# MARKET STRUCTURE AND INSIDER TRADING. Keywords: Insider Trading, Stock prices, Correlated signals, Kyle model

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6 6 WASSIM DAHER AND LEONARD J. MIRMAN v = v A The Equilibrium Concept. In this section, we introduce the equilibrium notion used in the paper. To begin, we define conditions to be satisfied for a Bayesian-Nash equilibrium for this model. Then we restrict our search to a linear equilibrium. The equilibrium conditions are: (i) Competition between market makers drives their expected profits to zero. (ii) The strategy for each informed trader is chosen to maximize expected profits. Note that in this paper, we assume that condition (i) which is called Market Efficiency is fulfilled and derived from unmodeled reasons. A Bayesian-Nash equilibrium of this model is a vector of four functions [y(.), X 1 (.), X 2 (.), P (.,.)] such that: (1) (2) (3) (4) (a) Profit maximization of the manager, E Z [((a by)yz P (X 1 (z) + X 2 (z) + U))X 1 (z)] P Z a.s. E[((a by )y z P (X 1(z) + X 2 (z) + U))X 1(z)], for any level of output y produced by the firm and any alternative trading strategy X 1 ; (b) Profit maximization of the owner, E Z [((a by)yz A P (X 1 (z) + X 2 (z) + U))X 2 (z)] P Z a.s. E[((a by)yz A P (X 1 (z) + X 2(z) + U))X 2(z)], for any alternative trading strategy X 2, (c) Semi-strong market efficiency, the pricing rule P (.,.) satisfies, P (q, R) = E[v q, R], almost surely. An equilibrium is linear if there exists constants µ 0, µ 1, µ 2 such that, q, r, P (q, r) = µ 0 + µ 1 q + µ 2 r. Note that conditions (1) and (2) define optimal strategies of the insiders, while condition (3) guarantees the zero expected profit for the market maker. 3. Optimality Results We have a game of incomplete information because the market makers, unlike the insiders do not know the realization of Z. Now, we solve the maximization problem of each insider, given the assumption of linear equilibrium. The manager chooses (y, X 1 ) to maximize expected profit, given Z = z. So, Max (y,x 1 ) G 1 where G 1 = E Z [v P ]X 1 + A.X 1 = E Z [v P ]X 1. This is equivalent to, Max [(a by)z µ 0 µ 1 (a by)z µ 2 (x 1 + X 2 (z))]x 1, (y,x 1 ) P Z a.s.

7 MARKET STRUCTURE AND INSIDER TRADING 7 The first order necessary conditions are, (5) and We obtain X 1 (z) = (a by)(y µ 1)z µ 0 µ 2 X 2 (z) 2µ 2 X 1 Z(a 2by + µ 1 b) = 0. y = (a + µ 1b) (6). 2b The owner chooses an order X 2 to maximize his expected utility, given Z = z. So, Max X 2 G 2 where G 2 = E Z [v P ]X 2. This is equivalent to Max x 2 [(a by)yz A µ 0 µ 1 (a by)z µ 2 (X 1 (z) + X 2 )]x 2 The first order necessary condition is, (7) X 2 (z) = (a by)(y µ 1)z µ 0 A µ 2 X 1 (z) 2µ 2 Solving for X 1 and X 2 from (5) and (7), we obtain, (8) and (9) X 1 (z) = (a by)(y µ 1)z µ 0 + A 3µ 2 X 2 (z) = (a by)(y µ 1)z µ 0 2A 3µ 2 Lemma 1. Let X = X 1 + X 2, X is normally distributed and, therefore, R is normally distributed. Proof: By adding (8) and (9), we find that X is a linear function of Z, which implies that it is normally distributed. Since X 1 and X 2 are independent of U (by assumption), X is also independent of U and thus the sum R, is normally distributed. Lemma 2. The random vector B = (v, q, R) is normally distributed and, therefore, E[v q, R] = µ 0 + µ 1 q + µ 2 R. Proof: First, note that y is deterministic, hence, v and q are normally distributed. Second, every linear combination of the elements of B, is normally distributed. Thus, B is a normal random vector. Finally, we can apply Theorem 1 in the Appendix to the random vector B and then the result is proved. Let us now determine the coefficients µ 0, µ 1 and µ 2. Lemma 3. Under competition in the stock sector, we have, (i) µ 0 = v µ 1 q µ 2 R,

