Wassim Daher
email: daher.w@gust.edu.kw.

Fida Karam
email: karam.f@gust.edu.kw.

Leonard J Mirman

Joseph Abdou

Bernard Cornet

Jean-Marc Bonnisseau

Insider Trading with Different Market Structures

Keywords: JEL classification: G14, D82 Insider Trading, Cournot, Stackelberg, Correlated signals, Kyle model

 with two insiders under three different market structures: (i) Cournot competition among the insiders, (ii) Stackelberg game between the insiders and (iii) Monopoly in the real market and Stackelberg in the financial market. We show how the equilibrium outcomes are affected by each of the market structure. Finally we perform a comparative statics analysis between the models.

Introduction

Insider trading is considered as one of the most notorious methods of stock fraud, in which an individual involved with a company whose stock is publicly traded, shares or sells information with regard to an event involving that company. Typically, this information is not only unavailable to the public, but is also expected to alter the behavior of the company's stock, resulting in profit opportunities of illegal nature. Insider trading has had some landmark incidents or scandals that helped define exactly what it is, starting with the Chase National Bank case in 1929. Newer scandals can also be cited such as the Enron Corporation Case in 2004 and the New Castle Funds case in 2010. All these scandals reflect the different types of inside-relationship and outside-connection of corporations.

The recurrence of insider trading scandals has renewed the academic interest in this topic. Much of the theoretical research has centered on two issues: the public desire of implementing regulations and laws which prohibit insider trading [START_REF] Dow | Informed Trading, Investment and Welfare[END_REF], [START_REF] Leland | Insider Trading: Should it be Prohibited[END_REF], [START_REF] Manove | The Harm from Insider Trading and Informed Speculation[END_REF]) and the dissemination of information that was captured by [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF] in his seminal work on insider trading.

The work of [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF] has constituted the cornerstone for subsequent works on the informational effect of insider trading. In the Kyle model, there is an insider who knows the value of the stock and a market maker who only knows the distribution of the values of the stock, gets information from the total noisy stock order flow, and sets the stock price in a way that his expected profits are zero . The main result is that the stock price reveals half of the inside information, regardless of the parameter values. When the marker maker is allowed to observe, in addition to the total order flow, another signal of the value of the asset, like in [START_REF] Jain | Insider Trading with Correlated Signals[END_REF], the stock price becomes more informative and the insider's profits decrease with respect to [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF]. Jain andMirman (2000, 2002), Daher andMirman (2006, 2007), [START_REF] Wang | Stackelberg Financial-Leader in Insider Trading Model[END_REF] and [START_REF] Wang | Stackelberg Real-Leader in an Insider Trading Model[END_REF], then explored, the impact on information revelation, of various types of speculative markets in the spirit of [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF] and [START_REF] Jain | Insider Trading with Correlated Signals[END_REF], by modeling the financial and real sectors together. The idea behind the introduction of the real sector is that the insider's information is due to his participation in the real activities underlying the financial assets that are the object of insider trading. In [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF], the insider (or trader) is also the manager of the firm acts as a quantity-setting monopolist in the real sector. Cournot competition in the real sector is added to this model in [START_REF] Jain | Effects of Insider Trading under Different Market Stuctures[END_REF]. Both papers show that the amount of information incorporated in the stock price, which is the same in both papers, increases with respect to [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF], but is the same as in [START_REF] Jain | Insider Trading with Correlated Signals[END_REF]. In other words, the introduction of the real sector does not affect the amount of information revealed. However, the insider's profits are, in both papers, lower than [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF]. The insider's profits and his compensation scheme are lower in [START_REF] Jain | Effects of Insider Trading under Different Market Stuctures[END_REF] with respect to [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF], due to Cournot competition in the real sector. Daher andMirman (2006, 2007), [START_REF] Wang | Stackelberg Financial-Leader in Insider Trading Model[END_REF] and [START_REF] Wang | Stackelberg Real-Leader in an Insider Trading Model[END_REF] extended [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF] and ( 2002) to include competition in the financial sector. Indeed, there are different types of insiders in the firm, some without any managerial responsibilities (the president and the members of the board of directors, for example), with the objective of maximizing their profits from trading the stock of the firm whose inside information they possess. Therefore, the competition among insiders is another form of competition that influence the amount of information disseminated in the stock price. [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF] and (2007) show that Cournot competition in the financial sector between the owner and the manager increases the amount of information incorporated in the stock price (which is the same between the two papers) with respect to [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF] and (2002). The profits of the manager sometimes increase and sometimes decrease, relatively to Jain andMirman (2000, 2002), depending on the variances of the exogenous variables. However, the profits of the manager and the owner are less in the Cournot-real case [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF] than in the monopoly-real case [START_REF] Daher | Market Structure and Insider Trading[END_REF]. [START_REF] Wang | Stackelberg Real-Leader in an Insider Trading Model[END_REF] introduce Stackelberg competition in the real market to the model of [START_REF] Daher | Market Structure and Insider Trading[END_REF] and support the conclusion that competition in the real sector does not affect the degree of information revelation. In [START_REF] Wang | Stackelberg Financial-Leader in Insider Trading Model[END_REF], the manager in the lower ladder of the organizational hierarchy takes the order from the owner and makes the decisions in the Cournot-real sector. He acts as a Stackelberg-follower in the financial sector to the owner who is high on the organizational hierarchy and who knows the manager's reaction function. The authors show that Stackelberg competition in the financial sector increases the amount of information revealed and the owner's profits, in comparison with [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF]. The manager's profits may decrease or increase depending on the exogenous parameters of the model.

In this paper, we offer an in-depth study to the effect of the financial market structure on the revelation of information. We start by a natural extension of [START_REF] Jain | Insider Trading with Correlated Signals[END_REF] to include Cournot duopoly in the financial market (Model I). We find that each insider looses the market power and partially controls the stock price. Hence, the stock price reveals more information with respect to [START_REF] Jain | Insider Trading with Correlated Signals[END_REF]. The unconditional profits of each insider also decreases. Those results are similar to Daher and Mirman (2007), when compared to [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF]. We then try to add a little more reality to Model I, and assume that one of the insiders, the owner, is high in the organizational hierarchy and chooses the second insider, the manager, to serve his purpose. In other words, we model Stackelberg competition in the financial market, where the owner is the leader and knows the reaction function of the manager. This type of financial information asymmetry between the owner and the manager is widely observed in real life. We show that with Stackelberg competition in the financial market (Model II), the manager trades less and hence earns less than in the Cournot case. However, the owner, due to her role as leader, trades more than in the Cournot and earns more profits. We also notice that the price reveals more information in the Stackelberg than in Cournot structure. We finally take into consideration the real activities of the firm whose stock is traded and assume that the owner, who acts as a leader in the financial model, does not have any managerial responsibilities. Instead, the manager who acts as a follower in the financial market is a quantity-setting monopolist in the real market (Model III). We show that the introduction of the real market does not affect the amount of information revealed with respect to Model II. The same result holds in [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF], when compared to Jain and Mirman (1999).

The paper is structured as follows: in Sections 2, 3 and 4, we present respectively Model I, II and III with their comparative statics. In Section 5, we conclude.

Model I : The Cournot Case

Let (Ω, F, P) be a probability space, all the random variables are defined with respect to this probability space. In this model we introduce Cournot competition between two insiders into the model of Jain and Mirman (1999). 2Consider an economy with one financial asset, the stock of the firm. There are three types of agents trading in the financial market. First, there are two risk-neutral rational traders who know the realization z of z, the value of the stock. Second, there are (non-rational) noise traders, representing small investors with no information on z. The aggregate noise trade is assumed to be a random variable ũ, which is normally distributed with mean zero and variance σ 2 u . Finally, there are K(K ≥ 2) risk-neutral market makers who act like Bertrand competitors.

