Financial markets with asymmetric information: An expository review of seminal models

FINANCIAL MARKETS WITH ASYMMETRIC INFORMATION: AN EXPOSITORY REVIEW OF SEMINAL MODELS

KAVOUS ARDALAN

ABSTRACT The purpose of this paper is to make an expository review of the seminal models of the rational expectations equilibrium models in the finance literature. The staggering explosion of the complex research in this area of the finance literature has come to existence when the perfect market assumption of homogeneous information is relaxed and prices play the role of aggregating and transmitting information in the financial market. This expository review, therefore, brings out: (1) the seminal models (i.e., the models with a contribution to theory, rather than application of an existing theory); (2) the essential structure and theoretical contribution of each model in relation to the other models; and (3) the informational role of prices (i.e., fully-revealing versus partially-revealing nature of the models). This paper systematically brings out the seminal rational expectations equilibrium models in the following three categories. The first category being comprised of the pioneer models, Grossman (1976) and Grossman and Stiglitz (1980). The second category being composed of the price-taking competitive models, a direction initiated by Hellwig (1980). The third category including the non-price-taking non-competitive models, a direction initiated by Kyle (1984, 1985).

Kavous Ardalan, School of Commerce and Administration, Laurentian University, Sudbury, Ontario, Canada P3E 2C6; Tel: (705) 675-1151 ext 2156; Fax: (705) 673-6518; E-Mail: KArdalan @Nickel.Laurentian.CA. Direct all correspondence to:

International Review of Economics and Finance, 7(1): 23-51 Copyright © 1998 by JA! Press Inc. ISSN: 1059-0560 All rights of reproduction in any form reserved. 23

KAVOUS ARDALAN

24

I.

INTRODUCTION

The purpose of this paper is to make an expository review of the seminal models of the rational expectations equilibrium (REE) models in the finance literature. 1 The staggering explosion of research in this area of the finance literature has come to existence when the perfect market assumption of homogeneous information is relaxed and prices play the role of aggregating and transmitting information in the financial market. This expository review, therefore, brings out: (1) the seminal models (i.e., the models with a contribution to theory, rather than application of an existing theory); (2) the essential structure and theoretical contribution of each model in relation to other models; and (3) the informational role of prices (i.e., fully-revealing versus partially-revealing nature of prices). This paper systematically brings out the seminal REE models in the following three categories. The first category being comprised of the pioneer models, Grossman (1976) and Grossman and Stiglitz (1980). The second category being composed of the price-taking competitive models, a direction initiated by Hellwig (1980). 2 The third category includin~ the non-price-taking non-competitive models, a direction initiated by Kyle (1984, 1985)." We aim at seminal models,4 and concentrate on the informational role of prices in these models. 5 Two themes underlie our discussion of REE models. The first theme emphasizes the information asymmetry among traders, where prices either fully reveal or partially reveal the informed traders' signals. In the fully-revealing models, the information asymmetry exists before the equilibrium is reached, but vanishes at the equilibrium. In the partially-revealing models, the information asymmetry exists both before the equilibrium is reached and at the equilibrium. The second theme emphasizes the degree of competition among traders. In the REE models, markets are either competitive or non-competitive. In the competitive market models, there exists an infinite number of price-taking informed traders, whereas in the non-competitive market models there exists a finite number of non-price-taking informed traders.

A.

The Seminal Pioneer Models

This section consists of two parts. First, we briefly discuss the fully-revealing REE model of Grossman (1976), including the chief criticism of this model. Second, we describe how this model was extended by Grossman and Stiglitz (1980) to a partiallyrevealing, noisy rational expectations equilibrium (NREE) model. Grossman (1976) considers a financial market with two periods and n traders (indexed by i, i = 1..... n). Trader i has initial wealth Woi and can purchase two assets: a risk-free asset and a risky asset. At time 0 trader i maximizes the expected utility (E{Ui[.] }) of her end-of-the-period wealth (Wli), conditional on her information set li:

maxE{ Ui[Wlit [i] } = maxEI Ui[ (1 + r)XFi + Pl Xi[li ] } XFi XFi subject to the budget constraint:

(1)

Markets with Asymmetric Information

25

Woi = XFi + PoXi where XFi is the value of risk-free assets purchased in period 0 by trader i, X i is the number of units of the risky asset trader i purchased in period 0, r > 0 is the exogenous risk-free rate of return, P0 is the current price of the risky asset, and PI is the (random) exogenous payoff per unit on the risky asset in period 1. Grossman (1976) assumes that trader i's information set includes the signal Yi:

Yi = PI + £i where e i - N(O,o 2) and the ei's, i = 1..... n, are i.i.d. Also, trader i is assumed to possess a negative exponential utility function: ~

Ui(l/Vli ) = -e -aiWli

ai> 0

where a i is her coefficient of absolute risk aversion. Solving the above maximization problem, Grossman demonstrates that trader i's demand for the risky asset is:

x di = E [ P I [ I i ] - ( 1 + r)P 0 aiVar[P 1 [l i]

(2)

d

Observe that X 7 increases with trader i's expectation of the risky asset's payoff, and decreases with the asset's conditional variance, her degree of risk aversion, and the risk-flee return. P0 is an equilibrium price if: n

i=1 where X" is the total stock of the risky asset (note that it is non-stochastic and public knowledge). Observe that each trader i's demand for the risky asset depends upon her information set, which includes her signal of its payoff, Yi. Consequently, the equilibrium price will depend on the vector y - (Yl,Y2,..",Yn); i.e., P0 = P0(Y), where, in general, different signal vectors lead to different equilibrium prices. There are many different possible price functions, P0(Y)- For a particular P0(Y) to be an equilibrium Grossman (1976) requires that: n

~, X~i[P~(y),yi] = X

Vy.

i=l

That is, the total demand for the risky asset must equal the total supply for all possible signal vectors, y. Grossman then shows that:

X~i[Po(y),yi]

= E[P 1 [yi,Po(y)l- (1 + r)P;(y) aiVar[ P1 ]yi,p~(y) ]

(3)

26

KAVOUS ARDALAN

Note that trader i's information set, I i, consists of both her private signal, Yi, and the observed equilibrium price, Po(Y); i.e., I i = [yi,Po(y)]. Consequently, equations (2) and (3) are identical in equilibrium. Next, Grossman (1976) demonstrates that Po(Y) is an equilibrium if:

P~(Y) = ~0 + CZlY

(4)

where:

n Yi

5'=-En i=1 pa Y~7= 1 - O2~" ~o ~

(1 + no2)(1 + r)ZT= no

~1----

la-].1

2

(1 + no2)(1 + r)

PI - N(PI'O2) Observe that ~ is the mean signal value, and that the equilibrium price depends on trader i's signal, Yi, only through Y. The following remark establishes an important property of model's equilibrium. Remark 1. Each trader i may choose to ignore her own signal Yi because: (a) each trader i observes Po(Y), from which she can obtain ~ ; and (b) ~ and the combination of ~ and Yi are equivalent signals of P1Proof. See Appendix 1. [] The above paragraphs describe Grossman's (1976) fully-revealing REE model. 6 Unfortunately, this model possesses a troublesome feature: There may not exist an incentive for individual traders to acquire costly information and consequently, if information signals are costly, an equilibrium may not exist. This occurs because, when the price is fully revealing (i.e., when P*o(Y) reveals Y, and y is as precise a signal o f P 1 as is the combination of and Yi), each trader who purchases a costly signal realizes that her information set is no more precise than that of traders who chose not to purchase a signal. Since all traders realize that the marginal benefit of the costly signal is zero, having any number of traders choose to purchase the signal is inconsistent with equilibrium (i.e., implies non-optimizing behaviour). However, the situation in which all traders choose not to purchase the costly signal is also inconsistent with equilibrium. This occurs because, when all traders choose not to purchase the signal, the equilibrium price will not reveal any information and the marginal benefit of the signal may exceed its marginal cost. Thus, each uninformed trader would decide to purchase the signal. This dual tendency on the part of each trader i, prevents an equilibrium from being attained, and is referred to as the "Paradox of Price Efficient Markets." To solve the "Paradox of Price Efficient Markets," Grossman and Stiglitz (1980) develop a NREE model in which the price is not fully revealing. In their model noise occurs because the per capita supply of the risky asset, x, is random. 7 This could occur because of liquidity trading.

