage prices with
asymmetric information and explains how such prices may reveal information.
Section 5 describes the renement process without prices.
2 The model
We consider the basic model of a two time-period economy with private
information and nominal assets: the simplest tractable model which allows us
to present arbitrage. The economy is nite, in the sense that there are nite sets
I, S, and J, respectively, of consumers, states of nature, and nominal assets.
In what follows, the rst period will also be referred to as t = 0 and the
second period, as t = 1. There is an a priori uncertainty at the rst period
(t = 0) about which of the states of nature s 2 S will prevail at the second
period (t = 1). The non-random state at the rst period is denoted by s = 0
and if   S, 0 will stand for f0g [ .
Agents may operate nancial transfers across states in S0 (i.e., across the
two periods and across the states of the second period) by exchanging a nite
number of nominal assets j 2 J, which dene the nancial structure of the
model. The nominal assets are traded at the rst period (t = 0) and yield
payos at the second period (t = 1), contingent on the realization of the state
of nature. We denote by V j
s the payo of asset j 2 J, when state s 2 S is
realized, by V the SJ -return matrix V := (V j
s ), which does not depend upon
2the asset price q 2 RJ, and by V [s] its row vector in state s (for each s 2 S ). A
portfolio z = (zj) 2 RJ species the quantities jzjj (j 2 J) of each asset j (with
the convention that it is bought if zj > 0 and sold if zj < 0) and V z is thus its
nancial return across states at time t = 1.
At the rst period, each agent i 2 I has some private information Si  S
about which states of the world may occur at the next period: either this
information is kept, or it is possible to infer that the true state will be in a
smaller set i  Si. In both cases, agents are assumed to receive no wrong
information signal, that is, the true state always belongs to the set \i2ISi or
\i2Ii, hence assumed to be non-empty. A collection (Si)i2I of subsets of S,
such that \i2ISi 6= ;, is called an (information) structure and a structure (i),
such that i  Si for every i, is called a renement of (Si).
We summarise by [(I;S;J);V;(Si)i2I] the nancial and information charac-
teristics of the economy, which are xed throughout the paper and referred to
as the (nancial and information) structure.
We recall the following standard denitions.
Given the return matrix V and a nonempty set   S, the price q 2 RJ is
said to be a no-arbitrage price for the couple (V;), or the couple (V;) to be
q-arbitrage-free, if one of the following equivalent assertions, (i) or (ii), holds:
(i) there is no portfolio z 2 RJ such that  q  z  0 and V [s]  z  0 for every
s 2 , with at least one strict inequality;
(ii) there exists  = ((s)) 2 R
++, such that q =
P
s2 (s)V [s].
We denote by Q[V;] the set of no-arbitrage prices associated to (V;). By
convention, we shall also say that the couple (V;;) is q-arbitrage-free for every
q 2 RJ, that is, we let Q[V;;] = RJ.
3 Sequential elimination of arbitrage states
Given V , a nonempty subset  of S and q 2 RJ, we now dene the sets:
A(V;;q) = fe s 2  : 9z 2 RJ; q  z  0;V [e s]  z > 0;V [s]  z  0;8s 2 g
S1(V;;q) := nA(V;;q),
with the convention that A(V;;;q) := ; and S1(V;;;q) := ;, for all q 2 RJ.
Given V and an information structure (Si), the set A(V;Si;q) consists in
the so-called \arbitrage state(s)" (of the second period t = 1), that is, states
which grant agent i an arbitrage when his beliefs are represented by the set Si.
The rst stage of elimination of arbitrage states leads to the set S1(V;Si;q).
3However, the rened set S1(V;Si;q) may display new arbitrage states, that
is, there may exist states s 2 A(V;S1(V;Si;q);q) such that s = 2 A(V;Si;q).
Thus, the elimination process may need to carry on. It is dened sequentially,
hereafter, in two slightly dierent ways, which will be shown to be equivalent.
Given V , an agent i with (nonempty) private information set Si  S, we
dene, by induction on k 2 N and for every q 2 RJ, the sets Sk
i (q) as follows:
S0
i (q) = Si, and for k  1
S
k+1
i (q) = S1(V;Sk
i (q);q) := Sk
i (q)nA(V;Sk
i (q);q).
Similarly, we dene by induction on k 2 N, the sets S0k
i (q) as follows:
S00
i (q) = Si, and for k  1
S
0k+1
i (q) =