We assume, as in [START_REF] Jain | Insider Trading with Correlated Signals[END_REF], that the market makers observe two signals, a noisy signal about the value of the firm, denoted by q = z + ε, where ε is normally distributed with mean zero and variance σ 2 ε , and the total order flow signal. We assume that z, ũ and ε are pairwise independent.

Following [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF], the trading mechanism is organized in two steps. In step one, a linear pricing rule and optimal order rule are determined by the market makers and the insiders, respectively, as a Bayesian Nash equilibrium. The market makers determine a (linear) pricing rule p, based on their a priori beliefs, where p is a measurable function p : R × R -→ R. Each insider i(i = 1, 2) chooses a stock trade function xi = x i (z), where

x i : R -→ R is a measurable function. In the second step,3 the insiders observe the realization z of z, and submit their stock order to the market makers based on the equilibrium stock trade functions. The market makers also receive orders from the noise traders, all these orders arrive as a total order flow signal r = i x i (z) + ũ. The two signals are used by the market makers to set the price p = p(q, r), based on the equilibrium price function, to clear the market. The insiders know only the realization z of z and does not know the values of ũ, ε, r, z + ε before the order flow decisions is made. Moreover, each market maker does not know the realization z of z but only knows its distribution. Finally, the market makers cannot observe either x i , u or ε. This is a game of incomplete information because the market makers, unlike the insider, do not know the realization of z. Hence, we seek a Bayesian-Nash equilibrium. A Bayesian-Nash equilibrium is a vector of three functions [x 1 (.), x 2 (.), p(., .)] such that:

(a) Profit maximization of insider i, E(z -p(z + ε, xi + x-i + ũ))x i ≥ E(z -p(z + ε, x i + x-i + ũ))x i (1)
for any level of trading order x i decided by the insider i and x-i is the trading order of the other insider ;

(b) Semi-Strong Market Efficiency: The pricing rule p(., .) satisfies,

p(q, r) = E[z|q, r]. ( 2
)
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 Condition (1) defines the optimal strategies of the two insiders while Condition (2) guarantees the zero expected profits for the market makers. The stock price, set by the market makers, is equal to the conditional expectation of the asset given the available information. We restrict our study to linear equilibrium. The normal distributions of the exogenous random variables, together with the particular expression of the demand, enable us to derive and to prove the existence of a unique linear equilibrium. Next in Proposition 1, we characterize the unique linear equilibrium.

Proposition 1 In the insider trading Cournot model with two signals, a linear equilibrium exists and is unique. It is characterized by

x1 = x2 = (1 -µ 1 )(z -z) 3µ 2 (4)
µ 0 = (1 -µ 1 )z, µ 1 = σ 2 z σ 2 z + 3σ 2 ε , µ 2 = √ 2σ 2 ε σ z σ u (σ 2 z + 3σ 2 ε ) (5) V ar(z|q, r) = σ 2 ε σ 2 z + 3σ 2 ε σ 2 z ( 6
)
and

π i (z) = σ u σ 2 ε (z -z) 2 √ 2σ z (σ 2 z + 3σ 2 ε ) i = 1, 2 (7)
Proof: See Appendix A Discussion of the equilibrium In this model we extend the Jain and Mirman (1999) model with one insider (monopoly) and two signals to the case of two insiders (Cournot) and two signals. The strategic competition between the two insiders alters the equilibrium outcomes. Indeed, Lemma 1 whose proof is omitted, shows that on the one hand, Cournot competition in the financial market decreases the trades of each of the insiders as well as their conditional profits. On the other hand, Cournot competition increases the total order flow and therefore increases the amount of information revealed by the stock price. In the process, the price function coefficients are also affected by Cournot competition. Formally,

Lemma 1

The effects of Cournot competition on the equilibrium outcomes relative to the monopoly case are given by,4

µ 0 > µ JM 0 µ 1 < µ JM 1 xi < xJM π i < π JM ∀ i = 1, 2 (8)
V ar(z|q, r) < V ar(z|q, r) JM (9) First note, adding another insider to Jain and Mirman (1999), decreases each the insider's trades and thus their conditional profits. But the total insider trade is increased as are the total profits. Moreover, this result also occurs in Daher and Mirman (2007) as compared to [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF], in both of these models, real decisions are endogenous. Moreover, Tighe (1989) and Holden and Subrahmanyam (1992) obtained the same result when they introduced Cournot competition between the insiders to the monopoly model of [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF] where the market maker observes only the total order flow signal.

µ 2 < µ JM 2 if σ 2 z < 3 -2 √ 2 √ 2 -1 σ 2 ε and µ 2 > µ JM
Second, strategic competition among the insiders has a direct effect on information revelation. Indeed, with two insiders the stock price conveys more information than in the case of a single insider. Indeed, the expressions of the conditional variances measuring the amount of information revealed in the stock price in Jain and Mirman (1999) and in our model are:

V ar(z|q, r) JM = σ 2 ε σ 2 z + 2σ 2 ε σ 2 z and V ar(z|q, r) = σ 2 ε σ 2 z + 3σ 2 ε σ 2 z
The key difference between these two conditional variances is the coefficient of σ 2 ε in the denominator. The greater the value of this coefficient, the more the amount of information revealed in the stock price. The origin of this value is the aggregate orders of the insiders. The aggregate order in the Cournot case is greater than in Jain and Mirman (1999). 5 It should be pointed out that with additional real market structure together with Cournot competition among the insiders, [START_REF] Daher | Market Structure and Insider Trading[END_REF] showed that the stock price reveals more information than in the monopoly case studied in [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF], also with additional real market structure. Moreover, with only one signal ( the total order flow) Tighe (1989) and Holden-Subrahmanyam (1992) found that under Cournot competition among the insiders, the stock price reveals more information than in the monopoly case studied by [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF].

Third, Cournot competition in the financial market alters the price function coefficients. In the presence of Cournot competition among the insiders and when the market maker observes zero value for both signals, he still receives

5 x = x1 + x2 = √ 2σu σz (z -z) xJM = σu σz (z -z)
more information than in the monopoly case (µ 0 > µ JM 0 ). On the other hand, the response µ 1 of the market maker to the noisy signal is lower in this model than in [START_REF] Jain | Insider Trading with Correlated Signals[END_REF]. In order to understand this result recall the expression of µ 1 in Jain and Mirman (1999) and in our model, i.e.

µ JM 1 = σ 2 z σ 2 z + 2σ 2 ε and µ 1 = σ 2 z σ 2 z + 3σ 2 ε
which depends on the value of the firm, the noisy signal and the total order flow (through the coefficient of σ 2 ε ). Hence, as the number of insiders increases, the total order flow increases. Thus the coefficient of the noise in the denominator increases which lowers the value of µ 1 . This result reflects the fact that with two insiders there is more information in the order flow signal which gets more weights, and thus makes the value of µ 1 lower than in Jain and Mirman (1999).

