Insider Trading under Discreteness Massimo Scottiy

Abstract This paper analyzes a version of the static Kyle’s (1985) model of insider trading where both the distribution of the liquidation value of the risky asset and the distribution of the order ‡ow of noise traders are discrete. We derive necessary and su¢ cient conditions for the existence of perfect Bayesian equilibria where the insider’s strategy is increasing in the value of the asset, and show that such equilibria can be constructed if and only if the variance of the asset is not too extreme. The results in this paper are relevant in contexts where a discrete version of the static Kyle’s (1985) model might be a convenient modelling choice. I would like to thank Pierpaolo Battigalli, Paolo Colla, Amil Dasgupta, Susanne Griebsch and an anonymous referee for valuable advice. Financial support from the School of Finance and Economics at the University of Technology Sydney is gratefully acknowledged. All errors remain my own. y University of Technology Sydney, School of Finance and Economics, PO Box 123 Broadway NSW 2007 Australia. Email: [email protected]

Introduction Rochet and Vila (1994) and Bias and Rochet (1997) propose a variant of the static Kyle’s (1985) model of insider trading where both the distribution of the liquidation value of the risky asset (v) and the distribution of the order ‡ow of noise traders (u) are discrete rather than normal as in the original Kyle’s contribution.1 They notice that the discrete nature of the model is such that the existence of any equilibrium of the associated trading game depends on its out-of-equilibrium beliefs, and that this feature leads to equilibrium multiplicity. This is clearly in sharp contrast with respect to what happens under the assumption of normal distributions, where a unique linear equilibrium of the trading game always exists.2 Focusing on the case of discrete uniform distributions, Rochet and Vila (1994) construct a non-countable family of equilibria where the insider’s strategy is a discrete function strictly increasing in the value of the asset. Loosely speaking, this strategy of the insider’s can be thought of as the discrete counterpart to the strategy that the insider follows in the unique linear equilibrium of the normal model. However, as previously stressed, the existence of such equilibria in the discrete model crucially depends on a set of out-of-equilibrium beliefs that must be speci…ed so as to induce the insider to follow the desired strategy. In this paper, we extend the model proposed by Biais and Rochet (1997) to the more general case in which the distribution of the value of the asset is discrete but not necessar1

This variant of Kyle (1985) was …rst proposed by Rochet and Vila (1994) and later examined more in details by Bias and Rochet (1997). 2 For an analysis and a discussion of the uniqueness of the linear equilibrium in the original static Kyle’s (1985) model see Brunnermeier (2001) and Boulatov et al. (2005).

ily uniform and examine the issue of equilibrium existence under di¤erent values of asset volatility, as measured by the variance of the asset value. We …nd that equilibria in the spirit of those proposed by Rochet and Vila (1994) exist if and only if the variance of the asset value is not too extreme. Indeed, when the variance of the asset value is relatively low, the incentive compatibility constraints for the insider cannot be satis…ed, as the insider always has an incentive to behave as if he had observed a high value of the asset. On the other hand, when the variance of the asset value is relatively high, the incentive compatibility constraints can be satis…ed, but it is not possible to specify a price rule for out-of-equilibrium aggregate trades such that the insider always prefers to follow his equilibrium strategy rather than deviate to some orders that are not compatible with his equilibrium strategy. We will conclude by noticing that the out-of-equilibrium beliefs speci…ed in Bias and Rochet (1997) for the case of discrete uniform distributions fail to support the family of equilibria they characterize. The results in this paper are relevant in the context of market models where a discrete version of Kyle (1985) might be a convenient modelling choice. Section 1 describes the model. Section 2 analyzes the conditions under which an equilibrium in the spirit of that proposed by Rochet and Vila can be constructed. In this section, we will also characterize a family of perfect Bayesian equilibria for the case of discrete uniform distributions and show that the out-of-equilibrium beliefs speci…ed in Bias and Rochet (1997) fail to support the family of equilibria they characterize. This concludes the paper.

