## Information in Securities Markets: Kyle Meets Glosten and Milgrom

by Kerry Back, Shmuel Baruch
Citation
Title:
Information in Securities Markets: Kyle Meets Glosten and Milgrom
Author:
Kerry Back, Shmuel Baruch
Year:
2004
Publication:
Econometrica
Volume:
72
Issue:
2
Start Page:
433
End Page:
465
Publisher:
Language:
English
URL:
DOI:
PMID:
ISSN:
Abstract:

INFORMATION IN SECURITIES MARKETS: KYLE MEETS
GLOSTEN AND MILGROM

BY KERRY BACK AND SHMUELBARUCH

This paper analyzes models of securities markets with a single strategic informed trader and competitive market makers. In one version, uninformed trades arrive as a Brownian motion and market makers see only the order imbalance, as in Kyle (1985). In the other version, uninformed trades arrive as a Poisson process and market mak- ers see individual trades. This is similar to the Glosten-Milgrom (1985) model, except that we allow the informed trader to optimize his times of trading. We show there is an equilibrium in the Glosten-Milgrom-type model in which the informed trader plays a mixed strategy (a point process with stochastic intensity). In this equilibrium, informed and uninformed trades arrive probabilistically, as Glosten and Milgrom assume. We study a sequence of such markets in which uninformed trades become smaller and ar- rive more frequently, approximating a Brownian motion. We show that the equilibria of the Glosten-Milgrom model converge to the equilibrium of the Kyle model.

THE MOTIVATION for this paper is to understand the relationship between the two canonical models of market microstructure, due to Kyle (1985) and Glosten and Milgrom (1985). The Kyle model is a model of a batch-auction market, in which market makers see only the order imbalance at each auction date. Market makers compete to fill the order imbalance, and matching orders are crossed at the market-clearing price. Because orders are batched, there are no real bid and ask prices. In the Glosten-Milgrom model, orders arrive and are executed by market makers individually. In this model, there are bid and ask quotes, which are determined by the probability that a particular order is informed. Glosten and Milgrom assume the arrival rates of informed and un- informed traders are determined exogenously. Informed traders trade when chosen by this exogenous mechanism as if they have no future opportunities to trade. In other words, when trade is profitable, they trade as much as possible whenever possible. On the other hand, Kyle determines the optimal trading be- havior for the single informed agent in his model and shows that in equilibrium the agent will trade on his information only gradually, rather than exploiting it to the maximum extent possible as soon as possible.

The first contribution of this paper is to solve a version of the Glosten- Milgrom model with a single informed trader, in which the informed trader chooses his trading times optimally. We wish to give the trader a rich menu of times from which to choose, and we wish to avoid the likelihood of in- formed and uninformed trades arriving simultaneously, because the spirit of the model is that market makers execute orders individually. So, we work in continuous time. We assume uninformed trades arrive as Poisson processes. We show that the informed trader plays a mixed strategy, randomizing at each instant between trading and waiting (specifically, his equilibrium strategy is

K. BACK AND S. BARUCH

a point process with stochastic intensity). This means that informed and un- informed trades do arrive probabilistically, as Glosten and Milgrom assume. The bid-ask spread, which depends on the relative arrival rates of informed and uninformed trades, is endogenous in our model.

The second contribution of the paper is to show that this version of the Glosten-Milgrom model is approximately the same as the continuous-time version of the Kyle model, when the trade size is small and uninformed trades arrive frequently. The distinction between batching orders, as in Kyle's model, or executing individual orders, as in Glosten and Milgrom's model, turns out to be unimportant when orders are small and arrive frequently. We show that the trading strategies and profits of the informed trader and the losses of un- informed traders are approximately the same in the two models. We also show that the bid-ask spread in the Glosten-Milgrom model is approximately twice the order size multiplied by "Kyle's lambda."

Kyle shows that the equilibrium in his continuous-time model is the limit of discrete-time equilibria of a batch-auction market when the time intervals be- tween trades and the variance of uninformed trades at each date become small. Therefore, our convergence result shows that the discrete-time Kyle model is approximately the same as our version of the Glosten-Milgrom model, when the time intervals and the variance of uninformed trades are small in the for- mer, and the frequency of uninformed trades is large and the order size small in the latter.

The convergence of the informed trading strategy in the Glosten-Milgrom model to that in the Kyle model means that the frequency of informed trades in the Glosten-Milgrom model does not increase at the same rate as the frequency of uninformed trades, when the latter approaches infinity. In the limit, the informed trading strategy is absolutely continuous ("of order dt") whereas the uninformed trades are a Brownian motion ("of order fin),

even though the order size is the same for both types of traders in the Glosten- Milgrom model. In Kyle's derivation of the continuous-time equilibrium as a limit of discrete-time equilibria, this difference in uninformed volume versus informed volume is the result of the informed order being significantly smaller than the variance of uninformed trades at each date. Our results provide an alternative interpretation for the difference in volume: even though the order size for both types of traders is the same, the informed trader simply chooses to trade at a frequency that is lower than the frequency at which uninformed trades arrive.

We focus on mixed-strategy equilibria in the Glosten-Milgrom model be- cause pure-strategy equilibria cannot exist. We assume the asset value 5has a Bernoulli distribution with values normalized to zero and one. A pure strat- egy for the informed trader who knows fi = 1 would be a sequence of stopping times 71,r2,. .. , measurable with respect to public information, at which he buys or sells the asset. If in equilibrium the informed trader were to follow such a strategy and there were no order at time T,,then market makers would

assume that 5 =0 and would quote bids and asks at 0 thereafter. Consequently, when 7, is reached, the optimal strategy would be to abstain from the market at that instant and then buy an infinite amount beginning immediately afterward. We conclude that there can be no equilibrium in pure strategies.

In a mixed strategy equilibrium, an agent is of course indifferent among the various choices over which he randomizes. In our model these choices are to trade (either to buy or to sell or he may randomize over both) or to wait to trade. The willingness of the informed trader to wait to trade in this version of the Glosten-Milgrom model is analogous to his trading gradually on his information in Kyle's model.