Finally, the coefficient of the total order flow signal µ 2 is affected by the relationship between the exogenous variances σ 2 z and σ 2 ε . When the noise signal is too noisy, i.e. when σ 2 ε is large relative to σ 2 z , the market maker can not extract information from the noise signal and thus the model resembles to the Kyle type models with one signal, in which the market depth measure (the coefficient of the total order flow signal) decreases as the number of insiders increases (see [START_REF] Tighe | Three Essays on Insider Trading[END_REF]). However, when the noise signal transmits valuable information to the market maker, (when σ 2 ε is not relatively large with respect to σ 2 z ) together with the zero expected profit condition, the value of the total order flow coefficient increases with respect to the monopoly case. 6

Model II : The Stackelberg Case

In this model we introduce Stackelberg competition among the two insiders in the financial market. Specifically, we assume that one of insiders, the owner, is high on the organizational hierarchy and acts as a Stackelberg leader in the financial market. The second insider, the manager, is in the lower ladder of the organizational hierarchy and acts as a Stackelbergfollower in the financial sector to the owner who knows his reaction function before making any decision. Proposition 2 characterizes the unique linear equilibrium. Formally, Proposition 2 In the Stackelberg financial model with two correlated signals, a linear equilibrium exists and is unique. It is characterized by

x1 = (1 -µ 1 )(z -z) 4µ 2 , x2 = (1 -µ 1 )(z -z) 2µ 2 (11)
µ 0 = (1 -µ 1 )z, µ 1 = σ 2 z σ 2 z + 4σ 2 ε , µ 2 = √ 3σ 2 ε σ z σ u (σ 2 z + 4σ 2 ε ) (12)
V ar(z|q, r) =

σ 2 ε σ 2 z + 4σ 2 ε σ 2 z ( 13
)
π 1 (z) = σ u σ 2 ε (z -z) 2 √ 3σ z (σ 2 z + 4σ 2 ε ) , π 2 (z) = 2σ u σ 2 ε (z -z) 2 √ 3σ z (σ 2 z + 4σ 2 ε ) (14)
Proof: See Appendix B

In this model, we study the effects of Stackelberg competition in the financial market on the equilibrium outcomes. [START_REF] Wang | Stackelberg Financial-Leader in Insider Trading Model[END_REF] were also interested by Stackelberg competition among the two insiders but with the presence of Cournot competition in the real market.

Extending Jain and Mirman (1999) (monopoly) to the Stackelberg case has two effects. On the one hand, it enables us to study the effect of Stackelberg competition between the insiders as compared to the Cournot case studied in the first model of this paper. On the other hand, it allows us to perform a comparative static analysis with the Kyle-type models when Stackelberg structure is adopted.

It is well known in industrial organization theory that in the presence of Stackelberg competition, the leader produces more than in Cournot and thus makes more profits. The follower produces a lower quantity than in Cournot and earns less. 7 Lemma 3 provides a complete comparative static analysis between the Cournot competition model ( the model I presented in the previous section) and the present model. It shows that the leader (the owner) trades more and earns greater profits than in the Cournot setting and that the follower (the manager) trades less and makes lower profits. Moreover, Stackelberg competition increases the amount of information revealed in the stock price. Finally, the stock price coefficients are also affected by Stackelberg competition. Formally

Lemma 2

The effects of Stackelberg competition on the equilibrium outcomes, compared to the Cournot-duopoly case are given by,

8 x1 < xM1 1 , π 1 < π M 1 1 , x2 > xM1 2 , π 2 > π M 1 2 (15)
7 For more details, see Julien (2011). 8 The superscript M1 refers to Model I.

x1 + x2 > xM1 1 + xM1 2 , π 1 + π 2 > π M 1 1 + π M 1 2 (16)
µ 0 > µ M 1 0 , µ 1 < µ M 1 1 , V ar(z|q, r) < V ar(z|q, r) M 1 (17)
µ 2 < µ M 1 2 if σ 2 z < 4 √ 2 -3 √ 3 √ 3 - √ 2 σ 2 ε and µ 2 > µ M 1 2 otherwise (18)
First, Stackelberg competition between the two insiders reduces the manager's trades and thus conditional profits decrease with respect to the Cournot case. However, the owner, due to his role as a leader, trades more than in the Cournot case and earns more. Moreover, the total trade of the insiders as well as the total insiders' conditional profits are greater in Stackelberg than in Cournot. All these results are consistent with the industrial organization theory.

However, when a real market is introduced to the model as in [START_REF] Wang | Stackelberg Financial-Leader in Insider Trading Model[END_REF], i.e. when real decisions are made endogenously, Stackelberg competition among the insiders does not always decrease the follower's (manager) profits when compared to [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF], where Cournot competition is introduced in both the real and financial markets.9

Second, Stackelberg competition among the insiders increases the amount of information revealed in the stock price compared to the Cournot case. To understand this result better, recall the expression of the total order flow signal,

r = β + γ z + ũ, (19)
where the noise term ũ, in the total order flow signal is homoscedastic10 (see equation [START_REF] Wang | Stackelberg Real-Leader in an Insider Trading Model[END_REF]. Hence, the greater the deterministic part γ of the total order flow signal, the more informative is the total order flow signal. Adding a second informed trader to Jain and Mirman (1999) (i.e, model I ) or introducing Stackelberg competition between the two informed traders (model II), changes the values of the deterministic part of the total order flow signal. Since, the noise term (ε) in noise signal (q = z + ε) is heteroscedastic and multiplicative, any change in the financial market structure affects only the total order flow signal. Thus, such change alters the market makers' information about z only through the total order flow signal.

Comparing the total order flow signal coefficients of model I and II, we find that γ in model II is greater than γ in model I. 11 Consequently, adding Stackelberg competition in the financial sector increases the level of information incorporated in the stock price set by the market makers with respect to the Cournot case.

Finally, note that both types of competitions (Cournot or Stackelberg) in the financial market alter the stock price coefficients. We have already shown that Stackelberg competition in the financial market leads to a wider dissemination of information through the stock price with respect to Cournot competition. This explains the lower value of the stock price noise signal µ 1 in Stackelberg than in Cournot. Moreover, due to the negative relationship between the noise signal coefficient µ 1 and the intercept coefficient µ 0 (which results from the zero expected profits of the market maker), we infer that the value of µ 0 in Stackelberg is greater than in Cournot.

Note also that the order flow signal coefficient µ 2 depends on the exogenous variances of the model. Lemma 3 shows that when the noise signal is relatively noisy, the total order flow coefficient of the stock price is greater in Stackelberg than in Cournot. This result is due to the fact that in the case of a noisy signal, the market maker only extracts information from the total order flow signal. To highlight the relationship between the value of µ 2 , the exogenous variance, the market structure and the number of insiders, recall that the noise signal is heteroscedastic and multiplicative and given by q = z + ε

It should be pointed out that this signal can not be manipulated by the insiders. In other words, competition (Cournot or Stackelberg) among the insiders in the financial market, does not alter the coefficient of the noise term (it is always equal to one). Thus, the market maker extracts the same information from the noise signal regardless the competition structure in the financial market. However, introducing competition in the financial market, alters the deterministic part γ, of the total order flow signal (see equation [START_REF] Wang | Stackelberg Real-Leader in an Insider Trading Model[END_REF].