8 8 WASSIM DAHER AND LEONARD J. MIRMAN (ii) µ 1 = a b k, (iii) µ 2 = a2 σ ε (1 k) 3 k where k = 2bσ U 3 = 3 2a 2 σεσ 4 z bσ u(σ, z+6σ 2 ε) 2 2 σ 2 Z (σ 2 Z +6σ2 ε). Proof: See Appendix. Note that after calculating the value of µ 0 from Lemma (3) and substituting in (8), we get, X 1 (z) = (a by)(y µ 1)(z Z) + 3A (10) P Z a.s. 3µ 2 Let us now write the second order condition for each insider. For the manager, the second order conditions are, (11) (12) 2bX 1 (z)z > 0, µ 2 > 0, P Z a.s. and the Hessian determinant is positive. For the owner, (13) µ 2 > 0. Proposition 1. A linear equilibrium exists if and only if, [b(y µ 1 ) 2 (z Z) + 3A]z > 0, P Z a.s. Proof: Substituting the value of X 1 in equation (11), we obtain the result. Notice that Proposition 1 does not ensure the satisfaction of the second order condition for all possible values of Z. Indeed, since the condition in Proposition 1 is quadratic in Z, there exits values of Z such that the second order condition is not satisfied. Now, we present a necessary and sufficient condition (11) for the existence of the linear equilibrium for all Z. Lemma 4. A linear equilibrium exists if and only if, A = µ 0 = (a by)(y µ 1) Z. 3 Proof: We start by proving sufficiency. If A = µ 0, then the expression for X 1 in equation (8) becomes, X 1 (z) = (a by)(y µ 1)z 3µ 2 Thus, the second order condition, given in Proposition 1, is satisfied. Hence, from Proposition 1, there exists a linear equilibrium. Next, we prove the necessary condition. Indeed, since there exists a linear equilibrium, the second order condition of Proposition 1 is satisfied for all Z. The second order condition has the following form: cz 2 + dz > 0. Here c = b(y µ 1 ) 2 > 0 and d = 3A b(y µ 1 ) 2 Z. So, we must have d = 0 to satisfy the second order condition (11), and then the result is proved.

9 MARKET STRUCTURE AND INSIDER TRADING 9 Remark 3.1. First, note that A is positive. Second, if Z is a centered normal random variable ( Z = 0), there is no need, as in JM, for a compensation scheme to satisfy the second order condition. Indeed, if Z = 0, then A = 0. Hence, in this case, by Proposition 1, a linear equilibrium exists and is symmetric, in the sense that the two insiders have the same trading orders. It should be pointed out that the effect of the compensation scheme is to center the manager s (random) profit function around 0, this is also the case of the insider in JM. This has the effect of aligning the objective of the shareholder and the manager. However, there is no need for a compensation scheme for the owner since the owner has no effect on the output decision of the firm. Thus, the random profit function of the owner is centered at the mean Z, as is the case without production e.g., in Kyle. So when Z = 0, both the profit functions of the manager and the owner are centered around zero, the mean of Z, and there is no need for a compensation to the manager. Lemma 4 provides several interesting and important insights. First, as in Kyle but not JM, µ 0 is different from zero. This is due to the presence of competition in the stock sector, which changes the form of the price function. Second, recall that in Kyle, the price is uninformative and equal to the mean value of the asset if and only if the total order flow is zero. This result is due to the fact that the order flow has a zero mean. In contrast, in our model and like JM, the price is uninformative and equal to the mean value of the asset if the two signals are different from zero and equal to some constants. These constants are calculated in Proposition 2. In other words, as in JM the two signals convey information even when they are zero. This was not noted by JM. In both our model and in JM, the mean of the order flow is not equal to zero. Proposition 2. The price of the stock is uninformative, that is p = v, if Z + ε = Z and R = A Z µ 2 Proof: Since we deal with linear price, it is straightforward to prove this result. The next proposition presents the unique equilibrium of the model. Proposition 3. A linear equilibrium exists. The equilibrium is unique and characterized by the following values of X 1, X 2, y and P. (i) y = (a+µ 1b) 2b, (ii) X 1 (z) = (a by)(y µ 1)z 3µ 2 (iii) X 2 (z) = (a by)(y µ 1)z 3µ 0 3µ 2 (iv) X = X 1 + X 2 = 2(a by)(y µ 1)z 3µ 0 3µ 2 (v) P (q, R) = µ 0 + µ 1 q + µ 2 R, (vi) µ 0 = A = (a by)(y µ 1) 3 Z,