We show that in certain circumstances the informed trader in the Glosten- Milgrom model will randomize over all the alternatives available to him, in- cluding trading in the "wrong" direction. We call this phenomenon "bluffing." Black (1990) discusses bluffing by uninformed traders who trade as if they had news, using market orders, and then reverse their trades using limit or- ders. We do not allow the informed trader to use limit orders; nevertheless, he sometimes trades as if he had the opposite information. Huddart, Hughes, and Levine (2001) also show that informed traders may bluff, but in their model, traders are required to disclose their trades ex-post, and it is this disclosure that makes it profitable to bluff. We show that bluffing can occur even when trades are ex-post anonymous. We should emphasize that the trader does not profit from bluffing, so it is perhaps incorrect to think of it as "manipulating" the market. The trader is simply indifferent in equilibrium between trading in the normal direction, waiting to trade, and bluffing. Each of the latter alternatives increases the potential for future profits at the expense of current profits.

The Kyle version of our model is analyzed in Section 1 and the Glosten- Milgrom version in Section 2. Section 3 establishes the convergence result, and Section 4 documents that there is sometimes bluffing in the Glosten-Milgrom model. Section 5 concludes the paper.

1. KYLE MODEL

We consider a continuous-time market for a risky asset and one risk-free as- set with interest rate set to zero. A public release of information takes place at a random time 7,distributed as an exponential random variable with pa- rameter r. After the public announcement has been made, the value of the risky asset, denoted by 5, will be either zero or one, and all positions are liqui- dated at that price. There is a single informed trader who knows 5 at date 0. There are also uninformed (presumably liquidity motivated) trades that arrive as a Brownian motion with volatility a'. We will call these "noise trades." All trades are anonymous. Competitive risk neutral market makers absorb the net order flow, the competition ensuring that the transaction price is always the conditional expectation of 6,given the information in orders. The informed trader recognizes that his trades affect prices through the Bayesian updating

K. BACK AND S. BARUCH

of market makers. Our goal is to determine the optimal trading behavior of the informed trader and the equilibrium price adjustments. as a function of order flow, of market makers.

This model differs from the continuous-time model of Kyle (1985) in only two ways: the announcement date is random, and the asset value is zero or one, rather than normally distributed. The assumption of a random announcement date may be more or less reasonable than assuming a known announcement date, depending on the context. Our motive for the assumption is tractability: it means that we can eliminate time as a state variable. Our motive for the distribution assumption on the asset value is that it simplifies the Glosten- Milgrom-type model. It is actually a bit of a hindrance in the Kyle-type model1

Without loss of generality, we take the initial position of the informed trader in the risky asset to be zero. We denote by XI the number of shares held at time t. We let Z, denote the number of shares held by noise traders at date t, taking Z,, = 0. We require that X be adapted to the filtration gen- erated by Z and 5. This does not literally require that Z be observed by the informed trader, because, given strictly monotone pricing, Z can be inferred by the informed trader from the price process.

Setting Y = X + Z, the net order received by market makers at time t can be viewed as dYi rdX,+ ciZ,. Risk neutrality and competition between the market makers implies that the price at time t < r is

The initial price po is the unconditional expectation of 5,and we assume 0 < Po < 1.

We assume that'

for some stochastic process H (depending on 5). We will show that there is an equilibrium in which the rate of trade of the informed trader at time t is only a function of p, and 5. We will denote the rate of trade of the high-type informed trader by H,,(p,) and the rate of trade of the low type by -OL(p,).We will show that OH > 0 and HL > 0, so OH denotes the rate at which the high type buys the security and dL denotes the rate at which the low type sells. Whether the trader has good or bad news, his order rate in this circumstance is

'Back (1992) solves the Kyle model for more general distributions than the normal, but he still requires the distribution to be continuous. We do not know of any literature on the continuous- time Kyle model with discrete distributions.

'Back (1992) shows that optimal strategies in Kyle models have this property.

In this case, the high-type informed trader's expected profit is

and the low type's expected profit is

When the informed trader's strategy is as described in the previous para- graph, the competitive pricing assumption (1.1) implies a specific form for the price process. Given (1.3), the expected rate of informed trade at time t,given the market makers' information. is

Hence. the surprise or innovation in the order flow at time t is

where

In this model. we have the standard result that the revision in beliefs is propor- tional to the surprise in the variable being observed, which means that

for some function A. Fairly standard filtering theory (we derive this in the Ap- pendix, in the proof of Theorem 1) shows in fact that, for 0 < p < 1,

We need to augment (1.7) by defining the boundary behavior at 0 and 1. New information will not change beliefs that put probability zero or one on the asset value being high. so we specify that 0 and 1 are absorbing states for p.

We assume that the informed trader believes prices evolve in accordance with (1.71, for some functions A and 4 that he takes as given. In general-i.e., without imposing (1.3)-the expected profit of the informed trader is

K. BACK AND S. BARUCH

We need to rule out strategies that first incur infinite losses and then reap infinite profits, because in that case the overall profit is undefined."n fact, we need to ensure that expected profits are well defined, so we require expected total losses to be finite. We define a strategy dX, = 8, dt to be admissible for the high type if

where as usual 8; =max(0, -0,) and we define it to be admissible for the low type if

where 8; = max(0, 8,). In these constraints, the price process p is to be un- derstood as generated by (1.7) for given functions A and 4;hence, it depends on 8.

We now define an equilibrium to be a collection {A, 4,OH, OL) such that A and 6are locally Lipschitzhnd:

(a)
given the informed trading strategy (1.3), the solution to the price dy- namics (1.7), with absorbing boundaries at 0 and 1, satisfies the competitive pricing condition (1.1);
(b)
given the price dynamics (1.7), with absorbing boundaries at 0 and 1, the informed trader's strategy (1.3) maximizes his expected profit over all absolutely continuous strategies dX,= 0, dt satisfying the admissibility con- straint (1.9).

All proofs are provided in the Appendix. A key result in the proof of the following is that, as in Kyle (1985), beliefs of market makers converge over time to the truth.

THEOREM

1: Let N(.) denote the standard normal distribution function. For each 0 < y < 1, define A*(p)by

3This is a meaningful restriction. because it will be possible for the high type to push the price to zero and then buy an infinite amount at a zero price; however doing so will first generate an infinite loss. There will be no strategies that generate an infinite gain without first generating an infinite loss.

"This guarantees the existence of a unique strong solution to (1.7), since the absorbing bound- aries rule out the possibility of a finite explosion time. See Protter (1990, p. 199).

and the selling rate of the low-type informed trader by

Then {A*, 4*,0;, 0;) is an equilibrium. At any date t prior to the announcement date r, given the price p, = p at date t, the maximum expected profit achievable by the high-~pe informed trader during the period [t, r] is

and the maximum expected profit achievable by the low-type informed trader dur- ing the period [t, r]is

Moreover, A* is a continuous concave finction on [0,11,yzmetric about p = 112,and attaining a maximum at p = 112equal to Jm.