From equation 18, the variance of the stock value σ 2 z and the variance of the noise signal σ 2 ε are the determinants of the effect of the competition structure on the total order flow response µ 2 . When σ 2 z is relatively small with respect to σ 2 ε , the total order flow signal becomes more relevant to the market maker than the noisy signal. Thus, the value of µ 2 in the presence of Stackelberg competition is greater than the value of µ 2 in the presence 11 Indeed, from Proposition 1, we infer that xM1 = xM1

1 + xM1 2 = √ 2σu σz (z -z) while Proposition 2 shows that xM2 = xM2 1 + xM2 2 = √ 3σu σz (z -z).
of Cournot. However, when both signals provide relevant information to the market maker, the zero profit condition makes the value of µ 2 in the Stackelberg case lower than the value of µ 2 in Cournot. Now, we turn to the role of information sources in the Stackelberg case. In other words, we study the effects of information efficiency when the market maker observes two correlated signals instead of the total order flow signal alone. For this purpose, we present the benchmark model which is identical to our model except that the market maker only observes the total order flow signal. In other words, the benchmark model studies Stackelberg competition among the insiders in the Kyle-type models. Proposition 3 whose proof is omitted, characterizes the unique linear equilibrium of the benchmark model.

Proposition 3

In the Stackelberg model with one signal (the total order flow), a linear equilibrium exists and is unique. It is characterized as follows

x1 = (z -z) 4µ 1 , x2 = (z -z) 2µ 1 (20)
µ 0 = z, µ 1 = √ 3σ z 4σ u , p = µ 0 + µ 1 r (21) V ar(z|r) = 1 4 σ 2 z (22) π 1 (z) = σ u (z -z) 2 4 √ 3σ z , π 2 (z) = σ u (z -z) 2 2 √ 3σ z (23)
A simple comparison between Propositions 2 and 3 shows that adding a signal that is correlated with the value of the asset and that can not be manipulated by the insiders, increases the amount of information revealed by the stock price. However, the additional signal does not affect the trading strategies of each insider. In other words, with one or two signals, each insider i trades the same quantity xi (i = 1, 2). Moreover, adding a correlated signal to the informational structure always decreases the insiders' profits.

Model III : Monopoly in the real market

In this section, we study the effect of introducing decisions in the real sector on the results of model II, i.e., model III studies Stackelberg competition in the financial sector and monopoly in the real sector. It adds a real sector to model II and hence allows us to study the effect of the real market on equilibrium outcomes. Specifically, we consider an economy with one real good and one financial asset. The real good is produced by a monopolistic firm at no cost. The inverse demand function is linear and stochastic, i.e., q = (a -by)z a, b > 0 (24)

where a and b are positive constants and z is a random variable, normally distributed with mean z (assumed positive) and variance σ 2 z . The stock of the firm is publicly traded in a competitive financial market. The value of the stock is the net profits of the firm per share. We assume that the manager and the owner of the firm are the two insiders who trade in the stock market based on their inside information. Moreover the owner has no managerial responsibilities. As in the previous models, we suppose that there exist noise traders whose aggregate demand is denoted by the random variable ũ with mean 0 and variance σ 2 u . We follow Jain and Mirman (2000) by allowing the market makers to observe two signals. The first, from the real market, is denoted by q = (a -by)(z + ε) where ε is normally distributed with mean 0 and variance σ 2 ε . The second is the total order flow, i.e., r = x1 + x2 + ũ where x1 and x2 are the manager's and the owner's trading orders, respectively.

Each insider knows only the realization z of z and does not know the values of ũ, ε, r, z + ε before his order flow decisions are made. Moreover, neither market maker knows the realization z of z but both knows its distribution. Finally, the market makers cannot observe either x i , u or ε.

The value per share of the firm is the net profit of the firm per share, i.e., v = (a -by)yz a, b > 0 (25)

and the profit of the manager and the owner are respectively

ψ 1 = (v -A -p) x1 + A x1 (26)
ψ 2 = (v -A -p) x2 ( 27
)
where A is the manager's compensation scheme. 12 We assume as in model II and [START_REF] Wang | Stackelberg Financial-Leader in Insider Trading Model[END_REF], that the owner moves first because he knows the manager's reaction function before making any decision. Proposition 4 characterizes the unique linear equilibrium, the amount of information revealed by the stock price and the conditional expected profits of the two insiders.

2 otherwise ( 10 )

 210 Before analyzing the results, let us highlight the relationship between the present model and the existing models. As noted, our model adds a second insider to Jain and Mirman (1999) and hence we study the effect of Cournot competition in the financial market. Daher and Mirman (2007) also studied Cournot competition in the financial market with respect to[START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF], but both of these models incorporate decisions made in the real sector and therefor both models have two signals. Finally,[START_REF] Tighe | Three Essays on Insider Trading[END_REF] and[START_REF] Holden | Long-Lived Private Information and Imperfect Competition[END_REF] studied the Cournot extension of the Kyle model with only one signal the total order flow.

Documents de Travail du Centre d'Economie de la Sorbonne -2011.56

It should be pointed out that the results can be automatically generalized to n insiders, but for the sake of comparison we restrict our attention to the two-insiders case.

Random variables are denoted with a tilde. Realized values lack the tilde. The mean of the random variable is denoted with bar.

The superscript JM refers to Jain and Mirman (1999). 6 Documents de Travail du Centre d'Economie de la Sorbonne -2011.56

For more details, see[START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF], the discussion after Lemma 3, p. 540-541. 8 Documents de Travail du Centre d'Economie de la Sorbonne -2011.56

See Wang el al (2009) page 129, Proposition 4.

See Creane (1993). 10 Documents de Travail du Centre d'Economie de la Sorbonne -2011.56

Following Jain and Mirman (2000), the existence of such compensation scheme is to ensure the second order condition of the manager's maximization problem. For more details, see[START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF] and the extended works thereafter.

Documents de Travail du Centre d'Economie de la Sorbonne -2011.56

In model II, the manager's strategy is an affine function while in our model it is a linear function passing through the origin.

The superscript WWR refers to[START_REF] Wang | Stackelberg Financial-Leader in Insider Trading Model[END_REF].

Documents de Travail du Centre d'Economie de la Sorbonne -2011.56

 Proposition 4In the insider trading Stackelberg model with a monopoly structure in the real market, a linear equilibrium exists and is unique. It is characterized by

Proof: See Appendix C

The relationship between the two models, II and III, is similar to the relationship between Jain and Mirman (1999) and [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF]. Indeed, Jain and Mirman (2000) added a real market to Jain and Mirman (1999) and investigated the effect of making decisions in the real market on the equilibrium outcomes. They found that several variables are affected by the real market structure. Specifically, in equilibrium, the order strategy of the owner and the amount of information revealed are both independent of the real market structure.

We obtain similar results. First, the conditional variances in both models I and II are the same. In other words, introducing a real market in model II has no effect on information revelation. This is due to the non-randomness of the real output variable (y).

Second, simplifying the expression of the order strategy of the two owners in both models I and II (see equations 11 and 28), we obtain

Hence, the owner's strategy is independent of the real market structure.

Note that, the introduction of the compensation scheme, decreases the manager's strategy compared with model II. 13 Moreover, it should de pointed out that going from one signal to two signal does not alter the insiders' strategies.

We analyze the effect of the real market structure on the equilibrium outcomes, i.e., we compare model III to Wang et Cournot competition in the real sector generates no new information when compared to the monopoly case. This is due to the randomness of the real output variables in both the monopoly and duopoly case. Moreover competition in the real sector does not affect the trading strategy of the insiders. Jain and Mirman (2002) who studied Cournot competition in the real sector and monopoly in the financial market, obtained the same results compared to Jain and Mirman(2000) who modeled a monopoly in both the real and financial market. Similarly, Daher and Mirman (2006) studied a Cournot competition in both the real and financial markets. They showed that the insiders' trades, as well as the information revealed are not affected by the real market structure compared to the monopoly case in the real market studied by [START_REF] Daher | Market Structure and Insider Trading[END_REF].

