Insider Trading with Uncertainty about Information Sergey Zhuk Princeton University March 23, 2009

Abstract The paper considers a simple extension of the classical model presented in Kyle (1985). It is assumed that information about the value of the security changes dynamically and it is shown that because of this dynamic structure price impact of volume of trading becomes essentially non-linear. It also follows from the model that even if all underlying shocks are normally distributed, the distribution of price returns will have fat tails. Finally, it is shown that it is not possible to unambiguously compare illiquidity of two stocks and that empirically estimated Kyle’s λ, although still a useful measure of average illiquidity, can be somewhat imprecise.

1

Introduction

Market liquidity is a very active area of the Finance and Economics research. It is not very easy to give a precise definition of liquidity, but broadly speaking some asset is liquid when it can be easily traded. Some aspects of illiquidity (mostly referring to financial assets) are bid/ask spread, price impact, resiliency. Illiquidity often arises from imperfections of the market and we usually cannot deal with such issues within a standard general equilibrium framework. As a result, most work on the origins of illiquidity comes from the market microstructure literature, which studies specific mechanisms of how trading occurs in the market. Some of the recent surveys of the literature include Biais, Glosten, and Spatt (2005) and Hasbrouck (2007). More general information about markets and trading institutions can be found in Harris (2002). O’Hara (1998) and Brunnermeier (2001) discuss economic theory of

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market microstructure. Also, recently there was a lot of work on how liquidity and asset pricing are linked. If some stocks are more costly to trade then others, the question arises how this should affect prices. Amihud, Mendelson, and Pedersen (2005) survey both theoretical and empirical literature on the subject. Several potential sources of illiquidity are usually considered: transaction costs, inventory risk, private information and search costs.1 Here, we will focus only on private information. In his seminal paper, Kyle (1985) proposed a clever model explaining how private information can affect trading of a given security. His approach turned out to be very influential and inspired a lot of further work. In the simplest version of Kyle’s model two agents (informed trader and uninformed liquidity traders) trade some security with a competitive market maker. Informed trader has some private information about the value of the security, while uninformed traders trade for exogenous reasons and are insensitive to the price. Market maker can choose the price of the trade, but, importantly, she does not know as much as informed trader about the value of the security and she cannot distinguish order flows from informed and uninformed traders. As a result, market maker has to infer information about the value of the security from the trading volume. Important feature of the model is that informed trader behaves strategically, that is, he take into account the fact that his trading affects prices.2 In the paper there also considered more elaborate versions of the model (with several trading rounds and with continuous time trading). In Kyle’s model for tractability it is assumed that value of the security and order flow from uninformed traders are normally distributed. It is shown that in equilibrium price of the security p linearly depends on the aggregate order flow y p(y) = p0 + λ · y Parameter λ increases when informational advantage of informed trader becomes more severe and decreases when uniformed traders trade more. Thus, Kyle’s model presents a very simple and tractable approach to the price impact of trading. Also, now coefficient λ can be used as a measure of illiquidity of the security (obviously, the higher is λ the more costly is to trade this security). Interesting feature of λ is that it can be relatively easily estimated 1 See Amihud, Mendelson, and Pedersen (2005) or Biais, Glosten, and Spatt (2005) for a broader discussion of this issue. 2 This is different from the other approach to dealing with private information suggested by Glosten and Milgrom (1985).