Concavity of A is an interesting property, because it explains the patience of the informed trader. In the model of Kyle (1985), the informed trader trades gradually as here, and A is constant. The main difference between the models is that in this model the trader faces the risk of losing his informational ad- vantage with probability rdt in each instant dt. Nevertheless, he is willing to trade gradually instead of capitalizing immediately on his informational advan- tage. The reason is that, if he abstains from trade, A(p,) will be a supermartin- gale, decreasing in expectation over time. Thus, the market becomes deeper on average when he abstains from trade, and this provides sufficient benefit to offset the risk of waiting to trade. The supermartingale property follows from concavity of A(.), Jensen's inequality, and the fact that the price process p is a martingale relative to his information set when the informed trader abstains from trading. The higher is r, the higher is the risk of waiting to trade, hence the greater must be the expected decrease in h when the informed trader ab- stains from trading. The theorem is consistent with this, because the degree of concavity of A is determined by its peak dm.

The formulas (1.11) and (1.12) follow from the filtering equations (1.6) and

(1.8) when we set 4 =0. Satisfying the filtering equations implies that the com- petitive pricing condition (1.1) holds. The formulas (1.13) and (1.14) for the

K. BACK AND S. BARUCH

value functions of the informed trader can be interpreted as follows (we give a different, rigorous treatment in the Appendix). Consider the high type and consider the strategy of purchasing an arbitrarily large quantity in an arbitarily short period of time. Given that the trader is indifferent between trading and waiting, this should be an optimal strategy. Ignoring the noise trades that ar- rive in this short period of time, the informed trader will move up the market supply curve dp =A(p) dx, generating profit

The formula (1.13) is simply the cumulative profit, starting at the given price p and purchasing the asset until p = 1, at which point there is no further profit to be earned.

The key to deriving the equilibrium is the analysis of the informed trader's optimization problem. In the proof, we do not assume the Bellman equation (we provide what is called a "verification theorem"). However. we did assume it when originally deriving the formulas in the theorem, and we will sketch our analysis here, because it provides some insight into the theorem. Denote the value function for the high-type informed trader by V and the value function for the low type by J. The stationarity of the problem facing the informed trader implies that these are strictly functions of the price p. Consider the high type. If he trades at rate 8 then the expected change in p given his information is Addt -A4 dt, the volatility of p is a2A'dt, and the probability that V drops to zero is I-dt. Therefore, the expected change in the value function is

The Bellman equation states that the sum of this and the expected instanta- neous profit from trade, namely (1 -p)8dt, should be zero at the optimal 0. Because this sum is linear in 8,the maximum can be zero only if the coefficient of 8 is zero and the remaining terms sum to zero. This means

Similarly, for the low type we should have

The boundary conditions h(0) = A(1) =0 are natural to impose, given that 0 and 1must be absorbing states. Assuming enough differentiability, the con- ditions (1.15)-(1.18) and these boundary conditions determine +, A, V, and J uniquely. To put this another way, there is only one + and h for which the con- ditions (1.15)-(1.18) admit a solution in V and J, and hence only one 4 and h for which the informed trader has an optimal trading strategy (assuming, in addition to enough differentiability, that the Bellman equation is a necessary condition for optimality). The key to analyzing (1.15)-(1.18) is to observe that if we differentiate both sides of (1.16) we obtain an equation in 4 and h and their derivatives and in V', V", and V"'. Each of the derivatives of V can be calculated in terms of h and its derivatives from (1.15), so we can eliminate V', V", and V"' and obtain a differential equation in 4 and A, namely

Similarly, (1.17) and (1.18) imply the following differential equation in d and A:

These two equations have a unique solution, obtained as follows. Simply sub- tract (1.20) from (1.19) to obtain

Rearranging (1.20) also gives

so we conclude that + = 0. With 4 =0, (1.19) and (1.20) are each equivalent to

We solved (1.21) in conjunction with the boundary conditions h(0) = h(1) =0 by separation of variables to obtain the formula (1.10) for A*.

The collection {A*, +*, OX, 0;) should actually be the unique equilibrium, under assumptions sufficient to guarantee the requisite differentiability and sufficient to guarantee that the Bellman equation is necessary for optimality. A result of this sort is given in Back (1992).

K. BACK AND S. BARUCH

2. GLOSTEN-MILGROM MODEL

We consider a model similar to Glosten and Milgrom (1985). except that trades take place at random dates. As in Easley, Kiefer, O'Hara, and Paperman (1996), we assume uninformed buy and sell orders arrive as Poisson processes with constant, exogenously given. arrival intensities P.However, we will endo- genize the arrival intensity of informed trades.

We denote the order size by 6. We denote the total number of buy orders by noise traders through time t by z; and the total number of sell orders by noise traders through time t by z;, and we set z, = z: -z;. The net number of shares bought by noise traders is then zr6. Similarly, we denote the num- ber of informed buys by x;, the number of informed sells by x; and the net informed orders by x, = x: -x;. Finally, we denote the number of net noise and informed orders by y, = z, + x,.The process J reveals the complete history of anonymous trades. The a-field generated by {y,l s 5 t] is denoted by Fi. As usual, we denote the left limit of y at time t by y, and set Ay, = I,-J;-. If there is a buy order at date t then Ay, = 1. and if there is a sell order then Ay, = -1.

Competition among the market makers implies that any transaction takes place at price p, rEIGIFl']. The posted ask and bid prices at time t < r are

ask, = E[G .F;-. Ay, = \$11,

bid, = E[G 1 F;-,Ay, = -11.

Here, .F:--U,_, denotes the information available to the market makers just before time r. We also denote the left limit of p at time t by p,_. The interpretation of p, is that it is the probability market makers place on the event 5 = 1, prior to observing whether there is a trade at time t, and hence it is the expected value of 5 prior to observing the order flow at time f.The ask and bid are of course the posterior probabilities (and expecthtions), given a buy and sell order respectively.

The integral Su7[21-askr]6dxT is the sum until the announcement date r of the profit per trade [G -ask,]6, summed over the various dates at which the informed trader buys (i.e., the dates at which dx: = I), and similarly for the integral L[bid, -516dx;.It is not entirely obvious that the expectation in (2.1) exists. As in the previous section, to guarantee that expected profits are well de- fined (in the extended reals), we rule out strategies that incur infinite expected losses. We define a strategy to be admissible if

(2.2a) E L7[l -bid,] dx; < m for the high-type trader and

for the low type.