Second, Cournot competition in the real market decreases the profits of the firm with respect to the monopoly case. Such a decrease in profits must be incorporated into the stock price function of the market makers. This explains the lower values of the price coefficients in Wang el al (2009) compared to model III. Also, the compensation scheme is affected by the monopoly structure in the real sector. Since the value of the compensation scheme is half the value of µ 0 , we deduce that the compensation scheme is greater in this model. In sum, competition in the real sector reduces the unconditional expected value of firm 1 and, thus, the coefficients of the stock price signals are less with the presence of Cournot competition in the real sector than in the monopoly case.

Finally, in the line with the industrial organization theory, we find that the amount of output produced in the monopoly case is less than in the Cournot case studied in [START_REF] Wang | Stackelberg Financial-Leader in Insider Trading Model[END_REF]. We also find that the insiders' profits under Cournot duopoly are less than those under monopoly . It should be pointed out that these results also hold in Jain and Mirman (2002) which studied Cournot competition in the real sector and monopoly in the financial market when compared to Jain and Mirman(2000) which modeled monopoly in both the real and financial market. Similarly, [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF] when studying Cournot competition in both the real and financial markets, obtained the same results as compared to the monopoly case in the real market studied by [START_REF] Daher | Market Structure and Insider Trading[END_REF].

Appendix

In this appendix we recall the Theorem that we use to prove all the results in the three cases of the paper.

Theorem 1 If the p × 1 vector Y is normally distributed with mean U and covariance V and if the vector Y is partitioned into two subvectors such that

Proof : See Graybill, Theorem 3.10 pp 63.

Appendix A: proof of Proposition 1

First note, in the Cournot case the equilibrium is symmetric, i.e. x1 = x2 .

Taking the first and the second order conditions of the insider maximization problem (equation 1), we find

and µ 2 > 0. Substituting the expression of µ 0 = z -µ 1 q -µ 2 r into equation (38), we obtain the insiders strategies in (4). Now, applying Theorem 1 to the normal random vector B = (z, q, r) with p = 3 and m = 1. By identification, we have

Where V 11 = σ 2 z , V 12 = (σ zq , σ zr ), V 21 = σ zq σ zr and V 22 = σ 2 q σ qr σ qr σ 2 r .

Note that

where D is the determinant of V 22 , that is D = σ 2 q σ 2 r -σ 2 qr .

Substituting the corresponding values for the variances and covariances, we find

Combining equations ( 41) and (42) we obtain

Now computing for the expression of D, one can find that

Arranging for (42) and (44), we obtain

Solving for ( 43) and ( 45) we obtain

Plugging ( 46) into (43) we obtain the expression of µ 2 in (5). Applying Theorem 1 to compute the conditional variance we find that

Finally, substituting the values of xi , µ 1 and µ 2 , we find the expression of the insiders conditional profits in (7) 2

Appendix B: proof of Proposition 2

In this appendix, the main change with respect to appendix A is the reaction curves derivation of each insider. The remainder is similar and thus omitted. Indeed, following Stackelberg structure, the manager moves second and hence his reaction curve is given by First note that the value of the firm is now ṽ and thus adjusting Theorem 1 with this new information and using structures detailed in appendices A and B, we derive equations ( 28) and (29). Second, the conditional variance becomes

V ar(z|q, r) = σ 2 z -µ 1 (a -by)y σ zq -µ 2 (a -by)y σ zr and equation (31) follows. Finally, equation (32) follows automatically after submitting all the required expressions. 2


PhD Lotte Gerritsen
email: l.gerritsen@donders.ru.nl

Indira Tendolkar

Barbara Franke

Alejandro Arias Vasquez

Sabine Kooijman

Jan Buitelaar

Guillén Fernández

Mark Rijpkema

BDNF Val66Met genotype modulates the effect of childhood adversity on subgenual anterior cingulate cortex volume in healthy subjects

Keywords: Brain-derived neurotrophic factor, Childhood adversities, Voxel-based morphometry, Imaging Genetics, Mood Disorders, Anterior Cingulate Cortex

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Introduction

The experience of childhood adversity (CA) constitutes a major risk factor for psychiatric disorders [START_REF] Kessler | Childhood adversity and adult psychiatric disorder in the US National Comorbidity Survey[END_REF]. Early life stress has also frequently been associated with structural brain alterations, particularly in those brain areas that mature late, such as the hippocampus and prefrontal cortex (PFC) [START_REF] Cohen | Early life stress and morphometry of the adult anterior cingulate cortex and caudate nuclei[END_REF][START_REF] Frodl | Interaction of childhood stress with hippocampus and prefrontal cortex volume reduction in major depression[END_REF][START_REF] Mcewen | Plasticity of the Brain in Relationship to Stress[END_REF]. Such structural brain alterations may be caused by a lack of neurotrophic factors, whose synthesis is reduced in the aftermath of stress [START_REF] Calabrese | Neuronal plasticity: a link between stress and mood disorders[END_REF]. For instance, it has been shown in rodents that stress reduces brain-derived neurotrophic factor (BDNF) levels in the hippocampus and PFC [START_REF] Roceri | Postnatal repeated maternal deprivation produces agedependent changes of brain-derived neurotrophic factor expression in selected rat brain regions[END_REF]. Also in humans, hippocampal and prefrontal BDNF levels are lower in depressed persons having committed suicide compared to control subjects (7;8).

There is however a large inter-individual variability in stress susceptibility. The leading explanation for this variability proposes that genetic variance modulates the effect of stress on the brain (9;10). Indeed, numerous studies have shown that persons with a specific genetic make-up are more prone to develop stress-related mental disorders after experiencing major life events like CA [START_REF] Caspi | Genetic sensitivity to the environment: the case of the serotonin transporter gene and its implications for studying complex diseases and traits[END_REF][START_REF] Hage | Vulnerability to depression: what is the role of stress genes in gene x environment interaction?[END_REF][START_REF] Xie | Interaction of FKBP5 with childhood adversity on risk for post-traumatic stress disorder[END_REF]. Moreover, recent studies suggest that gene by stress interactions may be involved in inter-individual differences in local brain volume [START_REF] Frodl | Childhood stress, serotonin transporter gene and brain structures in major depression[END_REF] and may also play a role in the onset of several psychiatric disorders, like bipolar disorder, schizophrenia, attention deficit hyperactivity disorder and depression (for review see [START_REF] Wermter | From nature versus nurture, via nature and nurture, to gene x environment interaction in mental disorders[END_REF]). In case of depression it has been suggested that gene by stress interactions may explain up to 60-70 % of the variance in disease susceptibility [START_REF] Hasler | Pathophysiology of depression: do we have any solid evidence of interest to clinicians?[END_REF]. A common functional polymorphism in the BDNF gene (rs6265, Val66Met) seems to be a relevant candidate for such an effect [START_REF] Gatt | Interactions between BDNF Val66Met polymorphism and early life stress predict brain and arousal pathways to syndromal depression and anxiety[END_REF]. This polymorphism leads to an amino acid substitution (valine to methionine) at codon 66 in the prodomain of the protein, which impairs intracellular trafficking and activity-dependent secretion of BDNF [START_REF] Egan | The BDNF val66met polymorphism affects activity-dependent secretion of BDNF and human memory and hippocampal function[END_REF] and has been implicated in the aetiology of depression [START_REF] Verhagen | Meta-analysis of the BDNF Val66Met polymorphism in major depressive disorder: effects of gender and ethnicity[END_REF]. Previously, Joffe et al. [START_REF] Joffe | Brain derived neurotrophic factor Val66Met polymorphism, the five factor model of personality and hippocampal volume: Implications for depressive illness[END_REF] were the first to show an interaction between the BDNF polymorphism and stress in healthy subjects; they found a significant relation between stress susceptibility and decreased hippocampal volume in Met-allele carriers, but no effect in subjects with the ValVal genotype. Additionally in another study by Gatt and colleagues, who studied the interaction between BDNF Val66Met and CA on local brain volume, differences in amygdala and hippocampal volume were found [START_REF] Gatt | Interactions between BDNF Val66Met polymorphism and early life stress predict brain and arousal pathways to syndromal depression and anxiety[END_REF]. These authors showed that, compared to ValVal homozygotes, a high number of childhood events was associated with a moderate volume reduction of hippocampus and amygdala in persons carrying the methionine (Met) allele of the BDNF polymorphism, but found no main effects of either BDNF or CA. However, these findings were obtained using volumetry of pre-specified regions and thus, it is unclear whether other brain areas are also affected by an interaction between BDNF and CA. Therefore, we sought to investigate to which extent BDNF genotype can modify the effect of CA on local gray matter changes using a whole brain optimized voxel-based morphometry (VBM) approach. By doing so we could localize any effects. We used data from two large cohorts of healthy subjects. This way we were able to replicate our findings, reducing the risk of type II errors. We also avoided possible confounding effects of disease, comorbidity, and treatment by investigating healthy subjects.