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from the observed financial data. In fact, several studies addressing the issue of whether illiquidity is actually priced (Brennan and Subrahmanyam (1996); Amihud (2002)) used Kyle’s λ as a measure of illiquidity of stocks. However, there seems to be an important limitation of Kyle’s model and of the most of the subsequent literature, which we would like to try to address in this paper. In Kyle’s model (even in the dynamic versions of one) informational structure is essentially static. Market Maker always knows that there is an informed trader, who at any given moment of time has certain information and is trading on it. No new information is released during trading, value of the security does not change. We would like to argue that this can be an important limitation. It seems reasonable to assume that news about changes in the value of the security can be released at random moments of time. Thus, at any given moment market maker cannot exactly know if there is someone who has a superior information. And this can significantly affect the behavior of the market, because now market maker has also to infer from the volume of trade the fact that some information was released. In the paper, we consider a very simple model that tries to model the dynamic structure of information. In fact, the idea is not entirely new. The model used here is almost exactly the one of Admati and Pfleiderer (1988). However, their focus is not on information, but on the changes of uniformed trading in order to explain intraday trading patterns. Most results of our paper come from the new assumption, that trading usually occurs significantly faster than new information is released. Even though considered model is very simple, it cannot be solved analytically and to find the equilibrium we had to perform numerical calculations. We are able to show that price impact of volume of trading becomes essentially non-linear. For small volumes of trade price remains almost unchanged and for large volumes there can be significant deviations. One implication of this is that it becomes impossible to unambiguously compare illiquidity of two given stocks: one stock can more liquid for small trades, the other can be more liquid for larger trades. We also show that even if underlying informational shocks are normally distributed, the distribution of price returns is going to have fat tails. The paper is organized as follows. In section 2 we discuss the model and define the equilibrium. Section 3 describes the numerical procedure that was used to find the equilibrium. Section 4 presents obtained results and their interpretation. Section 5 discusses potential problems and limitations and also outlines further development. Appendix contains proofs and figures.

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2

Model and Equilibrium

The Model Assume that we have one security in the market and this security is traded in T trading rounds t = 1, . . . , T . The value of the security evolves according to Vt = Vt−1 + vt , where vt is the shock realized in the period t (V0 is given). After all trading rounds everyone gets liquidation value VT for every share he holds. We will assume that every period with probability α some information is released about the security. In this case vt is distributed according to given known distribution with density fv (it can be normal as in the original Kyle’s model). With probability 1 − α nothing happens in this trading round and vt = 0. All vt are independent across trading rounds. There are three types of agents: informed trader (insider), uniformed liquidity traders and market maker. Informed trader trades for speculative reasons and she tries to maximize her profit. She observers shock vt before trading round t. Uninformed traders trade for some exogenous reasons that are outside of the model (for example, they can be long term investors). We will assume they in each trading round their demand ut is random and drawn from some known distribution with density fu . This demand is insensitive to price. All ut are independent across trading rounds. Market maker observes shock vt after trading round t. Thus, informed trader has one period informational advantage over market maker. Market Making is assumed to be competitive, that is, market maker makes zero profit. (Or, we can assume that there are several competing market makers which will lead to the same result). We will also assume that both insider and market maker are risk neutral. Trading proceeds as follows. Insider learns vt before each trading round and then she submits her demand xt , which can depend on all previous history of signals and prices. (Insider does not observe trading volume of uninformed traders). Then, demand ut of uniformed liquidity traders is realized. Market maker observers total demand yt = xt + ut and based on this demand and all information from past periods available to her, she chooses the price pt at which all trades will occur. Market maker does not know which part of demand comes from informed traders and which part comes from uninformed.

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The Equilibrium Strategy of informed trader can be represented as X = (x1 , . . . , xT ), where xt = xt (v1 , . . . , vt , p1 , . . . , pt−1 ).3 Strategy of market marker can be represented as P = (p1 , . . . , pT ), where pt = pt (y1, . . . , yt , v1 , . . . , vt−1 ). In equilibrium: • Informed trader maximizes her expected profit π(X, P ) =

T X

E[(VT − pt )xt ] → max X

t=1

(1)

• Market maker earns zero expected profit. Thus, price reflects all the information that is available to her. pt (y1, . . . , yt , v1 , . . . , vt−1 ) = E[VT |y1, . . . , yt , v1 , . . . vt−1 ]

(2)

Since informed trader has only one period informational advantage over market maker she has to trade on her information immediately. We can rewrite expected profit of informed trader as π(X, P ) =

T X

E[(VT − pt )xt ] =

t=1

T X

E[(vt − (pt − Vt−1 ))xt ]

t=1

Thus, if strategy of market maker is uniform across trading rounds (that is if pt − Vt−1 depends only on yt ) then optimal demand of informed trader will depend only on realization of current shock vt xt (v1 , . . . , vt , p1 , . . . , pt−1 ) = x(vt ) Now consider the market maker. In trading round t past shocks v1 , . . . , vt−1 are known to her and current trading volume yt = x(vt ) + ut depends only on realization of current shocks vt and ut . Thus, y1 , . . . , yt−1 do not provide any relevant information to the market maker. As a result, we get pt (y1, . . . , yt , v1 , . . . , vt−1 ) = E[VT |y1, . . . , yt , v1 , . . . vt−1 ] = " # T X = E V0 + vt |y1, . . . , yt , v1 , . . . vt−1 = t=1 3

Here and further we will assume that things like p1 , . . . , pt−1 for t = 1 denote just an empty list. That is, x1 = x1 (v1 ).