We assume the informed trader buys or sells at most one unit (i.e., 6 shares) at any point in time, since to trade multiple units would identify the trades as being informed. We search for an equilibrium in which the high-type informed trader buys the security in the instant dt with probability OH,(p,-)dt and sells with probability 8Hs(~r-)

dt and the low type buys and sells with probabilities OL5(pI-)dt and HLS(pt-)dt respectively. Specifically, we search for 9's such that the stochastic processes

and

OHy(pr-)ds -(1 -5)I'dL,(p5-)ds

are martingales, relative to the informed trader's information. In this circum- stance, the expected profit is

[1 -ask,] 60HB(pt_)dt -E [I -bid,] 60w,5(pr-)dt

for the high-type informed trader and

for the low type.

We will now solve for the bid and ask prices, assuming (2.3) and (2.4) are martingales. The probability of a buy order arriving in an instant dt is pHHB(p)dt + (1-p)OLB(p)dt +P dt,where the three terms refer to the three possible sources of an order: the high-type informed trader, the low-type in- formed trader, and uninformed traders. The ask price is the sum of the value- weighted conditional probabilities of the order coming from each of the three possible sources, namely,

K. BACK AND S. BARUCH

Thus, the ask price at time t is a(p,_). where

Similarly, the bid price must be b(p,.), where

Observe that to have a > p we must have OH, > BLHand in particular OHR> 0. Similarly, b < p implies dLJ> OHs > 0. Considering only the jumps to the ask or the bid, the expected change in p in an instant dt is

The process p is a martingale, so this expected change must be canceled by the expected change in p between orders. We conclude that between orders dp, = f (pl_)dt, where

Summarizing, the evolution of p must be given by"

As in the previous section, we take 0 and 1to be absorbing points for p.

The informed trader takes f, n, and b to be given functions and maximizes his expected profit subject to (2.8) and the boundary conditions that 0 and 1 are absorbing. As in the previous section, we denote the value fu~lction for the high-type informed trader by V and the value function for the low type by J. Because T is exponentially distributed, the value functions depend only on p,for t < T (they jump to zero at 7).

As observed above, to have n > p and b c p, we must have > 0 and dLy> 0, which means that it must be optimal for the high type to buy and the low type to sell at each time t. Because p jumps to a(p)when there is a buy

'The existence of a unique strong solution to this stochastic differential equation is guaranteed by the differentiability assumption in Theorem 2, which implies that ,f.a. and h are continuously differentiable, hencc locally Lipschitz. See Protter (1990. p. 199).

order and falls to b(p)when there is a sell order, this optimality implies

Also, it can never be better than optimal for the high type to sell or the low type to buy, and if the high type sells with positive probability or the low type buys with positive probability, then it must be optimal to do so. This is equivalent to

(2.11) V(p) 2[b(p)-116 + V(b(p)), with equality when HHS > 0,

(2.12) J(P) > -4~16+ J(a(p)), with equality when OLB > 0.

It must also be optimal to refrain from trading at each time. During an in- stant dt the announcement occurs with probability r dt and if the announce- ment occurs, then the value function becomes 0. An uninformed buy order will arrive with probability p dt, in which case p will jump to a(p) and the value function for the high type will jump to V(a(p)). Similarly, with probability /3 dt the value function will jump to V(b(p)). Finally in the absence of an an- nouncement or an order, p will change by f (p)dt and the value function will change by Vf(p) f (p)dt. For it to be optimal for the high type to refrain from trading, all of these expected changes in V must cancel. Applying this logic to the low-type informed trader also, we have

The natural boundary conditions (which hold also in our version of the Kyle model) are J(0) = V(l) = 0 and J(l) = V(0) = oo.

The following establishes the sufficiency of these conditions for equilibrium. The key step in the proof is the replication of the result of Glosten and Milgrom (1985) that beliefs of market makers converge over time to the truth.

THEOREM2: Let a, b, f, V, J, OHB, OHS, OLB, and OLS satisjj (2.5)-(2.14), with OHB > OLB and OLS > OHS. Assume the trading strategies are admissible. Assume V is a nonincreasing and J is a nondecreasingfunction of p,and V and J satisfi the boundaly conditions

lim V(p) = limJ(p)= cc and lim V(p) = limJ(p)= 0.

pi0 P-1 P-P-0

Assume the functions OHS, OHB, OLB, and OLS are contin~~ously

differentiable on (0, 1). Then the informed trading strategy is optimal and, for all t, p, = E[GlFi].

We can solve conditions (2.5)-(2.14) numerically. We consider a discrete grid on (0,l) and use a variation of value iteration to compute the value function at each p in the grid. Our numerical method is explained in Appendix B.

3. CONVERGENCE

We now consider the convergence of the Glosten-Milgrom equilibria to the Kyle equilibrium, when the trade size becomes small (6 + 0) and the noise trades arrive more frequently (P -+ m). We will take P = ~"(26~).

This im- plies that the expected squared noise trade per unit of time is 2P6' = r2,as in the Kyle model. In fact, these assumptions imply that the process of cumulative noise trades in the Glosten-Milgrom model converges weakly to the Brownian motion in the Kyle model (see, e.g., Ethier and Kurtz (1986)).

We assume the existence of an equilibrium in the Glosten-Milgrom model for each 6. We denote the equilibrum value functions by & and J;, and the equilibrium ask and bid prices by a, and bs. The equilibrium 8's also depend on 6 but our notation will not indicate it.

The following verifies the convergence of the value functions indicated by Figure 1. Part (b) establishes the relationship between the bid-ask spread in the Glosten-Milgrom model and "Kyle's lambda."

THEOREM

3: Assume the equilibria satisfj (2.5)-(2.14). Assume V, atzd Js converge in the topology of CLconvergence on compact subsets of (0,l) to func- tions V and J that are strictly monotone (V decreasing and J increasing) and satisjj the boundaly conditions

lim V(p)= limJ(p) = cx! and V(p)= lim J(p) = 0.

P-0 p41 p-io

Then:

(a) V and J must be the value functiorzs from our version of the Kyle model defined in (1.13) and (1.14);

(b) for each p E (0,I),

lim u,(P) -p = lim p -b8(~)

= h*(p),

S-+O 6 8-0 6

where A* is the market depth parameter (1.1O)from our version of the Kvle model;

(c) for each p # 1/2, there exists &p) such that for all 6 5 \$(p),either > 0 or ~HS(P)> 0.

