Econometrica,Vol.53,No.6 (November,1985) CONTINUOUS AUCTIONS AND INSIDER TRADING BY ALBERT S.KYLE A dynamic model of insider trading with sequential auctions,structured to resemble a sequential equilibrium,is used to examine the informational content of prices,the liquidity characteristics of a speculative market,and the value of private information to an insider. The model has three kinds of traders:a single risk neutral insider,random noise traders, and competitive risk neutral market makers.The insider makes positive profits by exploiting his monopoly power optimally in a dynamic context,where noise trading provides camou- flage which conceals his trading from market makers.As the time interval between auctions goes to zero,a limiting model of continuous trading is obtained.In this equilibrium,prices follow Brownian motion,the depth of the market is constant over time,and all private information is incorporated into prices by the end of trading. 1.INTRODUCTION How QUICKLY IS NEW PRIVATE INFORMATION about the underlying value of a speculative commodity incorporated into market prices?How valuable is private information to an insider?How does noise trading affect the volatility of prices? What determines the liquidity of a speculative market?The purpose of this paper is to show how answers to questions like these can be obtained as derived results by modelling rigorously the trading strategy of an insider in a dynamic model of efficient price formation. In the particular model we investigate,one risky asset is exchanged for a riskless asset among three kinds of traders:a single insider who has unique access to a private observation of the ex post liquidation value of the risky asset; uninformed noise traders who trade randomly;and market makers who set prices efficiently (in the semi-strong sense)conditional on information they have about the quantities traded by others.Trading is modelled as a sequence of many auctions,structured to give the model the flavor of a sequential equilibrium as described by Kreps and Wilson [4]. At each auction trading takes place in two steps.In step one,the insider and the noise traders simultaneously choose the quantities they will trade(in effect, placing "market orders").When making this choice,the insider's information consists of his private observation of the liquidation value of the asset,as well as past prices and past quantities traded by himself.He does not observe current or future prices,or current or future quantities traded by noise traders.The random quantity traded by noise traders is distributed independently from present or past quantities traded by the insider and independently from past quantities traded by noise traders.In step two,the market makers set a price,and trade the quantity which makes markets clear.When doing so,their information consists of observations of the current and past aggregate quantities traded by the insider and noise traders combined.We call these aggregate quantities the"order flow." The author thanks the Centre of Policy Studies,Monash University,and the Yale School of Organization and Management,Yale University,for their hospitality and support when this research was undertaken.The author also thanks Paul Milgrom,Peter Hartley,Phil Dybvig,and Avinash Dixit for useful comments. 1315
1316 ALBERT S.KYLE Market makers do not observe the individual quantities traded by the insider or noise traders separately,nor do they have any other kind of special information. As a consequence,price fluctuations are always a consequence of order flow innovations. The informed trader,who is risk neutral,is assumed to maximize expected profits.He acts as an intertemporal monopolist in the asset market,taking into account explicitly the effect his trading at one auction has on the price at that auction and the trading opportunities available at future auctions.The prices determined by market makers are assumed to equal the expectation of the liquidation value of the commodity,conditional on the maket makers'information sets at the dates the prices are determined.Thus,market makers earn on average zero profits.Since they cannot distinguish the trading of the insider from the trading of noise traders,the noise traders in effect provide camouflage which enables the insider to make profits at their expense. By assuming that the relevant random variables are normally distributed,the model acquires a tractable linear structure.This makes it possible to characterize explicitly a unique"sequential auction equilibrium"in which prices and quantities are simple linear functions of the observations defining the relevant information sets.In the limit as the time interval between auctions goes to zero,the discrete- time equilibrium converges to a particularly simple limit which we call a"con- tinuous auction equilibrium."This equilibrium corresponds to what one obtains when the model is set