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= V0 +

t−1 X

vt + E[vt |y1, . . . , yt , v1 , . . . vt−1 ] =

t=1

= Vt−1 + E[vt |yt ] = Vt−1 + p(yt ) where by p(yt ) we denote E[vt |yt ]. That is, in fact pt − Vt−1 depends only on yt . In this paper we are going to restrict ourselves only to such kind of uniform equilibria. Since in the considered model shocks ut are vt are uniform and independent across trading rounds and strategies of agents depend only on current realizations of shocks, we can omit time subscripts and consider a generic trading round. The model becomes essentially static. That is, we can describe it as follows. The value of the security v is with probability α distributed according to known distribution with density fv and with probability 1 − α is equal to 0. Demand from uniformed traders u is drawn from known distribution with density fu . Informed trader knows v and submit his demand x(v). Market maker chooses the price of trading p(y) based on the total volume y = x(v) + u. In equilibrium informed trader maximizes her profit and market makers price contains all information that is available to her. π ¯ (x(v), p) = E[(v − p(x(v) + u))x(v)] → max

(3)

x(v)

p(y) = E[v|y = x(v) + u]

(4)

Thus, in a way, we came to the original static model described in Kyle (1985). The only difference here is that value of the security is no longer normally distributed, but has some special distribution. But we would like to argue that this distinction is important and that this special kind of distribution arises because situation is essentially dynamic and new information is released slower than it is incorporated into prices. When analyzing the equilibrium we will be particularly interested in p(y), which we will call price impact or price response function. This function is important because it essentially describes the behavior of the market. And from the point of view of an outside investor who is willing to trade this security, this function will contain most of the relevant information regarding costs of trading this security. In the considered model it is assumed that trading occurs in rounds. In reality it usually proceeds continuously. But, we can think of a trading round here as of an interval of time τ of continuous trading, having the property that it usually takes around τ units of time for new information 6

to be incorporated into prices (or in other words informed trader has an average informational advantage of τ units of time). Assume also that new information is released approximately once in the interval of length T¯. Then, we can think of the parameter α as of probability that something happens during the interval of length τ , which would be approximately equal to the ratio of this time intervals α ≈ Tτ¯ . Based on this intuition we would like to argue that α can be relatively small and that the values of α for which results are presented in the paper are somewhat reasonable. As shown in the appendix, equilibrium conditions (3) and (4) are equivalent to the following expressions in terms of given density functions fv and fu . ˆ x(v) = arg max x

fu (y − x) · (v − p(y))x dv

´ α v · fu (y − x(v)) · fv (v) dv p(y) = ´ α fu (y − x(v)) · fv (v) dv + (1 − α)fu (y)

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Numerical solution

Even though the considered model is extremely simple and quite close to original Kyle’s version, it is hard to expect that we will be able to find an analytical solution (since we loose normality of the underlying distributions). Therefore, we are solving the model numerically.4 For numerical solution we assume that fu and fv are densities of standard normal distribution N (0, 1). The fact, that we use unitary variance is not restrictive since we can always scale the obtained results. If x(v), p(y) are equilibrium strategies for fu ∼ N (0, 1), fv ∼ N (0, 1) and x ¯(v), p¯(y) are equilibrium strategies for fu ∼ N (0, σu2 ), fv ∼ N (0, σv2 ) then the following relationships will hold

 x ¯(v) = σu · x  p¯(y) = σv · p

v σv



y σu



To avoid dealing with infinite integrals we are making some further simplifications. We assume that fv is a density of a cutoff of normal distribution on the interval [−vmax , vmax ] and also assume that market exists only as long as 4

MATLAB was used to perform the calculations. The code is available on request.