Part (b) means that for small 6 the change in the conditional expectation of the asset value when a buy order arrives is approximately 6A*(p),and the change in the conditional expectation upon receipt of a sell order is ap- proximately -6A*(p). This is fully consistent with Kyle's (1985)interpretation of A as the market depth parameter. Figure 3 illustrates the convergence of (a, -p)/S to A*.

Part (c) means that for sufficiently small 6, either the low-type informed trader buys the asset with positive probability, or the high-type informed trader

K. BACK AND S. BARUCH

that if market makers are competitive and risk neutral, then transactions prices will be a martingale relative to their information (there is no negative serial correlation due to "bid-ask bounce") and prices in the long run will reflect the information of better informed traders. Kyle focuses on the "depth" or "liquid- ity" of the market, which depends on the amount of private information rela- tive to the volume of uninformed trading. To derive the equilibrium depth, Kyle solves for the equilibrium strategy of an informed trader. In contrast, Glosten and Milgrom assume that the arrival rates of informed and uninformed traders are determined exogenously. However, Kyle makes the simplifying assumption that the market is organized as a series of batch auctions, which is not charac- teristic of most markets. The contribution of this paper is to show the consis- tency of the Kyle and Glosten-Milgrom models. We solved for the equilibrium strategy of an informed trader in a version of the Glosten-Milgrom model, and we showed that the equilibrium of the Glosten-Milgrom model is approx- imately the same as the equilibrium of the Kyle model, when the trade size is small and uninformed trades arrive frequently.

Our version of the Glosten-Milgrom model is less tractable than the continuous-time Kyle model, and we were only able to solve for the equilib- rium numerically. Our results provide justification for using the more tractable continuous-time Kyle model, not just as an approximation to a discrete-time batch-auction market, but also as an approximation to a market in which indi- vidual trades are observed and executed by market makers, as in the Glosten- Milgrom model and as most markets are actually organized. Conversely, our results provide justification for using the Glosten-Milgrom model with ex- ogenously imposed probabilistic arrival of informed and uninformed trades, because we showed that probabilistic arrival is consistent with equilibrium be- havior when an informed trader is allowed to optimize his trading times.

John M. Olin School ofBusiness, Washington University in St. Louis, St. Louis, MO 63130, U. S.A.; back@olin. wustl.edu and David Eccles School of Business, Universig of Utah, Salt Lake City,UT 84112, U. S.A.;finsb@business.utah.edu .

Manuscript recezr,edJunuay, 2002;final re~islonrecezved June. 2003

APPENDIX A: PROOFS

PROOFOF THEOREM1: For convenience, we will write v = 5. First we will veritji the filtering equations (1.6)-(1.8). Consider the system of stochastic differential equations

where A is defined in (1.8) and 4 in (1.6). We consider this system on {t :O < p, i1). Set fi, = E[vjFT]. We need to show that p, = 5,.

Define

Set z, = Z,/u and y, = Y,/u.Then, substituting for A and 4 from (1.8) and (1.6), we can write the system of stochastic differential equations as

PI(^ -Pi)[0~(pr)

+ OL(P~)]

(A.1) dp, =

u {dh

(A.2) dq;= h,dt +dz,.

Note z is a Wiener process. Define g, =vh,.Because v is zero or one, we have v2= v and therefore

Set h, =E[h,lq]and j, =E[~,IF:].From the Fujisaki-Kallianpur-Kunita Theorem (Rogers and Williams (2000, Section VI.8)), we have

dCr=[j,-.ir,h,]{dy,-11,dt].

Directly from the definitions. we have

Therefore,

From the definition of a conditional expectation of a 0-1 random variable, 0 and 1must be ab- sorbing boundaries for fi.

Now consider OH = 0; defined in (1.11) and OL = 6'; defined in (1.12). These functions are continuously differentiable, hence locally Lipschitz and bounded on each compact subset of (0, 1).This implies there is a unique solution of the system (A.l)-(A.3) up to any finite explosion time (Protter (1990, p. 199)). But the local boundedness of the 0's implies that y can explode only at a hitting time of the boundaries for p. Hence there is a unique solution of (A.l)-(A.3) on (t:0 < p, < 1). But if (p,,y,, C,) is a solution of (A.l)-(A.3), then clearly (p,, y,, p,) is also a solution. Moreover, the boundaries are absorbing for both p and ij; therefore, p, =C, for all t.

The definition of 0; and 0; imply (1.6) and (1.8) with +* =0, so we conclude that the price dynamics stated in the theorem, dp = A* dY, imply p, =E[uI(Y,),,,] for t < T as required. It remains to show that 0;; and 0; are optimal. We will do this using the value functions given in

(1.13) and (1.14) and simultaneously show that they are the true value functions.

We will prove optimality for 0;. The proof of optimality for 0; is identical. So assume v = 1. Let V(p)denote the value given in (1.13);i.e.,

V(p)=l'k-!da.

(A.4)

h*(a)

K. BACK AND S. BARUCH

We will write h = A* through the remainder of the proof. Obviously, (A.4) implies (1.15). the first part of the Bellman equation. Furthermore. the formula for Aimplies (1.31),so we have

Integrating by parts and using the fact that A(1)= 0 gives

i'il-u)A (u)du = (1-p)A (pi -hip).

1

(A.5) rV = -u2[h+(1 -p)A'].

2

When we combine (A.5) with (1.15) we obtain (1.16), the second part of the Bellman equation. Furthermore, differentiating both sides of (1.10) and using the fact that limp_,,, A(p)= 0, we deduce that limp,,, A'(p)= x.Therefore, (A.5) implies limp.,,i V(p)= x.

Let X be an arbitrary admissible strategy of the form dX = H, dt. The price dynamics dp, = A(p,)dY,hold only for t < T (because at time T the price jumps from p,_ to v) and for t less than the first date, if any, at which p, hits the boundary. Consider a different process defined by j, = p,, and dj, = A(j,)[O,dt + dZ,]for all t > 0 and 0 < j3, iI. We maintain the condition that 0 and 1 are absorbing states. We have of course that j, = p, for t < T.

By the definition of an admissible strategy.

Since T is exponentially distributed with parameter r and independent of j and H and since pi= j, for t .c r,we have

In particular,

exists and is finite a.s. This implies that

(A.6) e?ld( 1-jl,)HJ,du

Lrn

is well defined (though possibly equal to +w). Let T denote the first time at which j hits the boundary; i.e., T = inf{tI p, = 0or I). As usual, let t A T denote min{t, T).Applying It6's lemma to the function c-"V(j,) gives

= sfhi (-rv+ 10~. ) du +if'rr't'AV'dZu.