Methods

Subjects

This study was part of the Brain Imaging Genetics (BIG) study at the Donders Institute for Brain, Cognition and Behavior of the Radboud University Nijmegen (Medical Centre) [START_REF] Franke | Genetic Variation in CACNA1C, a Gene Associated with Bipolar Disorder, Influences Brainstem Rather than Gray Matter Volume in Healthy Individuals[END_REF]. All participants were of Caucasian descent and were screened prior to study participation using a self-report questionnaire for the following exclusion criteria: a history of somatic disease potentially affecting the brain, current or past psychiatric or neurological disorder, medication (except hormonal contraceptives) or illicit drug use during the past 6 months, history of substance abuse, current or past alcohol dependence, pregnancy, lactation, menopause and MRI contraindications.

The sample was made up of two independent cohorts based on scanner used.

These were analyzed separately, to provide an independent replication of the results.

Tesla cohort

Two hundred seventy five healthy subjects were scanned using a 1.5 Tesla MRI scanner.

The participants were aged 18 -50 years, with a mean age of 23.7 years (SD= 6.0). The majority was female (63%) and had a university or equivalent degree (61%).

3 Tesla cohort Two hundred ninety three healthy subjects were scanned using a 3 Tesla MRI scanner.

Participants were 18 -50 years old, with a mean age of 23.1 years (SD= 4.7). The majority was female (64%) and had a university or equivalent degree (73%).

Assessment of childhood adversity

CA was assessed using an adapted version of the List of Threatening Life Events developed by Brugha and colleagues with the aim of assessing the stress of life events [START_REF] Brugha | The List of Threatening Experiences: a subset of 12 life event categories with considerable long-term contextual threat[END_REF]. This inventory encompasses life events which are likely to occur relatively frequently and score relatively high on long-term threat. Participants were asked whether they had experienced a list of predefined events before the age of 16. This list comprised events like death of a parent, severe illness of self or spouse, moving house and sexual or physical abuse. Presence of CA was defined as having experienced at least one CA before the age of 16 years. Additionally, we also conducted analyses with CA dichotomized into presence of 2 or more CA versus no or 1 CA, as in an earlier study, which showed that only in persons with a history of 2 or more CA differences in brain volumes were found [START_REF] Gatt | Interactions between BDNF Val66Met polymorphism and early life stress predict brain and arousal pathways to syndromal depression and anxiety[END_REF].

Genotyping

Genetic analyses were carried out at the Department of Human Genetics of the Radboud University Nijmegen Medical Centre, in a laboratory which has a quality certification according to CCKL criteria. High molecular weight DNA was isolated from saliva using Oragene containers (DNA Genotek, Ottawa, Ontario, Canada) according to the protocol supplied by the manufacturer. The BDNF 198-G>A (rs6265) polymorphism (Val66Met) was genotyped using Taqman® analysis (assay ID: C_11592758_10; Applied Biosystems, Nieuwerkerk a/d IJssel, The Netherlands). Genotyping was carried out in a volume of 10 μl containing 10 ng of genomic DNA, 5 μl of Taqman Mastermix (2×; Applied Biosytems), 0.375 μl of the Taqman assay and 3.625 μl of H 2 O. Genotyping was performed on a 7500 Fast Real-Time PCR System and genotypes were scored using the algorithm and software supplied by the manufacturer (Applied Biosystems). The genotyping assay had been validated before use and 5% duplicates and blanks were taken along as quality controls during genotyping. Testing for Hardy-Weinberg equilibrium did not show deviations from the expected genotype distribution (p > 0.05). Subjects homozygous and heterozygous for the Met-allele were merged into a group of Metcarriers and compared to homozygous Val-allele carriers.

Mood state

To avoid confounding by current mood state we adjusted the analyses for the negative affect scale of the Positive Affect and Negative Affect Schedule (PANAS) [START_REF] Isgor | Delayed effects of chronic variable stress during peripubertal-juvenile period on hippocampal morphology and on cognitive and stress axis functions in rats[END_REF].

MR Image acquisition

All scans covered the entire brain and had a voxel size of 1x1x1 mm 3 .

Tesla cohort

All images were acquired at 1.5T Siemens Sonata or Avanto scanners (Siemens, Erlangen, Germany), using small variations to a standard T1-weighted 3D MPRAGE sequence (TR 2300 ms, TI 1100 ms, TE

MR Image conversion

Raw DICOM MR imaging data were converted to NIFTI format using the conversion as implemented in SPM5 (http://www.fil.ion.ucl.ac.uk/spm/software/spm5/).

Brain Volume

Normalizing, bias-correcting and segmenting into gray matter, white matter and cerebrospinal fluid was performed using the VBM5.1 Toolbox version 1.19 (http://dbm.neuro.uni-jena.de/vbm/) in SPM using priors (default settings). This method uses an optimized VBM protocol [START_REF] Ashburner | Voxel-based morphometry--the methods[END_REF] as well as a model based on Hidden Markov Random Fields (HMRF) developed to increase signal-to-noise ratio [START_REF] Cuadra | Comparison and validation of tissue modelization and statistical classification methods in T1-weighted MR brain images[END_REF]. Total volume of gray matter (GM), white matter (WM) and cerebrospinal fluid was calculated by adding the resulting tissue probabilities. Total brain volume was defined as the sum of white matter and gray matter volume.

Voxel-Based Morphometry (VBM)

Diffeomorphic Anatomical Registration using Exponentiated Lie algebra (DARTEL) registration toolbox in SPM (26) was applied. First, all images were realigned to templates created from 556 in-house datasets. Second, Jacobian scaled ('modulated') images were calculated and subsequently transformed to MNI space using affine transformation. Finally, all data were smoothed with an 8 mm FWHM Gaussian smoothing kernel. Analysis was performed in SPM using the GM images. After grouping all datasets according to genotype, images with poor quality or artifacts were identified using the outlier analysis routine implemented in the VBM toolbox. Images that showed a deviation of more than 1.5 times the interquartile range from the median were discarded from further analysis.