10 10 WASSIM DAHER AND LEONARD J. MIRMAN (vii) µ 1 = a b k, (viii) µ 2 = a2 σ ε (1 k) 3 k where k = 2 = 3 2a 2 σεσ 4 z 3bσ U bσ u(σ, z+6σ 2 ε) 2 2 σ 2 Z (σ 2 Z +6σ2 ε), µ 1 (0, a b ) and µ 2 > Comparative Statics In this section, we present several comparisons between the results of our model and the results of JM. Note that these comparisons highlight the effects of a change in the market stucture on the real and financial sectors as well as the decisions of the insiders Equilibrium variables and coefficients. In this paragraph we compare the coefficients and the variables of the unique equilibrium of Proposition 3 to those of JM. Note that the notation jm refers to the JM model. We, then have the following results, Lemma 5. Comparing our result to JM, we have, (i) µ jm 0 = 0 < µ 0, (ii) µ 1 < µ jm { 1, µ2 > µ jm (iii) 2, when σ2 ε is small relative to σz 2, µ 2 < µ jm 2, otherwise, (iv) y < y jm, (v) X 1 < X jm. Proof: Recall that µ jm 0 = 0 which is less than µ 0 (by Lemma 3 and the fact that Z is positive). Comparing µ jm aσz 1 = 2 b(σz 2 +4σ2 ε) with µ 1 given in Lemma 3, it follows that µ 1 < µ jm 1. A similar comparison between µ jm 2 and µ 2 results in (iii). Note that from Proposition 3, it follows that, y = (a + µ 1b) and y jm (a + µjm 1 = b). 2b 2b It is clear that both real outputs have the same expressions but by (ii) of this Lemma we get the result. For (v), substituting for y, µ 1 and µ 2 from Proposition 3 into the expression for X 1 yields 2σU (14) X 1 (z) = z 2σ Z which is less than the trading strategy of the monopolist in JM, where, (15) X jm (z) = σ U σ Z z, P Z a.s. Lemma 6. Comparing the Owner s trading strategy and the compensation scheme to JM, we have,

14 14 WASSIM DAHER AND LEONARD J. MIRMAN greater than µ 1. Hence, from the expression of the compensation schemes we can infer that the compensation in JM is less than in this paper. Remark 4.1. First, if we compare our stock price s coefficients µ 1 and µ 2 in JMC, we find that µ 1 takes the same value in both models. Indeed, compare equation (37) in the appendix to the expression of µ 1 in Proposition 2, Section 3 in JMC (after correcting this expression). In these two models, both the structure of the coefficient µ 1 as well as the output of the two firms differ. These differences cancel each other and, thus, the two models yield the same µ 1. Second, note that the compensation scheme of the manager in our model is always greater than JMC. Indeed, recall the expressions of the compensation scheme in JMC, A c = (a bµc 1 )2 Z, 18b where the superscript c denotes Cournot. Comparing A c and A, since the two models have the same µ 1, we infer that the compensation of the manager under the competition in the real sector is always less than in our model. Next, we analyze the effects of the demand parameters (a and b) on the equilibrium stock pricing rule, real output and the trading of each insider. Proposition 4. The effect of the demand parameters on the equilibrium outcome, is, (i) The coefficient µ 0 changes with demand parameters if and only if a2 b changes. (ii) The coefficients µ 1 and µ 2 vary positively with a and negatively with b. (iii) Real output varies positively with a and negatively with b. (iv) Manager and owner s trading are independents from a and b. (v) Stock price changes if and only if a2 b changes. Proof: (i), (ii), (iii) follow from the expression of µ 0, µ 1, µ 2 and y stated in Proposition 3. The result in (iv) follows from the expressions of X 1 and X 2 presented in (14) and (16). Finally, substituting the expressions of µ 0, µ 1 and µ 2 on the expression of equilibrium stock price and using the fact that q = Z + ε and R = X 1 + X 2 + U we obtain the result in (v). The general properties of the stock pricing function are the same as in JM Informativeness of stock price. In this section we examine the extent to which the stock price reveals information. This issue was originally studied by Kyle (1985) as well as Rochet and Vila (1994). They found that the stock price reveals exactly half of the inside information, regardless of the parameter values of the model. Moreover, recall that in Kyle s model with more than one informed trader, the amount of information revealed increases in the number of traders, 3 but is still a constant and is independent of the parameter values of the model. This independence result, is due to the fact that the insider has only one option, i.e., either to buy or sell stock. In particular, the insider(s) in these 3 In fact for two traders the amount of information revealed is exactly 2. See Tighe (1989). 3

15 MARKET STRUCTURE AND INSIDER TRADING 15 use the direct method instead models has no effect on the value of the firm. It is shown in JM that allowing the manager to have an effect on the value of the firm, together with the availability of a second signal to the market maker, changes the result. In particular, in JM, the amount of information released by the insider is greater than half, and, depends on the parameters of the model. In this paper, we show that the stock price reveals more information than in JM, since competition in the stock sector makes the order flow more informative than in JM. On the other hand the informational effect of the real signal is the same as in JM. Recall that the addition of a signal from the real sector, has the effcet of increasing the amount of information recieved by the market maker. However, as discussed below, competition in the stock market has no affect on the informational content of this signal. Following JM, we adopt the same measure of information i.e., the conditional variance of Z, given the information of the market maker. This can be written as, 4 V ar(z q, R) = σz 2 µ 1 σ Zq µ 2 σ ZR = (y µ 1) σ 2 3y Z. 4 See Theorem 1 in the Appendix.