+ ju.~'~rc

-

* 0

Substituting T.*'(j)= (j-l)/A(j)from (1.15)and rV(p)= u'A(fij2V''(p)/2

from (1.16)gives

In conjunction with the nonnegativity of V this implies

The second-term on the right-hand side of (A.8)almost surely has a finite limit as t +m,because it is an L2-bounded martingale (in fact, the L' norms are bounded by 1/2r).We have already seen that the first term has a well-defined limit. Hence, we have

and both integrals must be finite.

Consider the states of the world, if any, in which T < oo.We argue that we must have jT =1. If jT =0,then the left-hand side of (A.7) is infinite, because V(0)=m.For the right-hand side to be infinite, we must have

which implies that /:(I -j,)O,,du = -x,which is precluded by admissibility. Therefore, jT = 1. Because j, = jT = 1 for u 1T, (A.9)implies in this case that

Of course, if T = cc,then (A.9) is the same as (A.10),so we conclude that (A.lO)holds almost surely. Taking expectations throughout (A.lO)yields

because the stochastic integral, being an L2-bounded martingale, is closed on the right by its limit. Using again the fact that T is exponentially distributed with parameter r and independent of j and 0 and the fact that p, = j, for t < T,we have

So, we conclude that V(po)is an upper bound on the expected profits.

We now consider the strategy 0 = 0; (= 0") specified in the theorem. This strategy is certainly admissible because there are no losses for the high type when he never sells. We will show that the upper bound V(p,) is attained by this strategy. The key is to show that j, -t 1,implying that e-"V(F,) -t 0. Observe that ~3 is a submartingale on the informed trader's filtration, because d@,=A(jr)[OH(jr)

dt +dZ,] and OH > 0.Because the submartingale p is bounded, it converges a.s, to some p,. We have

Epx = r+x lim E lrA(J?,)O~(J?,,)

lim EB. =pa +r-x du

K. BACK AND S. BARUCH

The first equality is due to the bounded convergence theorem. The second equality is due to the fact that E i:A(j,,) dZ,, = 0 for all t. which follows from the boundedness of A. The third equality is due'& Fubini's theorem and the monotone convergence theorem. The finiteness of the improper integral on the right-hand side and the nonnegativity of the integrand im- plies E[A(p,)8,1(j,)]+ 0.Now Fatou's lemma gives us EIA(p,)Oii(p,)] = 0. \vhich implies p, E 10. I) as.

We have

and it is easy to show that

Since p = j on the event [t iTI. this implies

Now by iterated expectations and the independence of T and p we obtain E[p,(l-p,)]= E[(v-p,)']. Since p, E {0,1).p,(l -p,) 4 0 as. The bounded convergence theorem implies E[(v-i,)'I -t 0.Therefore p, -t v in the L1norm. This implies that there is a subsequence that converges to u as. However, every subsequence converges as. to p,. Therefore, p, = u. and in the case at hand p, = 1.

Now we reconsider the argument leading to (A.lO). The inequality in (A.lO) is due to the possibility that limsup,-, e-"V(p,) > 0. However for this strategy we have e-"V(j,) 4 0,so we have equality in (A.lO) and taking expectations gives

This establishes optimality and that V is the value function. Q.E.D.

PROOFOF THEOREM2: Without loss of generality we will take 6 = 1. It follows from (2.5)-(2.8) that p, = E[vl/;l. We will establish the optimality of the informed trading strategy. Consider the high-type informed trader; i.e., 13= 1. Let s=xi -x be an arbitrary admissible strategy. In analogy to the proof of Theorem 1. let j denote the solution of jo = p,, and

so j, = p, for t iT. Let T = inf{t ( p, = 0 or 1). By the definition of an admissible strategy.

Since T is exponentially distributed with parameter r. and independent of j~and x,we have

(A.11) E JTlb(p,,-1 -11dr-= r -11d,~-.

# E I' e

In particular,

exists and is finite a.s. This implies that

X

(A.12) e-'"[l -a(j,,)]dri+ 1 e-"'[b(f3)-lldx

is well defined (though possibly equal to +m).

Using the chain rule for Lebesgue-Stieltjes integrals and the law of motion for dj, and sub- stituting from (2.13), we have

The last two stochastic integrals on the right-hand side are L2-bounded martingales, the bound- edness being due to the bounds on V(a)-V(p)and V(b)-V(p)given by (2.9) and (2.11). Hence they converge almost surely to finite limits. We will denote the limit random variables by M and N. Rearranging, substituting for V(a)-V(p)from (2.9),and taking limits therefore gives

=V(po)+M+N -lim e-r""T'~(j,,T),

I-Y

assuming lim,,, e-'"'T' V(jI,T) exists. In any case, the nonnegativity of V implies that the left- hand side is dominated by V(po)+M +N. In fact the condition b-15V(p)-V(b)from (2.11) and the existence of the limit (A.12) implies

and both integrals on the left-hand side must be finite.

Consider the states of the world, if any, in which T < m. We claim that we must have \$T = 1. If jr=0,then the left-hand side of (A.13) equals +m at t = T by the assumed boundary condition V(0)=xs.For the right-hand side to be infinite, we must have

which, given that 1 -b? V(h)-V(p),implies

K. BACK AND S. BARUCH

However. this contradicts admissibility. Hence, we conclude that jjT = 1. The formulas (2.5) and

(2.6) imply a(1) = h(1)= 1.Because 1 is an absorbing point for p, we have a(p,,) = h(p,,)= 1 for u > T. Therefore (A.14)implies

(A.15) e-'"[l -a(j,,)] dx' + I' e-ru[b(\$,l_)-11dx 5 V(pil)+ M + N.

If T = m, (A.14) is the same as (A.15), so we conclude that (A.15) holds almost surely. Since the limit of an L2-bounded martingale closes it on the right, taking expectations through- out (A.15) yields

Using again the fact that T is exponentially distributed with parameter r and independent of j and x and the fact that p, = j, for t < T.we conclude that

Therefore. V(po)is an upper bound on the expected profits.