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total demand is not too big (y it lies in the interval [−ymax, ymax ]). Although, it might be arguable whether these assumptions are in fact reasonable, we would like to claim that as long as ymax and vmax are large enough, they should not affect equilibrium strategies in the relevant for us regions of variables. In the most calculations we used the values ymax = 11, vmax = 20. For this values (and for α = 0.05) probabilities that y and v will get out of bounds in equilibrium were only 3.6769 · 10−16 and 5.5072 · 10−89 correspondingly, which are fairly small numbers. The numerical calculations are organized in the following way. We start with some initial value for the price response function p0 (y). (In the calculations we used original Kyle’s solution, that is p0 (y) = y2 ). Then, assuming that we have k-th iteration pk (y) we calculate optimal response of the informed trader to this function according to the formula ˆ ymax fu (y − x) · (v − pk (y))x dy. (5) x(v) = arg max x

−ymax

After that, we calculate next iteration of price response function pk+1 (y) according to equilibrium pricing rule of market maker ´v v · fu (y − x(v)) · fv (v) dv α −vmax max pk+1 (y) = ´ vmax (6) α −vmax fu (y − x(v)) · fv (v) dv) + (1 − α)fu (y) If iterative procedure converges to some function p(y) we claim we found an equilibrium (we assume that convergence is achieved when difference between two successive iterations is smaller than chosen εp ). To calculate equilibrium strategy of informed trader x(v) we can use formula (5) again.

4

Results and Interpretation

Price Response Function Equilibrium strategies and price response function for α = 0.05 are presented in the figures 1 and 2. We see than now price depends on trading volume non-linearly. When volume of trade is small, prices almost do not change. But in rare events when we have high volume of trade price response becomes very significant. Thus, we might say that most of the time stock behaves like very liquid, but in the event of high volume, it becomes suddenly illiquid. The intuition of why such things happen can be the following. In the situation considered market maker has to infer not only what kind of information informed trader has, but she also has to infer the fact that informed 8

trader has some additional information. So, if volume is relatively small market maker may tend to believe that it is quite likely that no information was released (prior α is relatively small here). On the other hand, if market maker observers high volume of trade she might think that it is likely that something happened in the market, so she might adjust price more significantly. Figure 3 presents a graph of dependence of conditional probability that new information is released on the volume of trade to support this intuition. We might be also interested in how exactly parameter α affects equilibrium price response function of the market maker and strategy of informed trader. Results for comparative statics are shown in the figures 4 and 5. As α gets smaller the flat region around zero in price response function becomes larger and more pronounced. Also, if α = 1, we come to the original result by Kyle.

Distribution of Price Changes One way to describe behavior of the market is to look at the distribution of price changes between trading rounds. ∆pt = pt+1 − pt = vt + p(yt+1 ) − p(yt ) It is usually observed that returns (or price changes) of financial assets have distributions with fat tails. Given that now we know equilibrium price response function and strategy of informed trader we can calculate the kurtosis of the distribution of price changes (which is a measure of how fat are the tails of the distribution). Table below shows how relative kurtosis E(∆p4 ) ( E(∆p2t)2 − 3) depends on α. t

α 1.00 0.90 0.70 0.50 0.30 0.20 0.10 0.05 0.02 0.01

kurtosis of distribution of ∆pt 0.0000 0.1087 0.4560 1.1602 2.9815 5.4242 13.2424 29.8533 82.6337 174.5732 9

We see that as α gets smaller, kurtosis significantly increases. Thus, even though all informational shocks here are normally distributed, because these shocks are spread across time, the distribution of price returns is going to have fat tails.

Measuring Liquidity In the original Kyle’s model, when price response function was linear, it was straightforward that we can use the slope of this function λ as a measure of illiquidity of the stock. The higher is λ the more trading costs potential investor will incur. Unfortunately, things become more complicated in the considered model. Figure 6 shows two price response function for different values of parameters. For the first stock (solid line) new information is released more often, but every piece of information is less valuable. We see that for small volumes of trade second stock will be more liquid (price is less affected by trading), but if someone wants to trade a significant volume of shares (in the given period of time) than the first stock becomes more liquid. Thus, we would like to argue, that even if we consider only price impact of the volume (and abstract from such things as bid/ask spread) we still cannot precisely state which stock more liquid, this will depend on the needs of every specific investor. Now the question arises how to compare illiquidity of two stocks in such situation. One possible approach might be to start with some exogenous distribution (fe ) of possible trades of outside investors and calculate their average trading costs in this security ˆ Costs = fe (y) · yp(y) dy = E e [y · p(y)] If we normalize this expression by variance of trading volume we will get Costs ∼