17 MARKET STRUCTURE AND INSIDER TRADING 17 multiplies Z + ε, while the effect on the stock signal is only a multiplier of Z. However, the signal from the real sector, that is q, cannot be manipulated by the manager after observing the realization, z, of Z, since the output chosen by the manager is deterministic and, thus, is independent of Z. Second, recall that in JM, the manager can manipulate the total order flow, since the total order flow is the sum of his own trade and the noise trades. In contrast, to JM, under competition in the stock sector, the manager is less able to influence the total order flow (recall that the two insiders share the same information). It should be noted that this discussion shows that the coefficients µ 1, µ 2 do not, in general completely reflect the informational content of their respective signals, since, although the real signal in the two models have the same informational content, µ 1 < µ jm 1. This point was made in JMJ, The stock pricing coefficients µ 1 and µ 2 depend in a complicated way on the variances of the two signals as well as their covariances with the value of the firm. Another complicating factor in interpreting µ 1 and µ 2 is that some of these variances and covariances are also functions of µ 1 and µ Finally, the intuition for Proposition 5, can also be seen by comparing equations (18) and (19), the conditional variances in our model and in JM, respectively. There are two influences on the conditional variance. The first is the effect of the information contained in the order flow and real signals. This effect shows up in the functional form of the conditional varaince. Hence, for the same µ 1, the functional form of JM is greater than in our model. The second is in the values of µ 1 and µ jm 1. Since, µ 1 < µ jm 1, the two effects compliment each other. Remark 4.2. In JMC, even in the presence of the competition in the real sector, it is shown that the stock price reveals the same level of information as in JM. 5 That is, (a µ c 1 b) 2(a + 2µ c 1 b)σ2 Z, where the superscript c denotes Cournot. This reduces to, σ 2 ε σz 2 + σz. 2 2σ2 ε This is due to that fact that, the real signal, as discussed above, contains no added information and the order flow in JM and JMC are the same. In this case the value of µ 1 in JMC is influenced by the fact that there are two firms controlling the real signal and the coefficient µ 2 comes from a signal with the total output which is only being traded. However, this difference is accounted for when taking the conditional variance in both models, which is the same in the two models since the informational content of the signals are the same. 5 Indeed, they find that stock price reveals more information than JM. But, the fault comes from the calculation of µ 1 in the Appendix which must be equal to the µ 1 of this model. After correcting the expression of µ 1 and substitute it in the conditional variance, we obtain the same expression (equation (20)) as in JM.

18 18 WASSIM DAHER AND LEONARD J. MIRMAN Remark 4.3. Note that, if the market maker receives only one signal, the total order flow, then the level of information revealed is the same as in Kyle (1985) going from one to two traders. 6 Proposition 5 highlights the difference between competition in the real sector and competition in the stock sector with respect to the informativeness of the stock price. Indeed, under competition in the stock sector, the stock price reveals more information than the competition in the real sector. This is due to the fact that the change in market structure in JMC has no informational value i.e., more information is not generated in the equilibrium of JMC although there are changes in the equilibrium values of the model. While the changes due to competition in the stock sector affects both the real and the trade signal, although only the change in the stock signal affects the informativeness of the stock price. Note that the amount of information revealed varies with σ 2 Z and σ2 ε, a result similar to JM and contrast to Kyle Insiders Profits Manager s Profit. In this section, we present two results related to the manager s profits. First, we show that the manager s profits in our model are not always less than in JM but they are greater than in JMC. Second, we prove that the profits of the manager in JMC are less than in our paper, which are less than in JM, when the market maker receives only one signal, the total order flow. In order to prove these results, we calculate the profits of the manager in the presence of two insiders. Substituting for y, X 1, µ 0, µ 1, µ 2 from Proposition 3, in the manager s profit function, we obtain, (22) which reduces to, (23) G 1 (z) = (a bµ 1) 4 z 2 144b 2 µ 2 G 1 (z) = 3 2a 2 (σ 2 ε) 2 z 2 σ U 2bσ Z (σ 2 Z + 6σ2 ε) 2. P Z a.s. Recall the insider s profits in JM and JMC are respectively, (24) and (25) Π jm (z) = (a bµjm 1 )4 z 2 64b 2 µ 2 Π c (z) = (a bµ 1) 4 z 2 324b 2 µ 2 6 See Tighe (1989). Indeed, it is sufficient to calculate the conditional variance with the variables in Proposition 8 of the next section. It is easy to check that the level of information is For more details, see JM (2000).