27

Markets with A s y m m e t r i c Information

Grossman and Stiglitz (1980) has the same structure as Grossman (1976) with the following three exceptions. First, all traders now have the same degree of absolute risk aversion, a i = a, Vi. Second, some traders acquire a costly information signal, Yi (where now Yi = Y, V i), and become "informed" while the rest, the "uninformed," do not acquire a signal. Third, as mentioned above, the per capita supply of the risky asset is random. Each trader i again, at time 0, maximizes the expected utility of her end-of-the-period wealth, conditional on her information set. Grossman and Stiglitz (1980) show that the demands for the risky asset of the informed, X~/ and uninformed traders, ~ u , are: ~g

Xff/[y] = y - (1 + r)Po(Y,X ) 2 ac e

xaU •

[po(Y,X)] =

EtPlleo(Y,X)]-(1

+ r)Po(Y,X)

(5a)

(5b)

a Var[PllPo(y,x)]

Observe that (5a) is very similar to (2), and that (5b) is very similar to (3). Grossman and Stiglitz (1980) allows each trader to decide whether or not to acquire, at a given cost, one unit of information (i.e., one signal). As more traders choose to become informed, the marginal value of information declines (since the price reveals progressively more information to the uninformed). In equilibrium, the proportion of traders who choose to become informed is such that the marginal benefit of the information signal equals its marginal cost. Grossman and Stiglitz (1980) let ~, be the fraction of traders who choose to become informed. They define an equilibrium price function, Po(Y,X;~,), such that, for all (y,x): d

,

~,x I [Po(Y,X;3,),y] + (1 -)Qxd[p~(y,x;~,)] = x

(6)

where xtd and x d denote the informed and uninformed traders' per capita demand for the risky asset, and x denotes its per capita supply. If y, e, and x are mutually independent, and normally distributed, then the equilibrium price is: Po (y,x;)Q = fll + fl2w(Y ,x;)')

where: 2 a~e w(y,x;)~ ) = y - ----~-x

~r~>O

w(y,x;~,) = x

~rk=0

and fll and f12 are real numbers which may depend on ~,.8

(7)

KAVOUS ARDALAN

28

The variable w is equal to the signal, y, less a term which increases with a (the coefficient 2 of ARA), a e (the variance of the signal's error term), and x (how much the true per capita supply of the risky asset exceeds its expectation; this is the model's noise term), and decreases with ~,. If 1 > k > 0, the price system conveys information about y, but it does so imperfectly. The uninformed traders attempt to infer y from P0, but the noise term, x, prevents P0 from fully revealing y. If ~, = 0, the price contains no information about y (since no-one purchases the signal). If k = 1, all traders are equally informed (since all traders purchase the signal). The following remark establishes an important property of the model's equilibrium. Remark 2. The informed traders' information set is superior to the uninformed traders' information set. Proof. See Appendix~2. [] The existence of the unpredictable element, x, in the price masks some of the informed traders' information and prevents full revelation of their information to the uninformed. Grossman and Stiglitz (1980) assume that neither informed nor uninformed traders observe x. Uninformed traders cannot learn y by inverting P0 (Y, x; k) because they cannot distinguish variations in the price due to changes in the informed traders' information, y, from variations in the price due to changes in aggregate supply, x (this is evident from (5b) and (7), and also from noting that x and y are independently distributed). However, the price reveals some of the informed traders' information to the uninformed traders. Consequently, information asymmetry persists even in equilibrium. In this section we made an expository review of Grossman's (1976) fully-revealing REE model and Grossman and Stiglitz's (1980) partially-revealing NREE model. In the fully-revealing model, the information asymmetry exists before the equilibrium is reached, but vanishes at the equilibrium. In the partially-revealing model, information asymmetry exists both before the equilibrium is reached and at the equilibrium.

B. SeminalCompetitive Models Hellwig (1980) argues that Grossman's (1976) informed traders are "slightly schizophrenic," since "on the one hand agents are aware of the covariance between the price and their own signals and actions. On the other hand they behave as price takers." To avoid this inconsistency, Hellwig considers a "large" market, in which individual traders have no influence on the price. Hellwig's (1980) model is similar to Grossman's (1976), with two exceptions. First, in Hellwig traders vary in the precision of their information. Second, and more importantly, he considers the limit of the model's equilibrium as the number of traders grows to infinity (thus obtaining a competitive equilibrium). When there are n traders, Hellwig (1980) assumes the random vector (PI, X, e 1..... e n) has a multivariate normal distribution, with mean (P1 ,X,0 ..... 0) and variance: 9

2 ( Var[ P 1], Var[ X],~21 ..... ae. ).

Markets with AsymmetricInformation

29

As in Grossman (1976) each trader i maximizes the expected utility of her end-of-the-period wealth, to obtain her demand for the risky asset. Hellwig (1980) proves the existence of a linear rational expectations equilibrium: n

PO = rtO+ ~

~iYi - TX

(8)

i=1 where: n

7C° -

1 1 -m VaTP1]i~l.~ii_~t(Eo_TX)i2.,lai[~n_V.,:i

n

~ -- 7Ci

2 2 + ? 2 V a r [ X ] - 7c2i ~Ei2 ] lrCkO~k

=

~n

"~ ~i-

2 2

2

2

k= l~kt~e.k + Y V a r [ X ] - ~ilgl3e,

2

n

22

2

aic3e i Ek = lrCkt~ek + T V a r [ X ]

22

i = 1,...,n

- rc i 6~i

n ~=---- ~ i

i=1

1

2

n , V a r [ P l ] + ¢jei

( ~ - rr/) 2 - ( n - ~ i )

2 +~ n 2 2 + T2Var[X] i = l a i V a r [ P 1] + t~ei i= l a i [ E k = lXkZek E

2 2]

- ~ i (~Ei

2 _> (~Ej 2 then ~i < xj, where gi represents the sensitivity of the Observe that, if a i > aj or (~Ei equilibrium price to the signal Yi. That is, in the equilibrium price, the weight placed on a trader's signal decreases with her degree of risk aversion and increases with the precision of her information. Note that, as in Grossman and Stiglitz (1980), the random supply of the risky asset, X, plays an important role in both determining the equilibrium price and preventing the equilibrium price from being fully revealing. The sensitivity of P~ to X is represented by 3(, which increases with Vat[X]. As Var[X] (and consequently T) increases, the equilibrium price becomes less informative; i.e., it aggregates and reveals less of the traders' information. In the limit as Var[X] ---> oo, the equilibrium price reveals no information and each trader considers only her own signal. In the other limit, as Var[X] ---> O, the equilibrium price becomes fully revealing (as in Grossman, 1976). In order to remove the "schizophrenia" feature of Grossman's (1976) price taking, finite market, Hellwig (1980) considers the limit of the model's equilibrium as the number of traders grows to infinity. He obtains the competitive equilibrium: PO = gO + ~*Pl - T*x

(9)

KAVOUS ARDALAN

30 where:

el Var[x]A

+ Var[P1]YcAB

nO= Var[x]A + V a r [ P 1] V a r [ x ] B + V a r [ P 1]AB 2 ' re* =

V a r [ P 1 ] V a r [ x ] B + Var[P1]AB2 Var[x]A + V a r [ P 1] V a r [ x ] B + Var[P 1]AB 2 ' V a r [ P 1] Var[x] + V a r [ P 1l A B

~* =

Var[x]A + V a r [ P 1] Var[x] B + Var[P 1] A B 2 ' 1 a - I ~dkt ,

airJe i x ~ per capita supply.