E e [y · p(y)] E e [y 2 ]

ˆ The average value Now let us consider empirically estimated Kyle’s λ. of such estimate is going to be ˆ = E[λ]

E[y · p(y)] E[y 2 ]

which looks very similar to the expression above. However, the distribution of trades here is specific to the given market. So, if, for example, in some market 10

ˆ is going to overstate the large trades are very likely than estimated Kyle’s λ ˆ importance of such trades. Thus, even though empirically estimated Kyle’s λ seems to be still a relevant measure of average illiquidity, it can be somewhat ˆ depends on α. imprecise. Figure 7 shows how estimated λ

Profits of Informed Trader Another possibly interesting thing to look at is how expected profits of informed trader (which are equal to expected loses of uniformed traders) depend on α. Figure 8 shows overall expected profits and figure 9 expected profits from a period when some information was available to informed trader. We see that although for small α average overall profits are lower (which is reasonable since there are less trading opportunities), informed trader is able to earn more on each piece of information that is available to her.

5

Problems, Limitations and Further Work

In the considered model (as well as in the original Kyle’s model) it is assumed that informed trader cannot observe the volume of trade of uniformed. And this creates several potential problems. First of all, in such situation informed trader cannot be sure at which price his trade is going to be executed. He can expect to earn money on average, but it might happen that he will lose on a particular trade (which is a little bit unrealistic, because usually a trader can control the price of execution). Second, it seems that for such structure it is hard to prove any results about existence and uniqueness of equilibrium. And third, because of the unknown volume, price response function in the obtained results is not steadily increasing but has some small oscillations (see figure 4 for small values of α). We can consider an alternative specification where uninformed volume is observed by informed trader. And, in fact, in such case we can prove that for bounded distributions equilibrium actually exists and unique.5 Also, although it is not included in this paper, for such specification we were able to obtain similar results (but they become somewhat less pronounced). However, there is one possible objection. In such situation informed trader is always going to trade against uninformed volume, even if no new information is released. But, we would like to argue that in reality informed traders are more likely to trade when they have some information rather than when they do not have one. The reason for that might be that any trader cannot be 5

see Rochet and Vila (1994).

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sure if there are other traders in the market who have superior information. Possible way to deal with this is to consider a model with several informed traders and where only some of them learn new information if it is released. We have yet to do that. Another interesting extension would be to try to consider a more realistic model where informed trader does not have to trade immediately in his information, but the more he waits the more likely is that other market participants learn the same information.

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6

Appendix

Expressions for Equilibrium Conditions Probabilities. Denote by A the event that information is released in the current trading round (by A¯ that it is not released). Probabilities of this events are ¯ =1−α P r(A) = α, P r(A) Mutual density of v and y = x(v) + u given that event A happened is ρA (y, v) = fu (y − x(v)) · fv (v) Density of y given that event A happened is equal to ˆ ˆ fA (y) = ρA (y, v)dv = fu (y − x(v)) · fv (v) dv Density of y given that event A did not happen (in this case y = u) is fA¯ (y) = fu (y)

Market Maker. Since Market Making is competitive than p(y) = E[v|y] which means that E[p(y)h(y)] = E[v · h(y)] for any measurable h(y). Now we have E[p(y)h(y)] = ˆ ˆ ˆ = P r(A) dydv p(y)h(y)fA (y, v) + (1 − P r(A)) dy p(y)h(y)fA¯ (y) = ˆ =

 ˆ  ˆ dy h(y) · p(y) α dv fA (y, v) + (1 − α) dy fA¯ (y)

Analogously (if A¯ happened then v = 0), ˆ ˆ E[v · h(y)] = dy h(y) · α dv v · fA (y, v)

(7)

(8)

Since h(y) is an arbitrary measurable function, then expressions under the integrals in (7) and (8) should be equal. As a result,