A full-factorial ANCOVA was applied on the GM images using genotype and CA as factors and age, gender, total brain volume, scan protocol and PANAS negative scale as covariates. All statistical tests were family-wise error rate (FWE) corrected for multiple comparisons across the entire brain (p FWE <0.05) or across all voxels in a region of interest using a small volume correction (SVC; p SVC <0.05). Regions of interest (ROIs) for the amygdala, hippocampus and medial prefrontal cortex were based on a standardized neuroanatomical atlas (27;28). Here, we defined the medial PFC as encompassing the anterior cingulate cortex (ACC) (y > 10; [START_REF] Amodio | Meeting of minds: the medial frontal cortex and social cognition[END_REF]and the medial part of the inferior, middle and superior frontal gyri, sometimes referred to as 'medial frontal gyrus', that is to say Brodmann areas 24, 25 and 32, as well as the medial part of Brodmann areas 9 and 10 (30). For visualization purposes significant clusters were extracted using the toolbox MarsBar, version 0.42 [START_REF] Brett | Region of interest analysis using an SPM toolbox[END_REF]. First, the main effects of CA and BDNF genotype on local GM volume were evaluated. Second, the interaction between CA and BDNF genotype on GM was tested by adding an interaction term to the statistical models. In all analyses age, sex, total brain volume, PANAS negative scale and dummy regressors for the scan protocols were included as covariates. In case of a significant interaction, tcontrasts were created in SPM for post-hoc testing using p SVC < 0.05 for corresponding region of interest.

Results

There were no significant differences between the two cohorts on any of the sample characteristics (Table 1). Total brain volume measured at 3T appeared significantly smaller than that measured at 1.5T [START_REF] Jovicich | MRI-derived measurements of human subcortical, ventricular and intracranial brain volumes: Reliability effects of scan sessions, acquisition sequences, data analyses, scanner upgrade, scanner vendors and field strengths[END_REF].

No significant main effect of BDNF genotype on gray matter volume was found in either cohort, neither within the whole brain analysis (p FWE >0.05), nor within the regional specific ROI analysis (all p SVC >0.05). Within the 1.5T cohort, we found that a history of CA was related to less gray matter in the anterior cingulate cortex (ACC) (BA24) (p SVC = 0.02) and the medial orbitofrontal cortex (BA9) (p SVC = 0.05). However, these findings could not be replicated at 3T. The interaction between BDNF genotype and CA did not lead to significant effects observable in the whole brain analysis (p FWE >0.05). In the ROI analysis, however, we found an interaction effect in the subgenual ACC volume in both cohorts (BA 25, on 1.5 T p SVC = 0.006; on 3T p SVC =0.019; see Figure 1). Additionally, the interaction affected the volume of the medial orbitofrontal cortex (BA10) (on 1.5 T p SVC = 0.02) but this effect could not be replicated in the 3T cohort (on 3T p SVC =0.22). No interaction effects were found in amygdala or hippocampus. For more detailed description of results see Table 2.

Using simple t-tests with contrasts for each comparison Met-carriers were found to be more vulnerable to the effect of CA than homozygous Val-carriers (Figure 2). Within the 1.5T cohort, Met-carriers with a history of CA had less gray matter in the subgenual ACC than Val-homozygotes with a history of CA (T=3.33; p SVC = 0.007) and Met-carriers without a history of CA (T=4.70; p SVC <0.001). Similarly, within the 3T cohort, the comparison of Met-carriers and Val-homozygotes with a history of CA showed significantly less gray matter in the former (T=3.30; p SVC = 0.002). The difference between Met-carriers with and without CA did not reach significance within the 3T cohort, though (T =3.46; p SVC = 0.12).

In an attempt to replicate findings of a previous study [START_REF] Gatt | Interactions between BDNF Val66Met polymorphism and early life stress predict brain and arousal pathways to syndromal depression and anxiety[END_REF] we re-analyzed our data using two CA as the threshold for the presence of CA, however these analyses did not change the associations found in our previous analyses.

Discussion

Within two independent cohorts of healthy subjects we found that carriers of at least one Met-allele of the Val66Met polymorphism in BDNF who had a history of childhood adversity had decreased gray matter volume in the subgenual ACC, compared to Valhomozygotes and Met-carriers without a history of childhood adversity. Amygdala, hippocampus and medial PFC play a vital role in normal emotional processing and the maturation of these areas continues into adolescence [START_REF] Lenroot | Brain development in children and adolescents: insights from anatomical magnetic resonance imaging[END_REF].

Consequently, we hypothesized that these brain areas might be most vulnerable to environmental adversities during childhood and adolescence. The medial PFC is the last part of the brain to mature (34) and therefore might have been the most sensitive to CA as measured in our study. Within the 1.5T cohort we indeed found that persons with a history of CA showed a reduction of gray matter in the orbital prefrontal cortex as compared to persons without CA. This finding could, however, not be replicated within the 3T cohort and it remains therefore unclear whether this was a false positive finding or whether this non-replication was caused by higher levels of image distortion and signal dropout at 3T, which particularly affect the orbitofrontal and anterior temporal areas.

Consequently these areas are measured with greater reliability at 1.5T than at 3T (32;35).

The subgenual ACC is also part of the medial PFC and plays a role in emotion regulation [START_REF] Campbell-Sills | Functioning of neural systems supporting emotion regulation in anxiety-prone individuals[END_REF] and negative memory bias (37;38), as shown by functional neuroimaging studies. Reduced gray matter volume in the subgenual ACC has previously been found in persons with a history of CA [START_REF] Van Harmelen | Reduced Medial Prefrontal Cortex Volume in Adults Reporting Childhood Emotional Maltreatment[END_REF], as well, and volume reduction in this area is a consistent finding in mood disorders (for review see [START_REF] Drevets | The subgenual anterior cingulate cortex in mood disorders[END_REF]). Furthermore, the subgenual ACC has been shown to play a key role in the autonomic and neuroendocrine stress response and arousal associated with emotional states and behavior (41;42). Taken together, these results suggest that the subgenual ACC is implicated in the pathophysiology of mood and anxiety disorders and that decreased gray matter volume as detected in our study could be an indicator of increased emotional sensitivity as a result of genetic susceptibility interacting with CA.

Contrary to preceding studies among healthy subjects [START_REF] Bueller | BDNF Val66Met allele is associated with reduced hippocampal volume in healthy subjects[END_REF][START_REF] Pezawas | The brain-derived neurotrophic factor val66met polymorphism and variation in human cortical morphology[END_REF][START_REF] Frodl | Association of the brain-derived neurotrophic factor Val66Met polymorphism with reduced hippocampal volumes in major depression[END_REF][START_REF] Ho | Association between brain-derived neurotrophic factor Val66Met gene polymorphism and progressive brain volume changes in schizophrenia[END_REF] we did not find a main effect of the BDNF Val66Met polymorphism on regional brain volume. We had expected to find reduced gray matter volume in Met-allele carriers, irrespective of CA.

However, null findings have also been reported before (20;47;48). It thus seems that the effect of the Val66Met polymorphism on regional brain volume is dependent on additional factors that have not been properly controlled for so far. For instance, Szeszko et al. [START_REF] Szeszko | Brain-derived neurotrophic factor val66met polymorphism and volume of the hippocampal formation[END_REF] concluded that the effect of the Val66Met polymorphism on regional brain volume is pronounced in patients with schizophrenia. Our findings also suggest that the effect of the BDNF Val66Met polymorphism on regional brain volume is dependent on an additional factor, i.e. history of CA.