Observe that as the number of traders approaches infinity the weight gi (the sensitivity of P0 to Yi), in equation (8), converges to zero (i.e., the effect of individual, signals on P0 becomes negligible). Consequently, in a "large" market, individual traders do not affect the price. Thus, the price reflects only what is common to many signals, and the competitive price-taking assumption is no longer "slightly schizophrenic" ( as Hellwig (1980) argued it was in finite-trader markets, in which each trader's private signal realization affects the price). The following remark establishes an important property of Hellwig's (1980) model's equilibrium. Remark 3. The information set which consists of both the signal (Yi) and the equilibrium price P0' is superior to the information set which consists of only P0' Proof. See Appendix 3. [] Hellwig's (1980) competitive model, with an infinite number of informed, price-taking traders, has been the basis for a number of subsequent extensions and applications, several of which are discussed below. Verrecchia (1982) employs a model that is similar to Hellwig's (1980) to examine the information acquisition decision. Verrecchia, like Hellwig, considers a market with an infinite number of traders. However, unlike Hellwig, Verrecchia assumes each trader i can acquire a signal, and choose the precision of the signal according to a given continuous cost function: (c = c ( ~ ) , where c',c" > 0). Verrecchia also makes the simplifying assumption that $ = 0. Verrecchia's (1982) linear rational expectations equilibrium is identical to Hellwig's (1980) with one exception. Since $ = 0, in (9) above the second term in the numerator of n 0 vanishes. Verrecchia's (1982) two most important results.are that: (1) the informativeness of the price generally decreases with the cost of acquiring information; and (2) more risk-averse traders will choose less precise signals. /

Markets with Asymmetric Information

31

The following remark establishes an important property of Verrecchia's (1982) model's equilibrium. Remark 4. The information set which consists of both the signal (Yi) and the equilibrium price (P0), is superior to the information set which consists of only (P0)" Proof. See Appendix 4. [] Admati (1985) extends Hellwig (1980) to the case of many, n, risky assets. She assumes that the random vectors P1, x, and gi have an independent joint normal distribution. But, she makes no special assumptions concerning the variance-covariance matrices VAR[P1], VAR[x], and y2 10 Thus, her model admits general correlation patterns across assets payg i "

offs, supplies, and the error terms affecting each private signal. She considers a "large" market consisting of a continuum of traders indexed by i e [0,1]. Admati (1985) proves the existence of a linear rational expectations equilibrium: P0 = n0 + re*P1 + T*x

(13)

where:

It,0. = a;(0vaRl

,l

+ aE

VAR-I[x]E

+ Z-~i2)

1

x(VAR-I[p1]P1 + Y_.-e2VAR-I[x]2)

1

g~- I; aidi ~,-2 1 -2 . ei ----I; ai~ei dt where r is the risk-free rate, as before. The last two identities define two parameters of the model. The first one defines the average risk tolerance; and the second one defines a weighted average of the precision matrices, where the weights are the risk tolerance coefficients. Since Admati (1985) places no restrictions on the three variance-covariance matrices (asset payoffs, asset supplies, and the signal's error term), her model raises the possibility of numerous complex, and counterintuitive results occurring. For example, information about one asset (i.e., observing a signal of its value or observing its price) generally leads investors to update their beliefs regarding all n - 1 other risky assets. Consequently, one

KAVOUS ARDALAN

32

may find an asset whose equilibrium price is an increasing function of its own supply or a decreasing function of its own payoff. Further, one may observe an asset whose demand is an increasing function of its own price (i.e., a Giffen good). The following remark establishes an important property of Admati's (1985) model's equilibrium. Remark 5. In equilibrium, traders hold heterogeneous beliefs and P0 is not fully revealing. Proof. See Appendix 5. [] Grundy and McNichols (1989) consider the single-period model of Hellwig (1980) and examine the effect of allowing a second round of trade. They assume that the equilibrium price, P0' is a linear function of the average of traders' information, y, and the unobservable per capita supply, x. They argue that if a linear rational expectations equilibrium exists, the pricing equations at times 0 and 1 are: *

0

1_

2

P0 = t~0 + ct0Y + °~0x * 1_ 2 P1 = ~10 + ~ l Y + ~1 x

where P1 is the second-round price. Note that the realization of x is assumed fixed and unaffected by the trading process. Whenever: 1

1

@0 ~: @1 2 2 t~0 @1

the sequence of prices fully reveals y : Linear independence of the two pricing relations allows traders to solve the two equations for the two unknowns, ~ and x. Their analysis is based on the notion that a sequence of prices can reveal more about the average private signal and can thereby induce additional trade to attain further risk-sharing. In such equilibria, prices change even though it appears that no new external information has arrived; rather, the price changes reflect the further revelation of existing private information. Furthermore, rational traders' demands depend on the price history as well as the contemporaneous price. Grundy and McNichols (1989) show that a second round of trade leads to one of two types of equilibria. In one type of equilibrium, the average private signal will not be revealed by the price sequence, and in the second round the price remains unchanged and no trade takes place (which is the standard results of the previous models). In the second type of equilibrium, the average private signal will be revealed by the price change. Therefore, the equilibrium in the second period will be fully revealing. In the partially-revealing equilibrium, traders conjecture that the average private signal, Y, will not be revealed by the price sequence; in equilibrium, this conjecture is fulfilled, the price is unchanged, and no trade takes place in the second round. In the fully-revealing equilibrium, if traders conjecture that the average private signal will be revealed by the price change, then in equilibrium this conjecture is fulfilled. Brown and Jennings (1989) extend the model of Hellwig (1980) to two periods. They show that technical analysis has value in every "myopic"-investor economy, and find that

Markets with Asymmetric Information

33

the second-period price is dominated as an informative source by a weighted average of the fast- and second-period prices. Investors use the historical price in determining time-2 demands because the current price does not reveal all publicly available information provided by price histories, that is, investors use technical analysis to their benefits. Their results are consistent with those of Grundy and McNichols (1989). However, whereas in Grundy and McNichols (1989) traders receive private information only in the first period, traders in Brown and Jennings (1989) receive uncorrelated signals in both the first and second rounds. More importantly, in Grundy and McNichols (1989) the supply is assumed fixed between the two periods, whereas Brown and Jennings (1989) focus exclusively on the case in which supply variations occur in each period and obtain only the partially-revealing equilibrium price. Naik (1993) employs a continuous time market consisting of a large number of heterogeneous agents who are endowed with private signals which provide some information about the risky asset payoff. He proves the existence of a rational expectations equilibrium in which at every instant the agents use the information contained in the market prices without rendering their private information redundant. While showing that the market price is not affected by the noise in an agent' s private signal, Naik demonstrates that the intuition behind HeUwig's single period results holds in a multi-period setting as well. Brennan and Cao (1996) address the welfare gain of the opening of new markets, either by the more frequent opening of existing markets or by the opening of new markets in derivative securities. In the context of a noisy rational expectations model of the type of Hellwig (1980), they assume that knowledge about final asset payoffs emerges gradually over time and then consider the effect of allowing investors to trade more frequently. They show that, for a given number of market sessions, welfare is maximized when information is released in such a way that the variance of price change between successive market sessions is constant over time. Brennan and Cao demonstrate that, as the number of market openings increases without limit, the limiting payoff allocation is Pareto-efficient, subject to a smoothness condition on the public information flow. In this section we made an expository review of the seminal competitive models, a direction initiated by Hellwig (1980). We concentrated on the informational role of prices and emphasized the partially-revealing nature of equilibrium prices.