up heuristically in continuous time. In both the discrete model and the continuous limit,answers to the questions posed at the beginning of this paper are readily obtained.The informed trader trades in such a way that his private information is incorporated into prices gradually.In the continuous auction equilibrium where the quantity traded by noise traders follows a Brownian motion process,prices also follow Brownian motion.The constant volatility reflects the fact that information is incorporated into prices at a constant rate.Furthermore,all of the insider's private information is incorporated into prices by the end of trading in a continuous auction equili- brium.An ex ante doubling of the quantities traded by noise traders induces the insider and market makers to double the quantities they trade,but has no effect on prices,and thus doubles the profits of the insider. Perhaps the most interesting properties concern the liquidity characteristics of the market in a continuous auction equilibrium."Market liquidity"is a slippery and elusive concept,in part because it encompasses a number of transactional properties of markets.These include "tightness"(the cost of turning around a position over a short period of time),"depth"(the size of an order flow innovation required to change prices a given amount),and "resiliency"(the speed with which prices recover from a random,uninformative shock).Black [2]describes intuitively a liquid market in the following manner: "The market for a stock is liquid if the following conditions hold: (1)There are always bid and asked prices for the investor who wants to buy or sell small amounts of stock immediately. (2)The difference between the bid and asked prices (the spread)is always small
CONTINUOUS AUCTIONS 1317 (3)An investor who is buying or selling a large amount of stock,in the absence of special information,can expect to do so over a long period of time at a price not very different,on average,from the current market price. (4)An investor can buy or sell a large block of stock immediately,but at a premium or discount that depends on the size of the block.The larger the block,the larger the premium or discount. In other words,a liquid market is a continuous market,in the sense that almost any amount of stock can be bought or sold immediately,and an efficient market,in the sense that small amounts of stock can always be bought and sold very near the current market price,and in the sense that large amounts can be bought or sold over long periods of time at prices that,on average,are very near the current market price." Roughly speaking,Black defines a liquid market as one which is almost infinitely tight,which is not infinitely deep,and which is resilient enough so that prices eventually tend to their underlying value. Our continuous auction equilibrium has exactly the characteristics described by Black.Furthermore,these aspects of market liquidity acquire a new prominence in our model because the insider,who does not trade as a perfect competitor, must make rational conjectures about tightness,depth,and resiliency in choosing his optimal quantity to trade.Moreover,depth and resiliency are themselves endogenous consequences of the presence of the insider and noise traders in the market.Market depth is proportional to the amount of noise trading and inversely proportional to the amount of private information (in the sense of an error variance)which has not yet been incorporated into prices.This makes our model a rigorous version of the intuitive story told by Bagehot [1].Furthermore,our emphasis on the dynamic optimizing behavior of the insider distinguishes our model from the one of Glosten and Milgrom [3]. The plan of the rest of this paper is as follows:In Section 2,a single auction equilibrium is discussed in order to motivate the dynamic models which follow. In Section 3,a sequential auction equilibrium is defined,an existence and uniqueness result is proved,and properties of the equilibrium are derived.In Section 4,a continuous auction equilibrium is discussed heuristically,and in Section 5,it is shown that the continuous auction equilibrium is the limit of the sequential auction equilibrium as the time interval between auctions goes to zero. Section 6 makes some concluding comments. 2.A SINGLE AUCTION EQUILIBRIUM In this section we motivate our equilibrium concept by discussing a simple model of one-shot trading. Structure and Notation.The ex post liquidation value of the risky asset,denoted is normally distributed with mean po and variance So.The quantity traded by noise traders,denoted u,is normally distributed with mean zero and variance o.The random variables o and u are independently distributed.The quantity traded by the insider is denoted x and the price is denoted p. Trading is structured in two steps as follows:In step one,the exogenous values of and are realized and the insider chooses the quantity he trades x When