We will now show that the upper bound is attained by the strategy specified in the theorem. First we will show that under this strategy we have p, + 1 as., implying e-"V(j,) + O as., by virtue of the boundary condition V(1)= 0. The conditions (2.5)-(2.8) in conjunction with the conditions O < h < p < a < 1. BHy < HIS and OLH < HHR imply that

This is the expected change in p per unit of time conditional on fi = 1. In fact, for s < t.

where T -inf{u/j,,_E (0,1)).This implies that p, is a submartingale. Since it is bounded by one. it converges almost surely to an integrable limit p, (see Karatzas and Shreve (1988, p. 17)). In particular, if we take s = O and let t go to infinity, we get

(A.17) ~ 1 ~ k=E[pi6=~ ~

id11( 11~< X.

Since the expectation on the left-hand side is finite, the integral J{ k((i,,..) du is finite as.

If T = x,we conclude from the continuity and strict positivity of k that j, + p, E {p: k(p) = O}= 10. 1)as. If T < cx,then p, = p7 for all t T because 0 and 1 are absorbing, so in that case as well we have j, + p, E {O,1)as.

Similarly, we can show that, conditioned on fi = O. p,. is a positive supermartingale and p, -+ p, E 10, 1)a.s. The proof that p, = fi can now be completed in the same way as for Theorem 1.

Now we reconsider the arglment leading to (~.l5).The inequality in (A.15) is due to the possibility that limsup,_, e-"V(p,) =. O and the possibility that h -1 < V(p)-V(b).However, for this strategy we have e-"V(p,) 4 O and we have rlx-= 1only when h -1 = V(p)-Vib). Therefore (A.15) holds with equality and taking expectations throughout yields equality in (A.16), establishing that this is an optimal strategy. QE.D.

PROOF01:THEOREM3 MD THE COROLLARY:We adopt the hypothesis of the theoreni. First we will show that aa(p)-+ p and ha(p)-.p for each p. From (2.9), Vs(p)-I/,(us(p))+0, so V8(a6(p))+ V(p).Because the sequence {a6(p)]is bounded, each subsequence has a further subsequence with some limit q.By uniform convergence, V(q)= lim b\$(as(p))= V(p),which implies q = p by the strict monotonicity of V. Because this is true for each subsequence. the sequence (as(p)]must converge to p. The same argument using the convergence of J6 to J and equation (2.10) shows that b,(p)+p.

Now we will show that (a,(p)-p)/6 and (p-b8(p))/6have finite limits. Applying the mean value theorem to V,,we have

for some 5 between p and a6(p).Substituting from (2.9), this implies

Since as(p)4 p, we have (-. p and, by C' convergence, V,'([)+V1(p).Therefore

-il -U,(P))~

V'(p)= lim 6-0 as(p)-P

The factor 1-as(p)has the nonzero limit 1 -p, so (as(p)-p)/6 must also have a finite limit, which we will denote by A(p).Now we can write

Likewise, using (2.10), we can show that (p-b6(p))/6has a finite limit B(p)and

Now we will show that the limits A(y)and B(p)are equal. Applying the mean value theorem to V,again and using (2.11), we have, for some 6 between ba(p)and p,

Taking limits we obtain

Likewise, we obtain from (2.12) that

We conclude from (A.18) and (A.20) that A > B and from (A.19) and (A.21) that A 5B,so we must have A =B.

K. BACK AND S. BARUCH

Define A = A =B.Expand Vb(u)and v,(b)in (2.13) by exact second-order Tdylor series ex- panslons to obtaln

(A.22) rVa(p)=V,(p)[fa+P(a,+bn-2p)l

for some p < 5 < as(p)and bs(p)< 5 < p. Substituting P = &/26', we obtain from the previ- ous paragraph and the C' convergence that the last two terms each converge to

Set

It must have a limit because the other terms in (A.22) converge and the limit of V;(y) is nonzero. Denoting the limit by ~(p) -K(p)/A(p).we have

and defining d(p)=

The exact same reasoning applied to (2.14)yields

The system of equations (A.18)-(A.19) and (A.24)-(A.25) with A = B = A, is the system (1.15)-(1.18) that characterizes equilibrium in our version of the Kyle model. We now give the argument that we sketched in Section I leading to d = 0 and the differential equation (1.21). Note that from (A.18) and the fact that V is C', A must be C'.Differentiating (A.18) gives

and substituting this and (A.18) into (A.24) gives

This implies that the function 6+rr2~'/2

must be C1,because the other terms are C'.Differentiating again and substituting for V' from (A.18) gives

Repeating this argument for J gives Subtracting (A.29) from (A.28) yields

Dividing by -p in (A.29) also yields

Comparing these last two equations, we conclude that 4 = 0. We already noted that \$J + u2A'/2 must be C1. so actually it must be the case that A is C'. Moreover, equations (A.28) and (A.29) are each equivalent to (1.21).

Since A is nonnegative, the differential equation (1.21) implies A is concave. The limits limp,o A(p) and lim,,,, A(p) are therefore well defined. The boundary condition V(0) = x implies lim~up,~-, V'(p) = m. In view of (A.18) this implies lim,l,o A(p) = 0. Similarly, the bound- ary condition J(1) = 30 and (A.19) imply limp,i A(p) = 0.

The uniqueness of the solution to the differential equation (1.21) with these boundary condi- tions implies that A is the A' given by (1.10). This verifies part (b) of the theorem. Furthermore the boundary conditions V(1) = J(0) = 0 in conjunction with (A.18) and (A.19) imply that V and J are given by (1.13) and (l.l4), which is part (a).

It remains to prove part (c) and the Corollary. Fix any p E (0,l). Part (c) of the theorem is trivially true unless both HLR(p)/P + 0 and OHS(p)/P 0, so assume that these conditions hold:

4

i.e., adopt the hypothesis of the Corollary. Consider the quantity ~~(p) 0for each p.

defined in (A.23). We have shown that K8(p) From the definition (2.7) off and part (b) we have

Therefore, ~8 +O implies

The ask price can be written (see the derivation above (2.5)) as

where

and

The assumption BLB/P+ 0 implies ma + n, 1 and the fact that as -t p now leads to nS -+ 1

i

and rns + 0. From nS + 1,we conclude that

K. BACK AND S. BARUCH

Likewise. we can deduce from %H5/P+ (1 and hs+ p that

~HH\+ (1 -P)~L' +",

(A.32)

P

From (2.5),the definition of P, and (A.31), we have

Therefore, the result (as-p)/6 + A* implies

Likewise, (2.6). (A.32), and the result that (p-bs)/6+ A* imply

Write (A.30) as

~~(OHS-@~r)+ Sp(0i~-HHA) + ~(@Ls-HLH) +0.