C. SeminalNon-Competitive Models The early REE modelsl 1 possess an unsatisfactory property, dubbed the "schizophrenia" property by Hellwig (1980). Specifically, in these models each informed trader takes the equilibrium price as given, despite the fact that her trading influences the equilibrium price. Two approaches exist to deal with this problem. Hellwig (1980) proposes a "large market" model in which the number of informed traders approaches infinity, so that each informed trader becomes "small" in an appropriate sense. An alternative approach is proposed by Kyle (1984, 1985). He models a finite number of informed traders who take into account, when determining their trading strategies, the impact their trades will have on the equilibrium price.

34

KAVOUS ARDALAN

Kyle (1985) models a set of noise (liquidity) traders and a single (monopolistic) informed trader, all of whom submit market orders (for the single risky asset) to a market maker. The informed trader and the market maker are both assumed to be risk neutral. The orders are executed in a batch (i.e., in a one-shot auction), where all orders that arrive during the time period trade at the same equilibrium price. The most important feature of Kyle's model is that, in choosing the size of the market order to submit, the single informed trader takes into account the expected effect her order will have on the equilibrium price. The market maker, who trades from her own account to clear the market, sets the asset price equal to its conditional (on the net order flow) expected value. Kyle (1985) begins by analyzing a single auction. 12 Kyle assumes that the (ex post liquidation) value of the risky asset at the end of the period is/2' 1 - N(0, ~ ) 13 and the total quantity traded by noise traders is 2 - N(0, c~) . The random variables ~b1 and 2 are assumed to be independently distributed. The quantity traded by the informed trader is denoted by Yc and the equilibrium price is denoted by 12'0 . In K yle (1985), trading occurs in two steps. In step one, the informed trader observes a signal 5, where ~ -- ~b1 (without an error term in this model), but she does not observe 2. The informed trader's order is denoted by Yc = x(~), and her profit is denoted by = (~ - ~b0)~. In step two, the market maker observes the total order flow, J + 2, but not k or 2 separately. The market maker's pricing rule, denoted by P, is/50 = P(Yc+ 2). The risk-neutral market maker is assumed to behave competitively. Consequently, she sets P0 such that her expected profit, conditional on the net order flow, is zero (i.e., E[~IY¢ + 21 = 0 or E [ P I I & + 2 ] = P0)An equilibrium is defined as a pair (x,P) such that: (1) The informed trader chooses x to maximize her conditional (on ~ ) expected profit; and (2) The market maker chooses P0 such that her conditional (on the net order flow, Yc+ 2 expected profit is zero. The informed trader exploits her monopoly power (only she observes ~ ) by taking into account the effect k (which she chooses in step one) is expected to have on P0 (chosen by the market maker in step two), when determining ~c. In doing so, the informed trader takes P (the market maker's pricing rule, set in step two) as given. Kyle (1985) proves the existence of an equilibrium in which x and P are linear functions: x(3) = fl(6)

(14a)

P(.~+ 2) = ~()c+ 2)

(14b)

where: =

~, = ~

(°z/

(14

)

(14d)

Observe that in (14b) the equilibrium price is an increasing function of the net order flow, 2 + 2, because the order flow includes the informed trader's demand, ~, which the market

Markets with Asymmetric Information

35

maker expects to be positively correlated with P1 • Also, 3- (the signal/noise parameter) increases with a 6 and decreases w i t h t~z.14 That is, the slope of the market maker's price schedule increases with the expected value of the informed trader's signal, and decreases with the expected quantity of noise trading. The higher the expected value of the informed trader's signal (8 which, in this case, is also P1) the higher the current price set by the market maker. Also, the lower the expected quantity of noise trading the higher the current price, since the market maker expects to benefit less from trading with the noise traders and needs to cover herself against the expected losses to the informed trader. fl (the noise/signal parameter) increases with t~z and decreases with ~ . This occurs because the informed trader takes the strategy of the market maker as given. That is, the informed trader knows that 3- increases with 68 and decreases with 6 z. By (14b), a higher 3- means that the equilibrium price set at the current period is more sensitive to the order flow, (Yc+ ~). So, the informed trader trades less aggressively, i.e. chooses a lower ft. To summarize, when 3- is high, P is more sensitive to the net order flow and consequently, the informed trades less aggressively. Conversely, when 3. is low, the informed trades more aggressively because she knows that the equilibrium price is not that sensitive to the net order flow. That is, the strategy parameter of the informed trader (i.e., fl) is inversely related to the strategy parameter of the market maker (i.e., 3-). The following remark establishes an important property of Kyle's (1985) single auction equilibrium. Remark 6. In equilibrium, the price is not fully revealing. Consequently, the information asymmetry among the informed trader, the noise traders, and the market maker persists. Proof. See Appendix 6. [] The basic auction model of Kyle (1985) reflects several features of real financial markets, is relatively tractable, and does not possess the "schizophrenia" property discussed by Hellwig (1980). 15 Consequently, Kyle's model has been extended by a large number of authors. A brief survey of several of these extensions is now provided. Kyle (1984) extends the single auction equilibrium model of Kyle (1985) to include n informed traders, where the informed traders possess diverse (and noisy) signals about the asset's end-of-period value. Each of the n informed traders observes a private signal t~i = (/31 + Ei), i = 1..... n, where Ei is a random noise term. The noise traders trade a ran-

dom quantity, ~. Kyle assumes that the n + 2 random variables /31,~1..... ~n,~ are 2

2

normally and independently distributed with zero means and variances given by: cS,t~ei (~/i), and t ~

.16

Kyle (1984) assumes informed traders and the market maker follow linear strategies. The competitive risk-neutral market maker sets the price as a function of the net order n

flow, ~ = Y'i = lXi + "Z, but not of private observations (131 + ~i). Likewise, the ith informed trader chooses the quantity xi as a function of (/31 + ~i), but not of the net order flow.

KAVOUS ARDALAN

36

Kyle (1984) then proves the existence of a linear Nash equilibrium:

5Ci = fl

F

(15a)

(/31+gi)

ktie + tim /50 = ~,~

(15b)

where the market maker chooses the strategy parameter ~ and all informed traders choose the same strategy parameter fl, namely: 1

2 fl

=

n(ti

(15c)

ti _1

1

(

= 2 -1 n + l +

(15d)

The following remark establishes an important property of the model's equilibrium. Remark 7. In equilibrium, the price is not fully revealing. Consequently, the information asymmetry among the informed traders, the market maker, and the liquidity noise traders persists. Proof. See Appendix 7. [] Kyle (1984) shows that the expected profit of each informed trader, re, is given by: 7~ =

2 2.22 tie + ti~)tiz ti~ 2 2 4n[2ti~ + (n + 1 )%]ti~

Kyle demonstrates that it is by withholding some of their private information that informed traders are able to make profit on average. Admati and Pfleiderer (1988) use the single auction model of Kyle (1984, 1985) to analyze the interaction between strategic informed traders and strategic liquidity traders. They begin by assuming that there exist two types of liquidity traders, nondiscretionary and discretionary liquidity traders. Nondiscretionary liquidity traders must trade a particular random number of shares at a particular time• Discretionary liquidity traders also have liquidity demands, but can be strategic in choosing when to execute these trades within a given period of time (e.g., by the end of the trading day) to minimize their trading costs. Within this framework, Admati and Pfleiderer develop a theory to explain the patterns of volume and price variability in intraday transaction data. Admati and Pfleiderer (1988) consider a single asset traded over a span of time (e.g., a day) that is divided into T periods. They assume that the asset's value in period T is given by: T t=l