1318 ALBERT S.KYLE doing so,he observes o but not u.To accommodate mixed strategies,the insider's trading strategy,denoted X,assigns to outcomes of probability distributions defined over quantities traded.Since,however,mixed strategies are not optimal in what follows,the more intuitive interpretation of X as a measurable function such that=X()is justified.In step two,the market makers determine the price p at which they trade the quantity necessary to clear the market.When doing so they observe+but not or (or separately.While their pricing rule,denoted P,can be defined to accommodate randomization,an intuitive interpretation of P as a measurable real function such that p=P(+)is also justified. The profits of the informed trader,,denoted元，are given by亓=(i-p)x.To emphasize the dependence of and p on X and p,we write=(X,P), =(X,P). Definition of Equilibrium.An equilibrium is defined as a pair X,P such that the following two conditions hold: (1)Profit Maximization:For any alternate trading strategy X'and for any v, (2.1) E{亓(X,P)川i=}≥E{(X',P)川i=v} (2)Market Efficiency:The random variable p satisfies (2.2) (X,P)=E{创x+. This model is not quite a game theoretic one because the market makers do not explicitly maximize any particular objective.We could,however,replace the market efficiency condition in step two with an explicit Bertrand auction between at least two risk neutral bidders,each of whom observes the "order flow"+ and nothing else.The result of this explicit auction procedure would be our market efficiency condition,in which profits of market makers are driven to zero. Modelling how market makers can earn the positive frictional profits necessary to attract them into the business of market making is an interesting topic which takes us away from our main objective of studying how price formation is influenced by the optimizing behavior of an insider in a somewhat idealized setting.Kyle [5],however,discusses a model of imperfect competition among market makers,in which many insiders with different information participate. The insider exploits his monopoly power by taking into account the effect the quantity he chooses to trade in step one is expected to have on the price established in step two.In doing so,he takes the rule market makers use to set prices in step two as given.He is not allowed to influence this rule by committing to a particular strategy in step one:The quantity he trades is required to be optimal,given his information set at the time it is chosen.This requirement seems to be reasonable given anonymous trading and the strong incentives informed traders have to cheat given any other strategy they commit to.The insider is not allowed to condition the quantity he trades on price.A model in which insiders choose demand functions ("limit orders")instead of quantities ("market orders")is considered in Kyle [6]
CONTINUOUS AUCTIONS 1319 Fortuitously,our model has an analytically tractable equilibrium in which the rules X and P are simple linear functions,as we show in the following theorem: THEOREM 1:There exists a unique equilibrium in which X and P are linear functions.Defining constants B and A by B=()2 and =2(/)12,the equilibrium P and X are given by (2.3)X()=B(i-Po,P(x+)=po+A(元+i) Proof:Suppose that for constants u,A,a,B,linear functions P and X are given by (2.4) P(y)=u+ay,X(v)=a+Bv. Given the linear rule P,profits can be written (2.5)E{[i-P(x+)]xi=}=(u-4-Ax)x Profit maximization of this quadratic objective requires that x solve v-u-2Ax= 0.We thus have X(v)=a+Bu with (2.6)1/B=2入，a=-uB. Note that the quadratic objective(implied by the linear pricing rule P)rules out mixed strategies and also makes linear strategies optimal even when nonlinear strategies are allowed. Given linear X and P,the market efficiency condition is equivalent to (2.7) u+入y=E{创a+Bi+i=y}. Normality makes the regression linear and application of the projection theorem yields BΣo (2.8) 入= B2o+-Po=-A(a+Bpo). Solving (2.6)and(2.8)subject to the second order condition A>0 yields the desired result.Note that we have u=po,a=-Bpo,and the second order condition rules out a solution with B and A both negative.This completes the proof of the theorem. Properties of the Equilibrium.The equilibrium X and P are determined by the exogenous parameters Eo and o.To obtain a measure of the informativeness of prices,.defineΣ，byΣ，=var{lp}.A simple calculation shows that∑，=∑o; thus,one-half of the insider's private information is incorporated into prices and the volatility of prices is unaffected by the level of noise trading ou. The quantity 1/A measures the "depth"of the market,i.e.the order flow necessary to induce prices to rise or fall by one dollar.This measure of market liquidity is proportional to a ratio of the amount of noise trading to the amount of private information the informed trader is expected to have.In this sense,it