S0k
i (q); if A(V;S0k
i (q);q) = ;;
S0k
i (q)nfskg for some sk 2 A(V;S0k
i (q);q) if A(V;S0k
i (q);q) 6= ;:
The two sequences of nite sets (Sk
i (q))k2N and (S0k
i (q))k2N are decreasing,
that is, S
k+1
i (q)  Sk
i (q) and S
0k+1
i (q)  S0k
i (q) for every k. Hence, they must
be constant for k large enough and we let:
S
i (q) := \k2N Sk
i (q) (in fact equal to Sk

i (q) for some k large enough);
S
i (q) := \k2N S0k
i (q) (in fact equal to S0k

i (q) for some k large enough).
The following result shows that for every price q, the successive elimination
of arbitrage states leads agents to infer the same information sets, whether they
rule out the states of arbitrage one by one (and then, whatever the chronology
of inferences), or in bundles.
Theorem 1 Let [V;(Si)] be a given structure and q 2 RJ. Then, for every
i 2 I, S
i (q) = S
i (q), and this set is the (possibly empty) greatest subset  of
Si such that A(V;;q) = ;.
The successive elimination of arbitrage states, may be interpreted as a ra-
tional behavior. This behavior, referred to as the \no-arbitrage principle", does
not require any knowledge of the ex ante characteristics of the economy (en-
dowments and preferences of the other consumers) or of a relationship between
prices and the private information of other agents. This is the main dierence
between our model of asymmetric information and that of rational expectations.
We prepare the proof of Theorem 1 with two claims.
Claim 1 Given q 2 RJ and 1  2  S, then, S1(V;1;q)  S1(V;2;q):
4Proof of Claim 1 By contraposition. Suppose that there exists some ~ s 2
S1(V;1;q)  1  2, such that ~ s = 2 S1(V;2;q). Then, ~ s 2 A(V;2;q), that
is, there exists z 2 RJ such that  q  z  0; V [s]  z  0 for every s 2 2
and V [~ s]  z > 0. Since ~ s 2 1  2, we deduce that ~ s 2 A(V;1;q), which
contradicts the fact that ~ s 2 S1(V;1;q). 
Claim 2 For every   Si, such that A(V;;q) = ;, and every k 2 N, the
following inclusions hold:   Sk
i (q)  S0k
i (q).
Proof of Claim 2 By induction on k. The above inclusions are true for k = 0,
since S0
i (q) = S00
i (q) := Si. Assume now that the inclusions hold up to rank k.
Let   Si be such that A(V;;q) = ;. Then, from Claim 1:
S1(V;;q)  S1(V;Sk
i (q);q)  S1(V;S0k
i (q);q).
Since A(V;;q) = ;, we deduce that S1(V;;q) =  and, from the deni-
tions of Sk
i (q) and S0k
i (q), that S
k+1
i (q) := S1(V;Sk
i (q);q) and S1(V;S0k
i (q);q) :=
S0k
i (q)nA(V;S0k
i (q);q)  S
0k+1
i (q). Consequently,   S
k+1
i (q)  S
0k+1
i (q). 
Proof of Theorem 1 Let i 2 I and   Si be such that A(V;;q) = ;.
Taking k large enough yields, from Claim 2:   S
i (q)  S
i (q). We deduce
from the denitions of S
i (q) and S
i (q) that A(V;S
i (q);q) = A(V;S
i (q);q) =
; . These relations imply, rst, that S
i (q) = S
i (q) (take  = S
i (q) above)
and, second, that S
i (q) = S
i (q) is the greatest element (for the inclusion)
among the subsets  of Si, such that A(V;;q) = ;. 
4 Sequential procedures and price revelation
Given the structure [V;(i)], Theorem 1 shows the existence of a unique set,
denoted ~ Si(q), which is the greatest subset  of Si such that A(V;;q) = ;.
This section will compare the sets ~ Si(q) and Si(q), the information set revealed
to agent i by the price q 2 RJ, which was introduced in Cornet-De Boisdere
(2002). We recall that Si(q) is the unique (possibly empty) subset of Si, which
is the greatest subset of Si that is q-arbitrage-free. It is immediate to see that
Si(q)  ~ Si(q) and that both sets may be empty.
For arbitrary prices, the families (Si(q)), (~ Si(q)) may not be information
structures, that is, one may have \iSi(q) = ; or \i~ Si(q) = ;. To get informa-
tion structures, we now need to consider no-arbitrage prices, as in Cornet-De
Boisdere (2002), and we recall the following denitions. Given the structure
[V;(i)], the price q 2 RJ is said to be a no-arbitrage price for agent i if it is
a no-arbitrage price for the couple (V;i), and a common no-arbitrage price of
the structure [V;(i)] if it is a no-arbitrage price for every agent i 2 I, that
5is, if it belongs to the set Qc[V;(i)] = \iQ[V;i]. Alternatively, the struc-
ture [V;(i)] is said to be arbitrage-free (resp. q-arbitrage-free) if it admits a
common no-arbitrage price, that is, if Qc[V;(i)] 6= ; (resp. q 2 Qc[V;(i)]).
The price q 2 RJ is said to be a no-arbitrage price of [V;(Si)] if q is a com-
mon no-arbitrage price for some information structure (i) rening (Si). We
denote by Q[V;(Si)] the set of no-arbitrage prices of the structure [V;(Si)].
Theorem 2 Let [V;(Si)] be a given structure. Then, for every q 2 Q[V;(Si)]