Markets with AsymmetricInformation where the "St,t = 1 ..... T ,

37

are exogenous random shocks, which are i.i.d, with

In periods prior to T, information about F is revealed through both public and private sources. In each period t, the innovation 8t becomes public knowledge. Also, in period t each of the n informed traders observes the same signal (~t + 1 + et), where et - N ( 0 , ~ ), that is informative .about ~t + 1 • Informed traders must determine the size of their ma~l~et order in period t, x~t . This decision is made knowing fZt_ 1 = (to] ..... t°t- 1), the history of total net order flows; At = (El ..... ~t), the history of innovations; and the signal, ( ~ t + 1 + ~t)"

Each nondiscretionary liquidity trader must trade a given number of shares in a period. Each of the m discretionary liquidity traders is assumed to have a demand for shares which is determined in period T' and needs to be satisfied before period T", where T' < T" < T. Since discretionary liquidity traders are risk-neutral, they follow a trading policy so as to minimize the expected cost of trading. _i -j Let x t be the ith informed trader' s order in period t, Yt be the order of thejth discretionary liquidity trader, and let Zt be the total demand for shares by the nondiscretionary n

.i

m

-j

liquidity traders. Then the market maker must purchase ~t = Ei = 1Xt + Y'j = lYt + "Zt shares in period t and set Pt so that her expected profit is zero. t7 Admati and Pfleiderer (1988) prove that (when each trader takes the strategies of all other traders, as well as the market maker's competitive (linear) price-setting strategy, as given) the equilibrium, at time t, is characterized by: .i x t = f l t ( ~ t + 1 + ~t)

(16a)

t = P + E

(16b)

+

Z=I

where: 2 ~z,

[ ~t =

"

2-"

[n(%+

1 12 2

OZt =- V a r

(lOc)

+ zt Lj=I 1

o~ [- n

]2

(16d)

t43Zt~ V 8 -v 2

The reciprocal of ~ is the market-depth parameter, which is decreasing in (~z, the total variance of liquidity trades in period t. That is, the higher the expected (absolute) value of the liquidity trades, the deeper is the market. .

.

.

.

t

KAVOUS ARDALAN

38 •

2

.

However, when some hquidity trading is discretionary, ffz ~s an endogenous parameter. D~screttonary liquidity traders mlnmuz~ their expec,ted t~ansactmn costs, subject to meeting their individual liquidity demand. ~ , where ~ = ~ if thejth discretionary liquidity trader trades in period t and where ~Jt = 0 otherwise. The expected cost of trading is the difference between what the liquidity trader pays for the security and the security's expected value. Specifically, the expected cost to thejth liquidity trader of trading at time t between T' and T" is: .

.

.

.

.

.

.

.

.

t

.

Which simplifies to: gt(.ilr,) 2 . ThUS, for a given set of ~,t, the expected trading cost is min2 imized by trading in the period in which ~,t is the smallest. Since ~,t is decreasing in O z , if in equilibrium the nondiscretionary liquidity trading is particularly heavy in a particular period t, then ~,t will be set lower, which in turn makes discretionary liquidity traders concentrate their trading in that period. Foster and Viswanathan (1990) introduce repeated trading periods (days) into the continuous-time version of Kyle (1985) model, and allow new information (which lasts for some time) about the security to arrive every day. In their model, the private advantage of the informed trader is reduced through time by public information and the market maker's inferences from changes in the order flow. Information about the asset's price enters the model from either a public source or from a private signal observed by the informed trader. Public information is available in two ways: first, at the close of each trading day there is noisy signal of the asset's price; and second, every calender quarter the asset's price is announced (at the close of the day's trading). At the time of the quarterly announcement, there is no information asymmetry. Each day, the informed trader receives a signal, (st, of the asset's price next quarter. The expected value of the asset's price, given all past information, including the private signal, is (St, where: (st= (st_l + gt 9 is i.i.d, and distributed N(0,a~). Note that, this is the reduced form of a model in which the payoff next quarter is the sum of the daily shocks observed by the informed trader: (st-1 is the sum of past daily shocks, and et is the current shock. L e t X t denote the holdings of the asset by the informed trader. Then the instantaneous market order is denoted by dXt = x r Let Z t denote the holdings of the asset by the liquidity traders. Then their instantaneous net purchases or sales are denoted by d Z t = zt dv, where d v is a Wiener process and zt is the instantaneous order submission rate of liquidity traders. The market maker observes the total instantaneous net order flow, d W t - w t = x t + zt, a n d sets the price, I t , so as to make zero expected economic rents. Foster and Viswanathan (1990) prove the existence of a linear Nash equilibrium: w h e r e Et

xt = fit (St - P t ) d t

(17a)

Markets with Asymmetric Information

39

dPt = ~t (xt + zt)

(17b)

where: 1

2 [~z (Y~t- At )]~

(17c)

fit = A t + (~'t - A t ) ( 1 - t) 1

)~t = ~ \ oz J

(17d)

where ~t and A t are the variance of 5t (around Pt) at the beginning and at the end of day t, respectively. 18 Observe that the sensitivity of the price to the order flow, ~t, increases with the amount of information, released by the informed trader, ~ t - At, and falls with the amount of liquidity trading, ffz" From (17d), the informed trader transacts more intensely when there are more liquidity transactions to disguise her trades. Holden and Subrahmanyam (1992) extend Kyle's (1985) multi-period auction model to include multiple informed traders with long-lived information. They prove the existence of a unique linear equilibrium. They show that informed traders compete and trade aggressively. Even just two informed traders cause nearly all of their common private information to be incorporated into prices "quickly" and cause the depth of the market to become "large," provided the number of auctions is reasonably large. That is, they show that a market with multiple informed traders closely approximates a strong-form efficient market. Holden and Subrahmanyam (1992) further show that in the case of at least two informed traders, as the number of auctions is increased, by subdividing each time interval into even smaller subintervals, more information is revealed by any cutoff point in calendar time. In the limit as the number of auctions goes to infinity all information is revealed immediately. Market depth is small in the earlier periods when there is severe adverse selection and large in later periods when most of private information has already been revealed. Holden and Subrahmanyam (1992) also demonstrated that, in the limit in which the number of informed traders goes to infinity, all of their private information is revealed in the first trade. Thus, the perfectly competitive outcome is strong form market efficiency, irrespective of the total number of auctions in the game. Subrahmanyam (1991) introduces risk aversion into the single auction model of Kyle (1984, 1985). Informed traders, who are risk averse, respond less aggressively to an increase in liquidity trading than do risk-neutral traders. Thus, when information acquisition is exogenous, an increase in the variance of liquidity trades decreases price efficiency, unlike the risk-neutral case. Also, an increase in the number of risk-averse informed traders increases the aggregate risk tolerance of these traders, which tends to raise their aggregate profits. This causes the marginal effect of an increase in the number of informed traders on market liquidity to be negative despite greater competition between these traders, provided the number of informed traders is small. Therefore, with endogenous information acquisition, increased liquidity trading, which leads to an entry of more informed traders, can lead to decreased market liquidity.