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´ α dv v · fA (y, v) = p(y) = ´ α dv fA (y, v) + (1 − α)fu (y) ´ α dv v · fu (y − x(v)) · fv (v) ´ = α dv fu (y − x(v)) · fv (v) + (1 − α)fu (y) Informed trader. If no information is released than insider is not going to trade at all. Expected profit of the insider given v and given that event A has happened π ˜ (v) = EA,v [(v − p(y)) · x] = EA,v [(v − p(x + u)) · x] = ˆ = du fu (u) · (v − p(x + u))x du = ˆ =

dy fu (y − x) · (v − p(y))x dy

Insider maximizes her profit. Thus, ˆ x(v) = arg max dy fu (y − x) · (v − p(y))x x

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Figure 1: Price response function (α = 0.05) 4 α = 0.05 3

2

p(y)

1

0

−1

−2

−3

−4 −8

−6

−4

−2

0 y

15

2

4

6

8

Figure 2: Strategy of informed trader (α = 0.05) 8 α = 0.05 6

4

x(v)

2

0

−2

−4

−6

−8 −8

−6

−4

−2

0 v

16

2

4

6

8

Figure 3: Conditional probability of informed trading (α = 0.05)

1 0.9 0.8 0.7

Pr[A|y]

0.6 0.5 0.4 0.3 0.2 0.1 α = 0.05 0

−6

−4

−2

0 y

17

2

4

6

Figure 4: Price response function. Comparative statics 4

3

α=1 α = 0.5 α = 0.05 α = 0.01

2

p(y)

1

0

−1

−2

−3

−4 −8

−6

−4

−2

0 y

18

2

4

6

8

Figure 5: Strategy of informed trader. Comparative statics 8

6

α=1 α = 0.5 α = 0.05 α = 0.01

4

x(v)

2

0

−2

−4

−6

−8 −8

−6

−4

−2

0 v

19

2

4

6

8

Figure 6: Comparison of price response functions 3 α = 0.7 σv = 0.6 α = 0.1 σv = 1 2

p(y)

1

0

−1

−2

−3

−6

−4

−2

0 y

20

2

4

6

ˆ Figure 7: Estimated Kyle’s λ 0.7

0.6

0.5

λ

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5 α

21

0.6

0.7

0.8

0.9

1

Figure 8: Overall expected profits of informed trader 0.7

0.6

0.5

E[π]

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5 α

22

0.6

0.7

0.8

0.9

1

Figure 9: Expected profits of informed trader given that new information is released 1.8

1.6

1.4

E[π|A]

1.2

1

0.8

0.6

0.4

0

0.1

0.2

0.3

0.4

0.5 α

23

0.6

0.7

0.8

0.9

1

References Admati, A. R., and P. Pfleiderer (1988): “A Theory of Intraday Patterns: Volume and Price Variability,” The Review of Financial Studies, 1(1), 3–40. Amihud, Y. (2002): “Illiquidity and stock returns: cross-section and timeseries effects,” Journal of Financial Markets, 5(1), 31–56. Amihud, Y., H. Mendelson, and L. Pedersen (2005): “Liquidity and asset pricing,” Foundation and Trends in Finance, 1, 269–364. Biais, B., L. Glosten, and C. Spatt (2005): “Market microstructure: A survey of microfoundations, empirical results, and policy implications,” Journal of Financial Markets, 8(2), 217–264. Brennan, M. J., and A. Subrahmanyam (1996): “Market microstructure and asset pricing: On the compensation for illiquidity in stock returns,” Journal of Financial Economics, 41(3), 441–464. Brunnermeier, M. K. (2001): Asset Pricing under Asymmetric Information: Bubbles, Crashes, Technical Analysis, and Herding. Oxford University Press, USA. Glosten, L. R., and P. R. Milgrom (1985): “Bid, ask and transaction prices in a specialist market with heterogeneously informed traders,” Journal of Financial Economics, 14(1), 71–100. Harris, L. (2002): Trading and Exchanges: Market Microstructure for Practitioners. Oxford University Press, USA. Hasbrouck, J. (2007): Empirical market microstructure: the institutions, economics, and econometrics of securities trading. Oxford University Press, USA. Kyle, A. S. (1985): “Continuous Auctions and Insider Trading,” Econometrica, 53(6), 1315–1335. O’Hara, M. (1998): Market Microstructure Theory. Wiley. Rochet, J., and J. Vila (1994): “Insider Trading without Normality,” The Review of Economic Studies, 61(1), 131–152.

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