Our results fit nicely with the neurotrophic hypothesis of depression [START_REF] Duman | A neurotrophic model for stress-related mood disorders[END_REF]and thus suggest that stress (and depression) can lead to structural brain changes in turn leading to increased stress sensitivity. Therefore it is crucial that clinicians are aware of this potentially brain-damaging effect of depression and treat depressed patients as early and effectively as possible [START_REF] Hasler | Pathophysiology of depression: do we have any solid evidence of interest to clinicians?[END_REF]. Neurotrophic factors play a crucial role in neural plasticity during development and also during adulthood where they modulate axonal and dendritic growth and remodelling, membrane receptor trafficking, neurotransmitter release as well as synapse formation and function [START_REF] Lu | Neurotrophins and hippocampal synaptic transmission and plasticity[END_REF]. Indeed, the transcription and secretion of BDNF can be modified by environmental factors such as stress, which can lead to changes in neuronal plasticity and ultimately will affect the function and structure of certain brain areas. Our results could thus be explained by reduced neuronal plasticity in the subgenual ACC. Neither CA alone nor reduced BDNF secretion alone was sufficient to cause reduced subgenual ACC volume in our samples, the combination of the two factors was critical in causing an effect that we could measure with our approach. It is important to note that this was found in self-reported healthy subjects, suggesting that these structural brain abnormalities indeed may serve as a risk factor for mood disorders and not a consequence of disease or therapy. In line with this, structural brain changes have frequently been identified as endophenotypes for psychiatric disorders [START_REF] Hasler | Discovering endophenotypes for major depression[END_REF], because they can represent an elegant link between genes and psychiatric disorders.

We did not replicate the findings of a preceding study reporting smaller hippocampal and amygdala volumes in carriers of the BDNF Met-allele and a history of two or more childhood events [START_REF] Gatt | Interactions between BDNF Val66Met polymorphism and early life stress predict brain and arousal pathways to syndromal depression and anxiety[END_REF]. The differences in findings may be explained best by the methods applied. In the study by Gatt and colleagues [START_REF] Gatt | Interactions between BDNF Val66Met polymorphism and early life stress predict brain and arousal pathways to syndromal depression and anxiety[END_REF], VBM was applied to calculate volumes of pre-defined regions of interest (namely hippocampus, amygdala and medial PFC), whereas we applied a whole-brain approach and looked into pre-specified regions by limiting the search space for statistical testing, in addition. Another difference is the method for assessing CA. While we used a dichotomized version (one or more CA versus no CA), they looked into total number of CA (ranging from 0 to 5) and found that only in subjects who had experienced two or more CA there was an interactive effect between BDNF genotype and CA on hippocampal and amygdala volume. Possibly, we found no effect of CA in the hippocampus and amygdale, because within our cohorts a relatively small group had experienced two or more events before the age of 16 years (15%), which may have resulted in lack of statistical power.

Strengths of our study include the large number of subjects and the ability to replicate our own findings by using two separate cohorts, which decreases the chance of false positive findings. Also, all of our associations were corrected for family-wise error rate and thus the results appear to be very robust. Furthermore, we used a whole brain optimized DARTEL VBM approach, which is a fully automatic technique that allows an objective analysis of structural differences between groups across the entire brain. By surveying the whole brain, VBM provides a unbiased measure of highly localized regions that may not be investigated in hypothesis-based studies that employ more laborintensive manual ROI measurement techniques [START_REF] Barnes | A comparison of methods for the automated calculation of volumes and atrophy rates in the hippocampus[END_REF]. However, the sensitivity of the DARTEL VBM approach for detecting amygdala and hippocampal volume changes is probably less than the sensitivity of manual tracing of these structures. Possibly, we therefore found no changes in the amygdala and hippocampus in relation to CA and BDNF Val66Met polymorphism, as has previously been reported (3;4;54). History of CA was assessed by means of a questionnaire and it is important to acknowledge the subjectivity of self-reported CA. Retrospective assessment of CA may be subject to recollection bias; persons with higher depressive symptoms might recall more negative events than persons with less depressive symptoms. However, we believe that the influence of negative mood was limited, firstly, because our study was conducted among healthy subjects without psychopathology and secondly, because we adjusted our findings for negative affect and found that current negative mood did not critically affect our results. Also, although the idea that CA in combination with BDNF Val66Met polymorphism is associated with gray matter volume changes in subgenual ACC fits well with numerous preclinical studies, the possibility of reversed causality (e.g. gray matter volume changes leading to less effective coping with stress, which could lead to overreporting of CA) cannot be excluded by the current findings as our study had a crosssectional design.

In summary, the gene-environment interaction affecting local gray matter volume suggests that BDNF Met-allele carriers are particularly sensitive to CA. Given the important role of the subgenual ACC in the regulation of emotional behavior, this finding represents an important link in our understanding of how BDNF in combination with CA contributes to increased stress sensitivity. Based on our study it is however not clear whether Met-allele carriers are more prone to the detrimental effects of stress on the brain, or whether CA leads to decreased gray matter volume, irrespective of genotype, and normalization is impaired in Met-allele carriers, resulting in gray matter volume reductions in this group. Future longitudinal studies should therefore aim at unraveling the relation between the susceptibility of BDNF Met-allele carriers to stress on the brain.

Figure 1 .Figure 2 .

 12 Figure 1. Reduced gray matter in subgenual anterior cingulate cortex (ACC) as a result of an interaction of BDNF Val66Met genotype with childhood adversities (p SVC <0.05) .

  3.03 ms, 192 sagittal slices, field of view 256 mm). These variations included a TR/TI/TE/slices of 2730/1000/2.95/176, 2250/850/2.95/176, 2250/850/3.93/176, 2250/850/3.68/176, and the use of GRAPPA parallel imaging with an acceleration factor of 2. TI 1100 ms, TE 3.93 ms, 192 sagittal slices, field of view 256 mm). These variations included TR/TI/TE/slices of 2300/1100/3.03/192, 2300/1100/2.92/192, 10 2300/1100/2.96/192, 2300/1100/2.99/192, 1940/1100/3.93/176, 1960/1100/4.58/176, and the use of GRAPPA parallel imaging with an acceleration factor of 2.

	3 Tesla cohort
	All images were acquired at 3T Siemens Trio or TimTrio scanners (Siemens, Erlangen,
	Germany), using small variations to a standard T1-weighted 3D MPRAGE sequence (TR
	2300 ms,

Table 2 . Effects of BDNF Val66Met and childhood adversities on voxel-based morphometry within medial PFC

 2

		Cohort	Region	Brodmann	MNI coordinates	Cluster extended	Z-score
				area	X	Y	Z	(k)
	Main effect CA	1.5T	ACC	25	-5	-5	47	112	3.75 a
			mOFC	9	32	64	11	547	3.71 a
		3 T	ACC	24	10	17	45	50	2.66
			mOFC	10	22	41	40	14	2.50
	Interaction BDNF*	1.5T	ACC	25	-5	-5	47	300	4.41 a
	CA			24	3	-3	51	95	4.17 a
					8	-3	51	32	4.07 a

a p < 0.05 FWE for small volume correction (p SVC ) BA: Brodmann Area, BDNF: Brain-Derived Neurotrophic Factor, CA: Childhood Adversities; ACC: Anterior Cingulate Cortex; mOFC: medial Orbitofrontal Cortex

Acknowledgements: none

Conflict of interest:

All authors have no conflict of interest regarding any financial, personal or other relationships with other people or organizations within three years of beginning the work submitted that could have inappropriately influenced their work.

None of the authors' institutions have contracts relating to this research through which they or any other organization may stand to gain financially now or in the future.

Financial disclosure: none.