40

KAVOUS ARDALAN

Subrahmanyam (1991) also shows that market liquidity may be increasing in the precision of private information and the risk tolerance of the informed. The intensity of competition between the informed increases as they become more precisely informed or more risk tolerant, which leads to increased market liquidity if the number of informed traders is sufficiently large. Back (1992) examines the continuous-time version of Kyle (1985). He shows that there is a unique equilibrium pricing rule within a certain class. He obtains the pricing rule, in closed form, for general distribution of the asset value. General trading strategies are allowed. In equilibrium, the risk-neutral informed agent has many optima. He does not correlate his trades locally with the noise trades nor does he submit discrete orders. Caballe and Krishnan (1994) examine a multi-security financial market and in this way generalize Kyle's (1985) single-security model. They prove that there is always an equilibrium with a symmetric and positive definite matrix governing the relationship between the vector of prices and the vector of order flows. Hirshleifer, Subrahmanyam, and Titman (1994) analyze trading behavior and equilibrium information acquisition when some traders receive common private information before others. In their model, there are two trading dates and they consider three types of traders: (1) a continuum of competitive, risk-averse investors with finite aggregate mass, some subset of which discovers a private information signal about a security's terminal value before the other subset; (2) liquidity traders whose share demands are unmolded; and (3) risk-neutral market makers who absorb the net demands of the other traders at competitive prices. Hirshleifer, Subrahmanyam, and Titman solve for the linear equilibria of their dynamic model in closed-form. Their model implies that, under some conditions, investors will focus only on a subset of securities, a phenomenon called herding. Vives (1995a) analyzes the effect of traders' horizons on the information content of prices. In an independent work, Vives uses a model similar to Hirshleifer, Subrahmanyam, and Titman's (1994) model but with n periods. Vives finds a complete closed-form solution to the n-period dynamic model with long-term agents. Vives shows that short horizons enhance or reduce accumulated price informativeness depending on the temporal pattern of private information arrival. With concentrated arrival of information, short horizons reduce final price informativeness; with diffuse arrival of information, short horizons enhance it. Vives (1995b) examines the rate at which private information is incorporated into prices. He uses Kyle's (1985) model to show that asymptotic precision of prices is negatively related to the degree of risk aversion of traders and the amount of noise in the system. Vives also demonstrates that, in the presence of competitive market makers, the rate of convergence of prices to the value of the risky asset is faster than when competitive market makers are absent from the system. 19 In this section we made an expository review of the seminal non-competitive models, a direction initiated by Kyle (1984, 1985). We concentrated on the informational role of prices and emphasized the partially-revealing nature of equilibrium prices.

II.

CONCLUSION

This paper made an expository review of the seminal models of the rational expectations equilibrium in the finance literature. The staggering explosion of the complex research in

Markets with Asymmetric Information

41

this area of the finance literature necessitated the provision of an expository review essay that systematically brings out the seminal theoretical models, and exposes the essential structure and theoretical contribution of the models. This line of research has come to existence when the perfect market assumption of homogeneous information is relaxed and market prices play the role of aggregating and transmitting information. The paper, therefore, concentrated on the informational role of prices and contrasted the fully-revealing and the partially-revealing nature of equilibrium prices.

APPENDIX 1 R e m a r k 1. Each trader i may choose to ignore her own signal Yi because: (i) each trader i observes P0(Y), from which she can obtain ) ; and (ii) ~ and the combination of ~ and Yi are equivalent signals of P1-

Proof. To demonstrate (i), note that each trader i observes PO" The expression for PO Po(Y ) - 0~0

above (see (4)) can be inverted to obtain y O~1 a r e

, where the parameters ct0 and

known by all traders.

For (ii) we demonstrate that: E[ P 1 ly i] ~ E[ P ~19] = E[ P 11~,y i] and correspondingly that V a r[ P 1 ly i ] > V a r[ P 119 ] = V a r[ P 11~,yi]. The proof proceeds in three steps. 20 First, the posterior distribution of P1 (conditional on yi) is normal with mean and variance:

Var[ P 1] E[PllYi ] = E[P1] +

2(Yi- E[P1]) Var[P 1] + t~e

( Var[PllY i] = VarIPl] 1

Var[P1] ] ~ 2 ' Var[P 1] + t~eJ

Second, the posterior distribution of P1 (conditional on y ) is again normal, with mean and variance:

nVar[P 1] E[PI[~] = E [ P 1] +

,~(~- E[P1])

Var[P 1] + o e Var[Pl[ ~]

nVar[P1] I Var[P1] 1 Var[P1]+t~e) ~

-

-

2

"

KAVOUS ARDALAN

42

Third, the posterior distribution of P1 [conditional on ( y , Yi)] is also normal, with mean and variance:

n V a r [ P 1] E[P1 [Y,Yi] = E[P1] +

2(Y - E [ P 1 ] ) Var[P t] + G e

nVar[P1]

I

Var[Pll~,y i] = Var[P 1] 1 Var[P1]+°e ~ - 2 j;" By comparing the above three sets of equations we note that indeed Remark 1 holds true. []

APPENDIX 2 R e m a r k 2. The informed traders' information set is superior 21 to the uninformed traders' information set. Proof. The proof proceeds in three steps. (a) First, we examine the uninformed traders' beliefs. Their information set consists solely of the equilibrium price, P0 (y,x; ~,). Recall from (7) that, P~ = fll + fl2w(y,x; ~'), which can be rewritten as: P ~ ( Y , X ; k ) - fll W

=

where fll and r2 are parameters known by all investors. Note that, if an investor observes P o , she knows w (i.e., the equilibrium price reveals w to all traders). 2 aG e Also, recall from (7) that w(y,x; L) = y - ---~-x, which can be rewritten as: 2 aG e y = w(y,)G~,) + --~--x

for Z.>O

where: y = P1 + e (where: e - (0,G~)); x - N(O, Var[x]) ; and a, G 2 , and ~, are known 2 aG e by all traders. Note that w reveals a noisy version of y, where the noise term is: ~ x . L This means that uninformed traders (who observe P0 which reveals w) learn y to within

a random variable, with mean zero and variance

2 4 a GE

~2 Var[x].

Markets with Asymmetric Information

43

The uninformed traders' beliefs regarding P1 is:

EIP~IPo] =

Var[y] E[PllW(y,x;~,)] = E[y] + V~r[w](W- E[y])

2 Var[P IlPO] = Var[PllW(y,x;~)] = Var[y] + o e

Var[y] Var[w]

where: 2 4

Var[w] = Var[y] + a t~ eVar[x] ~2 (b) Second, we examine the informed traders' beliefs. All informed traders hold identical beliefs in equilibrium since they all observe the same signal (Yi = Y, Vi) and equilibrium price, P~ (y,x; L). Recall that y = P1 + e, or equivalently, P1 = Y - e where e - N(0, a~). Then, for informed traders: E[P 1 l y] = y, and Var[P 1 ly]

= 2.

(c) Third, we compare the beliefs of the uninformed and the informed traders. Consequently: Var[ P 11Po] - Var[ P 1ly] = Var[y] - Var[w] ear[y] " This completes the proof. Alternatively, invoke Blackwell's Theorem for immediate proof.

[]

APPENDIX 3 Remark 3. The information set which consists of both the signal

(Yi) and the equilibrium

price (P~), is superior22 to the information set which consists of only (Po)"

Proof The proof proceeds in two steps. First, the posterior distribution of P1 (conditional on (yi,Po)) is normal, with mean and variance: -~7*2Var[x](Yi- P1) + x * ( P ~ - ~ 0 - x'P1 + T'x) E[PllYi,Po] = P1 + t~e,

(9a)

1 + l_]~l*2ear[x] + x*2 VamP1] 2J

•

7*2Var[x]Var[P1](Yei2

Var[PllYi'Po] = 7*2Var[x]°2i + 7*2Var[x]Var[P1]+

Oei2"

(9b)

44

KAVOUS ARDALAN

Next, the posterior distribution of Pl (conditional on and variance:

) is again normal, with mean

~*Var[P 1] E[PIIP~)] = P1 + x,2Var[P1] + T *2 Var[x] (P0 -

g ~ - ~ * P 1 + T *'~)

T*2 Var[ x ] Var[ P 1] Var[P 11P~] = ~*2Var[P1] + T*2Var[ x] We now show that E[P 1 lyi,P~l :/: EtP 11Po] and Var[P 1

(10a)

(10b)

lyi,p~] < Var[P 11Po].