and every i 2 I, Si(q) = ~ Si(q) = S
i (q) = S
i (q).
The proof of Theorem 2 is a direct consequence of Theorem 1 and the follow-
ing proposition, which is also of interest for itself.
Proposition 1 Given a structure [V;(Si)] and a price q 2 RJ, the following
three conditions are equivalent:
(i) q is a no-arbitrage price, that is, q 2 Q[V;(Si)];
(ii) (Si(q)) is an information structure, (i.e., \iSi(q) 6= ;);
(iii) (~ Si(q)) is an information structure (i.e., \i~ Si(q) 6= ;), and ~ Si(q) is q-
arbitrage-free for every agent i at the rst period (t = 0), in the sense that there
is no portfolio z 2 RJ such that  q  z > 0, V [s]  z  0 for every s 2 ~ Si(q).
Moreover, if one of the above conditions holds, Si(q) = ~ Si(q) for every i.
Proof The equivalence [(i) , (ii)] is proved in Cornet-De Boisdere (2002).
[(ii) ) (iii)] From (ii) we rst deduce that ; 6= \iSi(q)  \i~ Si(q). Since
Si(q) 6= ;, from the denition, q 2 Q[V; Si(q)]. This implies q 2 Q[V; ~ Si(q)],
since Si(q)  ~ Si(q). Hence ~ Si(q) is q-arbitrage-free at t = 0.
[(iii) ) (ii)] It is clearly sucient to show that Si(q) = ~ Si(q) for every
i. Let i 2 I be given. By denition of ~ Si(q), for every ~ s 2 ~ Si(q), there is no
z 2 RJ such that  q  z  0;V [s]  z  0; for every s 2 ~ Si(q), and V [~ s]  z > 0.
That condition, together with the fact that ~ Sj(q) is q-arbitrage free for every
agent j 2 I at the rst period, implies that ~ Si(q) is q-arbitrage-free for agent i.
Consequently, ~ Si(q)  Si(q), from the denition of Si(q), whereas the inclusion
Si(q)  ~ Si(q) is immediate. Hence, Si(q) = ~ Si(q). This completes the proof. 
We point out that the above assertions (ii) and (iii) of Proposition 1 may
not be equivalent if we do not assume in (iii) that ~ Si(q) is q-arbitrage free for
every agent i at the rst period, as shown by the following counter-example.
Example. Consider two agents, ve states (S = f1;2;3;4;5g), private infor-
mation sets S1 = f1;2;3;5g, S2 = f1;4;5g, and the payo matrix:
V =
0
B
B
B B
@
 1 0 0
1 1 0
0 0 1
0 1 0
0 0 0
1
C
C
C C
A
6Then, for q = (1;1;0), S1(q) = ~ S1(q) = f2;5g, S2(q) = ;, while ~ S2(q) =
f4;5g. Thus, ; = S1(q) \ S2(q)  ~ S1(q) \ ~ S2(q) = f5g. 
5 Reaching the coarsest arbitrage-free renement
We denote by S the set of arbitrage-free renements of (Si). Given the
structure [V;(Si)]; we recall that there exists a unique coarsest element in S,
denoted by (Si[V;(Si)]) or simply (Si) when no confusion is possible, that is
(Si) 2 S and (i) 2 S implies i  Si, for every i. We refer to [1] for this
denition and, also, below for an alternative proof of the existence of (Si).
The purpose of this section is to provide an alternative process of inferences,
which does not rely on prices and leads agents to infer the coarsest renement
(Si) of (Si). Thus, given the structure [V;(Si)] and a renement (i) of (Si),
we let, for each i 2 I:
Ai(V;(i)) := fe si 2 i : 9(zj) 2 (RJ)I, such that V [e si]zi > 0;
P
j2I zj = 0
and V [sj]  zj  0;8j 2 I;8sj 2 jg;
S1
i(V;(i)) := inAi(V;(i)).
Then, we dene, similarly as in the previous section, two alternative inference
processes, by induction on the integer k 2 N. Namely, for each i 2 I, we let:
S0
i := Si, and, for every k  0, S
k+1
i := S1
i(V;(Sk
i ));
similarly, we let:
S
00
i := Si, and, for every k  0,
S
0k+1
i :=

S0k
i , if Ai(V;(S0k
i )) = ;, and, otherwise,
S0k
i nfsk
i g, for some arbitrary sk
i 2 Ai(V;(S0k
i )) 6= ;

Again, both sequences (Sk
i )k and (S0k
i )k are decreasing in the nite set Si,
hence, constant for k large enough. We denote their limits by (S
i ) and (S
i ).
Theorem 3 Let [V;(Si)] be a given structure. Then, for every i 2 I, S
i =
S
i and (S
i ) is the coarsest arbitrage-free renement of (Si).
The proof of Theorem 3 is similar to that of Theorem 1 and left to readers.
Again, Theorem 3 shows that the chronology of all agents' inferences will not
change the outcome. Whatever the individual paths of inferences, they always
lead to the same limit, namely the coarsest arbitrage-free renement (Si).
Reference
[1] Cornet, B., De Boisdere, L.: Arbitrage and price revelation with asym-
metric information and incomplete markets. Journal of Mathematical Eco-
nomics 38, 4, 393-410 (2002)
7

Arbitrage and price revelation with asymmetric information and incomplete markets.

Elimination of Arbitrage States in Asymmetric Information Models

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Elimination of Arbitrage States in Asymmetric Information Models

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