19 MARKET STRUCTURE AND INSIDER TRADING 19 which reduce to, (26) and (27) Π jm (z) = Π c (z) = 2a2 (σ 2 ε) 2 z 2 σ U bσ Z (σ 2 Z + 4σ2 ε) 2, 2a2 (σ 2 ε) 2 z 2 σ U bσ Z (σ 2 Z + 6σ2 ε) 2. P Z a.s. P Z a.s. A comparison between these expressions leads to the following result, Proposition 6. Π c < G 1 { G1 > Π Proposition 7. jm when σε 2 is small relative to σz 2, G 1 < Π jm otherwise The proof of Propositions 6 and 7 is obvious and therefore is omitted. Proposition 6 shows that the manager s profits with competition in the real sector are always less than the manager s profit with competition in the stock sector. To see how competition in real sector affects the firm s value, consider equations (22) and (25), representing the respective profits in our model and in JMC. Since, both models have the same µ 1, the functional form of the profits in JMC has the same numerator as in our model. However, the total order flow coefficients, that is µ 2, in both models, are related in such a way that makes the denominator in the profit function of the manager in JMC is greater than in our model. The reason for this result is that competition in the real sector reduces the firm s value, which is the financial asset of the stock market while competition in the stock sector does not always reduce the value of the firm. Hence, competition in the real sector reduces the value of the firm and, thus, the profits of the manager while competition in the stock sector has an ambiguous effect on the profits of the firm as as on the profits of the manager. Adding a second informed trader to JM does not always lower the manager s profits relative to JM. 8 Indeed, Proposition 7 shows that the profits of the manager in this model are greater than in JM when σε 2 is small, relative to σz 2 and they are less than JM, when, σε 2 is large, relative to σz 2. The intuition for this result is that when σ2 ε is large, relative to σz 2, the unconditional net profits of firm 1, is less in this model than in JM. In contrast, when σε 2 is small, relative to σz 2, the inverse is true, that is the net profits of firm 1 in JM is less than in this paper. Note that when σε 2 is large, relative to σz 2, the total order flow signal r becomes more informative relative to the real signal q and hence this result is similar to the Kyle type models going from one to two insiders (see Tighe 1989). 8 It should be pointed out that in JMJ, going from one to two insiders, the equilibrium is symmetric, i.e., the trading levels of the two insiders are equal and yield equal profits, which are less than the insider s profits in JMJ. This result is due to the absence of the real market, which is studied in this paper but not in JMJ.

20 20 WASSIM DAHER AND LEONARD J. MIRMAN In order to make a proper comparison, we now consider as Kyle (1985), the case when the market maker receives only one signal, the order flow. Everything else remains unchanged in our model. The profit function of the manager is G 1,k = E Z [v P ]X 1 + AX 1, where v = (a by)yz A and the profit function of the owner is, The stock price is set as follows, G 2,k = E Z [v P ]X 2. P (R) = E[v R] = µ 0 + µ 1 R. Solving for this equilibrium we get the following result, Proposition 8. If the market maker observes only the total order flow, the equilibrium variables become, (i) y = a 2b, (ii) X 1 (z) = (a by)yz 3µ 1 (iii) X 2 (z) = (a by)yz 3µ 0 3µ 1 (iv) P (R) = µ 0 + µ 1 R, (v) µ 0 = A = (a by)y Z, (vi) µ 1 = 3 2(a by)yσz 3σ U. Proof: Following the same method of section 3 for solving the equilibrium variables, and Kyle (1985) to derive the expression of µ 1, we get the result. We can now write the expression of G 1,k, G 1,k (z) = This leads to the following result, Proposition 9. G 1 < G 1,k Proof: the proof is straightforward. a2 z 2 σ U 12 2bσ Z P Z a.s. This result is similar to those obtained in JM and JMC. This is due to the fact that in these models the market maker receives more information 9 than in Kyle. Hence, he is able to get a better fit of the stock price to the actual value of the firm, thus, the manager s profits from trading, are less than in a Kyle type model. Consequently, the manager s profits when the market maker receives only one signal (the total order flow), are more than the manager s profits when the market maker receives the two signals. 9 The signal from the real sector and the total order flow.

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