To show that E[P 1lyi,Po] ~ E[P 11P0], note that, in general, the inequality holds true, unless in the especial case where: ~*{a2 + V a r [ P 1 ] - I } ( P ~ - ~ - ~ * ' I + T Y c )

Yi = P1 +

g*2Var[P1] + T*2Var[x]

Next, note that, from (9b) and (10b): 1

1 ,

Var[PllYi,PoI

1

Var[PIlP ;]

- -T>0

t~ei

which implies that:

Var[PllYi,P ~] < VartPtIP~] . This completes the proof. Alternatively, invoke Blackwell's Theorem for immediate proof.

[]

APPENDIX 4

Remark 4. The information set which consists of both the signal (Yi) and the equilibrium price (P~), is superior23 to the information set which consists of only (P~). Proof. The proof proceeds in two steps. First, from Verrecchia (1982) the posterior distribution ofP 1 (conditional on (yi,Pg)) is normal, with mean and variance:

1----T*2Var[x](Yi- P1) + g*(P~ + P1 ) 2

EtPllYi,P~] = P1 + oei

(11a)

1

Vat [PllYi ,P~] =

+

1]T*2Var[x]

+

/t*2

T *2Var[ x ] Var[ P1]~21 T*2Var[x]¢~2 i + T*2Var[x]Var[ P1] + lc*2Var[ P1]~2 i

(1 lb)

Markets with Asymmetric Information

45

Next, the posterior distribution of P1 (conditional on and variance:

) is again normal, with mean

~*Var[P1] E[PIIP~] = P1 + x.2Var[P1] + T,2Var[x](P~ - P1)

(12a)

q(*2Var[ x ] Var[ P 1]

Var[P llP~] = x*2Var[P1] + "~*2Var[ x]

(12b)

We now show that E[P 1lyi,Po] :/: E[P 11P~] and Var[P 1lYi,Po] < Var[e 11P~]. To show that E[P 1lyi,Po] ~ E[P 1[P~], note that, in general, the inequality holds true, unless in the especial case where: ~*I~2[e, + V a r [ P 1 ] - I } ( P ~ - P x )

Yi = P1 +

x*2Var[P1] + T*2Var[x]

Next, note that, from (1 lb) and (12b): 1

1

Var[P 1lYi,Po]

VartPllP~]

1

= --$>0

t~e,

which implies that:

Var[P llyi,PO] < Var[P lIPs]. This completes the proof. Alternatively, invoke Blackwell's Theorem for immediate proof.

[]

APPENDIX 5 Remark 5. In equilibrium, traders hold heterogeneous beliefs and Po is not fully reveal-

ing.

Proof. The distribution ofP 1 as assessed by trader i, using her private signal Yi and the equilibrium price vector Po is multivariate normal with variance-covafiance matrix and mean:24

E[PllYi,P*o] = Boi + BliYi+ B2iPo

KAVOUS ARDALAN

46

where:

BOi = VARtP 1lYi,eo] = I -

VAR[x]Z-~

+ at

x(VAR-I[p1]PI+~e2VAR-I[x]Yc) B. =

VAR[PIIyi,P~]E-E2

-~------2 -1 VAR[x]Z~2 ---I(vAR-I[p1] + Z ~2)}-1 B2i = r I+ Zei x{(I+?tVAR-I[x]~e2)-la~e2-1VAR-I[p1]+I}. First, note that note

that

Var([PllYi,P ~1 ~ Var[PllYj,P~]) $

*

E[PI[Yi,Po]~E[PI[yj,Po]

Var([Pllyi,Po] ~ Var[PllYj,Po] ) .

for

for i ~ j, since Z~2 ~ Z~2. Second, i

~ j,

since

Yi =~ Yj and

So, individual traders hold different beliefs in

equilibrium. Alternatively, invoke Blackwell's Theorer0 for immediate proof.

[]

APPENDIX 6 R e m a r k 6. In equilibrium, the price is not fully revealing. Consequently, the information

asymmetry among the informed trader, the noise traders, and the market maker persists. Proof. We examine the beliefs of the informed trader, the noise traders, and the market maker in turn, and then compare them. The informed trader observes ~ (where, in Kyle (1985), ~ = /31 without an error term), hence her beliefs can be represented by ~b1 - N ( P

1,0), E[PI[Oe]

= P1 and

Var[P1 I oe] = 0. Note that she learns nothing from observing the equilibrium price. The noise-traders observe neither the informed trader's signal (~) nor the net order flow (.~ + ~). However, they do observe the equilibrium price. Their (conditional on P0) beliefs can be represented by P 1 -

1 2 N(Po,~t~8).

The market maker observes (~ + ~) and sets P0. Her beliefs can be represented by 1 2

P 1 - N(P0,~c~).

Markets with Asymmetric Information

47

To see that the equilibrium price is not fully revealing, compare the beliefs of the informed trader, /91 - N ( P 1 , 0 ) , of the market maker, / 9 1 - N ( P 0 , ~ o ~ ) , and of the noise traders, /91-N(P0,~o~). Note that, in general, their beliefs are different and, 2 therefore, the equilibrium price is not fully revealing. As long as o z > 0, the equilibrium price does not fully reveal the informed's signal and consequently, their heterogeneous beliefs persists at the equilibrium. Alternatively, invoke Blackwell's Theorem for immediate proof. []

APPENDIX 7 Remark 7. In equilibrium, the price is not fully revealing. Consequently, the information asymmetry among the informed traders, the market maker, and the liquidity noise traders persists. Proof. We examine the beliefs of the informed traders, the market maker, and the liquidity noise traders in turn, and then compare them. The ith informed trader observes t5i = (/91 + el), i = 1..... n, and her beliefs can be represented by: 2 a6 E[/911/91 + Ei] = 2 2 (/91 + Ei) Oe + 08

Vi

2 2 Var[/91[/91 + ~i] -

~e°8 2" 2 0¢ + a~

Vi

The market maker observes the net order flow, ~ = zni= Hence, her beliefs can be represented by:

Et/9,l

l =

-i Var[ /911fv] = F K n f l C o v [ /91,P l ] where: F=

1

2

2 -i K = 06n[i(Cov)[P1,P1 ]

1Xi"+ ~

where x i = fl/gz1 .