From (A.34) the first term converges to -a2A*/(2(l -p)),and from (A.33) the second term converge5 to the same thing. Therefore,

Likewise, (A.30). (A.33), and (A.34) imply

The limit in (A.35) is the definition (1.12) of Or, and the limit in (A.36) is the definition (1.11) of 0;. This completes the proof of part (a) of the Corollary. Now write (A.30) as

6(1-~)(HLA-HHS) + ~~(OLB-4

-OHH)+ S(~HSHLB) 0.

From (A.34) the first term converges to m1A*/(2p),and from (A.33) the second term converges to -a2h*/(2(1-p)).Therefore,

Thus, for p > 112and sufficiently small 6,

and, for p < 112 and sufficiently small S.

which is part (b) of the Corollary. As noted before, part (b) of the Corollary implies part (c) of the theorem. Q.E. D.

APPENDIX B: NUMERICALMETHOD

We work with a grid of size n on [O, 11. We start with a guess for the value function V of the high type at each point on the grid, taking V(1)= 0. We interpolate linearly to compute V at all other points, except for p < lln. For p < lln,we extrapolate V as follows. We choose a normalizing function 5(p)that is also infinite at 0 and fit a quadratic function to V(p)/[(p)at p = lln, 2/n, and 3/n. We then calculate V(p)for p < l/n as (ap' + bp + cp)[(p).For the normalizing function, we use the value function (1.13)from the Kyle model. We also start with a guess f = 0 for the drift of p between orders, the equilibrium value of which is given in (2.7).

We work on the grid in the region p 2112 and use symmetry to define variables for p < 112. In accordance with part (b) of the Corollary, we conjecture (and confirm) that BLR(p)= 0 for p > 112.Note that in this circumstance equations (2.5)-(2.7)imply

where

The algorithm is as follows.

Step 1: We set J(p)= V(1- p). We compute a(p)on the grid for p > 112 from

V(p)= (1 -a(p))a+ V(a(p)),

and we compute b(p)from

We compute g(p)from (B.l).

Step 2: We check inequality (2.11).If it is satisfied. we set f(p) = g(p)If not, we set

for a suitably chosen constant E,.To estimate V'(p)in (B.4),we use a five-point approximation (Gerald and Wheatley (1999,p. 373)).Equation (B.2)is motivated by (2.13),which states that

which equals

when (2.9)holds and (2.11)holds as an equality. We then compute ? and .f by adding a small fraction of the errors in (2.13)and (2.14).Specifically, we set

(~.j) .f(p)= + P[J(a(p))+ ~(b(p))2J(p)l- rJ(p)).

J(P)+ E~{J'(P)~^(P)

Equations (B.4)-(B.5) employ the method that Judd (1998,p. 166) describes as "extrapolation."

Step 3: For p < 112,define P(p)=.f(l -p) and return to Step 1,with V = P and f = f. K. BACK AND S. BARUCH

When this converges. we have V, J. a, h, and f. We compute the 6's from a. b. and f via (2.5)-(2.7).

Our updating equation for the vaue functions can be understood as a type of value iteration. To interpret it in this wa!), consider a discrete-time model with period length It in which an uninformed buy order arrives with probability P At. an uninformed sell order arrives with the same probability. the announcement arrives with probability rIt, and no two of these events occur simultaneously. Assume that, if none of these events occurs. then p moves to p tf (p)At, which is consistent with (2.8). Assuming it is optimal to wait to trade and letting V denote the value function for the next period. the value function 6'for the current period will satisfy

If we take Lt = F:. approximate V(/I+ f(p)It) as V(p)+ V'(p)f(p)lt.and ignore terms of order (At)'. (B.6) is the same as (B.4). Value iteration is well known to c~n\~erge

in discounted dynamic programming problems (see. e.g., Judd (1998, p. 412)). and one would expect this discrete-time model to converge to our continuous-time model as It -.0. However, we are actually solving an equilibrium problem. updating the functions a, /I. and f' in each step, so we cannot appeal to standard results for convergence. Nevertheless. the algorithm did converge.

We iterated until the change in V (the maximum change across the grid points) was sufficiently small. We enforced (2.9)-(2.10). so the equilibrium conditions we need to check at the end are the inequalities (2.11) and (7.12) and the differential-difference equations (2.13) and (2.14).For the values of 6 = .5 and above shown in the figures. the inequalities hold strictly. as Figure 6 shows. For S = .2 and below. the inequality (2.11) holds as an equalip for large values of p and

(2.12) holds as an equality for small values of p, with a maximum error in all cases on the order of 10F4. In all cases. the error in (2.13) and (2.14) was less than lo-"' at each grid point. We did numerous robustness checks. We started with different initial estimates for I/, we used

different grid sizes (11 = 100, rz =400. and t7 = 1000). we used different methods of extrapolating V belo\\, 1,:n. and me tried both (B.1) and (B.6) (and the equation for j corresponding to (B.6)) as the updating equation. Provided the constants E, and c1 \Yere chosen well. the algorithm con- verged in all cases, and when it converged, it converged to the same limits.

Even though all the equilibrium conditions hold with a high order of accuracy. it appears from our plots that the 0's are not estimated very accurately when there is bluffing. This is probably an inevitable result of our estimation method, because we are estimating the 6's from f obtained from (B.3). The derivative V'(p)is small near one. where bluffing occurs. and even if V were known exactly, small errors introduced by the numerical computation of this derivatiw in (B.3) mould lead to relatively large errors in f and hence in the 0's.

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Infrequently Traded Stocks." Joiirnal of'Finnnce. 51. 1405-1436. ETHER. S.. AYD T. KURTZ(1986): Market. Processes. New York: John Wiley & Sons. GERALD. C.. WD J? WHEATLEY (1999): Applied 12'urnericctl Analysis. Reading, MA: Addison-

Wesley. GLOSTEN.L., 4UD I? MILGKOM(1985): ..Bid. Ask, and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders." Journal of Fi~~atzcial

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K. (1998): N~rrnericul Methods in Economics. Cambridge: MIT Press.

KARATZAS, I., AND S. SHREVE (1988): Brownian Motion and Stochastic Calculus. New York:

Springer-Verlag. KYLE,A. S. (1985): "Continuous Auctions and Insider Trading," Econometnca. 53, 1315-1335. PROTTER, P. (1990): Stochastic Integration and Differential Equations. New York: Springer-Verlag. ROGERS,L. C. G., AND D. WILLIAMS (2000): DifSl~sions, Murkov Processes and Martingales,

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