KAVOUS ARDALAN

48

2 P1 =

2

2 (•1 + Ei)

[ ° V a r [ _ ,{ -

2----------2+ " - T " - 5

% +% [% +%J 4 - i

C°v[PI'P1]=

~8

2 2" t~e + (IS

The noise traders observe neither the informed traders' signals, n t~i = 0bl + ~i), i = 1..... n, nor the net order flow, ~ = ~ i = 1Xi + ~" However, they observe the equilibrium price. Since the pricing rule of the market maker is assumed to be linear, as per (15b), observing the equilibrium price is equivalent to observing the net order flow. So, the information of the noise trader is equivalent to the information of the market maker, as provided above. To see that the equilibrium price is not fully revealing, compare the beliefs of the informed traders and that of the market maker and note that, in general, their beliefs are different, therefore, the equilibrium price is not fully revealing. In other words, in equilibrium they hold heterogeneous beliefs. Alternatively, invoke Blackwell's Theorem for immediate proof. []

NOTES 1. The author would like to thank Kevin Hebner and the anonymous referee for constructive and helpful comments. The author also wishes to thank his wife (Haleh), son (Arash), and Daughter (Camellia) for their support and patience. 2. The literature which follows Hellwig's (1980) direction includes: Verrecchia (1982), Admati (1985), Grundy and McNichols (1989), Brown and Jennings (1989), Naik (1993), and Brennan and Cao (1996). 3. The literature which follows Kyle's (1984, 1985) direction includes: Admati and Pfleiderer (1988), Foster and Viswanathan (1990), Holden and Subrahmanyam (1992), Subrahmanyam (1991), Back (1992), Caballe and Kdshnan (1994), Hirshleifer, Subrahmanyam, and Titman (1994), Vives (1995a), and Vives (1995b). 4. Kyle's (1989) seminal model considers rational expectations equilibria with imperfect competition. Since his model has not initiated a direction and formed a category in the literature, it has not been included in this review. 5. See Admati (1991) for a general, as opposed to specific and expository, review of the literature. Admati (1991) touches upon every aspect of the literature including implementation, learning, and existence of the rational expectations equilibrium, and the market microstructure literature. We focus on seminal rational expectations equilibrium models and emphasize the modeling contribution of each model and how it fits with the other models. Whereas Admati (1991) extensively reviews the literature, we intensively investigate

Markets with Asymmetric Information

49

the modeling contributions to the rational expectations literature. This leads us to the intensive treatment of the modeling aspects of the literature and the abstraction from the other aspects of the literature. 6. Grossman (1977, 1978, and 1981) provide additional fully-revealing REE models. Grossman (1977) focuses on the informational role of futures markets. He provides an example in which, the spot price together with the futures price, jointly reveals a two-dimensional signal. Grossman (1978) derives a version of the CAPM in which, initially, traders have diverse information. Grossman (1978) is a generalization of Grossman (1976) to many assets, a more general information structure, and no special form for traders' utility functions. Grossman (1981) demonstrates that, in a complete Arrow-Debreu economy, there always exists a fully revealing REE price. 7. Diamond and Verrecchia (1981) present a NREE model where traders possess diverse signals. As in Grossman and Stiglitz (1980), noise exists because the aggregate supply of the risky asset is random. However, the source of aggregate supply noise is explicit: traders' endowments in the risky asset are random (normal and i.i.d.) and the aggregate supply is the sum of these endowments. Note, however, that because of the distributional assumption about the traders' endowments, if the number of traders grows (but the variance of each trader's endowment is bounded), then by the law of large numbers per capita supply becomes constant, and Diamond and Verrecchia's model approaches Grossman's (1976) fully-revealing model. 8. See Grossman and Stiglitz (1980), page 397. For expositional simplicity we have set the expected value of x equal to zero, E[x] = O. 9. For consistency, this section employs Grossman's (1976) notation as much as possible. 10. The capital letters in VAR[.] and 5"-are used here to emphasize that they are matrices. 11. For example, Grossman (1976), Grossman and Stiglitz (1980), and Diamond and Verrecchia (1981). 12. For consistency, this subsection adopts Admati and Pfleiderer's (1988) notation. 13. For consistency with Admati and Pfleiderer (1988), we assume the mean to be zero. 14. Actually, there is a typographical error in Kyle (1985) as the scalar coefficient in (14d) is reported to be 2, rather than I . 15. Kyle (1985) also extends his ~'single auction model to both a sequential and a continuous auction model. In the sequential auction model, the informed trader must decide how intensely to trade on the basis of her private information, given the pattern of market depth expected at current and future auctions. In the sequential auction equilibrium, the price's informativeness increases over time as information is gradually incorporated into the price. The equilibrium price is "almost" fully-revealing by the end of trading. In the continuous auction, the time interval between auctions goes to zero, and the equilibrium price becomes fully-revealing. 16. For simplicity, we abstract from the public information. 17. Admati and Pfleiderer (1988) assume that the random variables: (~1 ,-- . ,~rn,~ 1 ..... ZT- 1 '~1 ..... ~T'E1 ..... ET- 1 )

are mutually independent and distributed multivariate normal, with each variable having a mean of zero.

50

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18. Equation systems (14), (15), (16), and (17) are each alphabetically ordered for ease of comparisons of each equation across equation systems. It is interesting to note how each equation can be directly obtained from the other by changing the assumptions. 19. Note that, the last three models in this section are competitive models. It might seem paradoxical to include them in this section, which is entitled, "Seminal Non-Competitive Models." However, in this section we include Kyle (1985) and its extensions. The last three models in this section are the limiting cases of Kyle's (1985) model, which happen to be competitive models. Here, the limiting case means that informed traders are individually infinitesimal and fall on a continuum, so that no informed trader can affect the price. 20. See Copeland and Weston (1988, p.190), Freund and Walpole (1987, p.233), Judge, Hill, Griffiths, LutkepoM, and Lee (1988, p. 50), and Chiang (1984, p.106). 21. Here, superiority is used synonymously with precision, which means lower variance. 22. Here, superiority is used synonymously with precision, which means lower variance. 23. Here, superiority is used synonymously with precision, which means lower variance. 24. This result is taken from Admati (1985).

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Grossman, S. (1977). The existence of futures markets, noisy rational expectations and informational externalities. Review of Economic Studies, 64, 431-449. Grossman, S. (1978). Further results on the informational efficiency of competitive stock markets. Journal of Economic Theory, 18, 81-101. Grossman, S. (1981). An introduction to the theory of rational expectations under asymmetric information. Review of Economic Studies, 48, 541-559. Grossman, S., & Stiglitz, J. (1980). On the impossibility of informationally efficient markets. American Economic Review, 70, 393-408. Grundy, B. & McNichols, M. (1990). Trade and the revelation of information through prices and direct disclosure. Review of Financial Studies, 2, 495-526. Hellwig, M. (1980). On the aggregation of information in competitive markets. Journal of Economic Theory, 22, 477-498. Hirshleifer, D., Subrahmanyam, A., & Titman, S. (1994). Security analysis and trading patterns when some investors receive information before others. Journal of Finance, 49, 1665-1698. Holden, C., & Subrahmanyam, A. (1992). Long-lived private information and imperfect competition. Journal of Finance, 47, 247-270 Judge, G., Hill, C., Griffiths, W., Lutkepohl, H., & Lee, T. (1988). Introduction to the theory and practice of econometrics. John Wiley and Sons. Kyle, A. (1984). Market structure, information, futures markets, and price formation. In G. Storey, A. Schmitz, & A. Sarris (Eds.), International agricultural trade: advanced readings in price formation, market structure, and price instability. Vestview Press. Kyle, A. (1985). Continuous auctions and insider trading. Econometrica, 53, 1315-1335. Kyle, A. (1989). Informed speculation with imperfect competition. Review of Economic Studies, 56, 317-356. Naik, N. (1993). On aggregation of information in competitive markets: the dynamic case. IFA Working,Paper 159-192, Institute of Finance and Accounting, London Business School. ~ Subrahmanyam, A. (i991). Risk aversion, market liquidity, and price efficiency. Review of Financial Studies, 4, 417--441. Verrecchia, R. (1982). Information acquisition in a noisy rational expectations economy. Econometrica, 50, 1415-1430. Vives, X. (1995a). Short-term investment and the informational efficiency of the market. Review of Financial Studies, 8, 125-160. Vives, X. (1995b). The speed of information revelation in a financial market mechanism. Journal of Economic Theory, 67, 178-204.