1

The Model

Consider the following trading game adapted from Rochet and Vila (1994). A risky asset is traded in a market by three classes of risk neutral players: noise traders, market makers, and an informed trader. Let v, u, x and p respectively denote the liquidation value of the asset, the quantity trade by noise traders, the quantity traded by the insider and the price set by market makers. Let us assume that Pr(v = 2) = Pr(v = Pr(v = 1) = Pr(v =

1) =

1 2

2) = w and

w, where w 2 0; 12 . Thus, we have that: E(v) = 0 and

V AR(v) = 6w + 1. Finally, let us assume that u can take on values 1 and

1 with equal

probability 12 , and that u and v are independent. In stage one of the game, the values of u and v are realized. The insider observes v, but not u, and submits a market order x 2 R. Let X(v) : f 2; 1; 1; 2g ! R denote the trading strategy of the insider. In stage two, market makers observe the aggregate trade z = x + u 2 R, but not x and u separately. Based on the observation of z, market makers compete a là Bertrand to supply z. The outcome of this competitive auction is a price, denoted with P (z), at which orders are liquidated. Because of risk neutrality and Bertrand competition, P (z) = E(vjz). At the end of the game, the insider receives x(v traders receive u(v

P (z)), and market makers receive z (v

P (z)), noise

P (z)).

De…nition 1 A perfect Bayesian (PB) equilibrium of the trading game described above is a pair (X(v); P (z)) such that: (i) Given P (z), X(v) maximizes the expected pro…ts of the

insider; (ii) Given X(v), P (z) = E(vjz) and is determined by Bayes rule, whenever possible.

Notice that the discrete structure of the game implies that in any equilibrium where the insider trades with positive probability, there always exists a non-empty set of aggregate trades for which E(vjz) cannot be computed via Bayes rule. Let a

= fz; z = X(v) + ug (i.e.

i

and

a

i

= fx; x = X(v)g and

respectively denote the set of the insider’s orders

and the set of aggregate trades that are consistent with the equilibrium play). Given strategy X(v), orders z 2 R

a

have zero probability to be observed in equilibrium. For these orders,

E(vjz) cannot be computed via Bayes rule and a set of out-of-equilibrium beliefs have to be speci…ed in order to determine E(vjz). In particular, out-of-equilibrium beliefs must be de…ned so that for every z 2 R

a

the resulting price rule P (z) satis…es part (i) of De…nition

1.

2

Equilibrium Existence

For the case in which w = 41 , Rochet and Vila (1994) propose the existence of a non-countable family of equilibria indexed by q 2 0; 25 where the insider’s strategy X(v) satis…es:

X(v) =

8 > > < 1 + q, v = 2 > > : 1

q, v = 1

; Xq ( v) =

X(v), q 2 (0; 1)

(1)

In this section, for the more general case in which w 2 0; 21 , we will analyze the conditions under which it is possible to support an equilibrium where the insider follows strategy (1).3 Let me …rst consider part (ii) of De…nition 1. Given (1), the set of equilibrium trades of the insider is

i

= f1

aggregate trades reads

q, a

1 + q, 1

= f 2

q,

q, 1 + qg and the corresponding set of equilibrium 2 + q, q, 2

q, 2 + qg. Thus, for any z 2

a,

a

straightforward application of Bayes rule allows us to derive the price rule corresponding to strategy (1), which reads:

P (z) = E(vjz) =

8 > > > 6w > > > < 1 > > > > > > : 2

1 z=q z=2

q ; P (z) =

(2)

P ( z),

z =2+q

Price rule (2) is the one de…ned for equilibrium orders z 2

a.

For orders z 2 R

a,

the

price rule cannot be computed via Bayes rule. As we will see, the critical point in the current analysis will be to de…ne price rule P (z) for every z in a way that the insider’s strategy (1) can be supported in equilibrium. To see the later point, let us turn to the analysis of part (i) of De…nition 1. Let B(v; x) = x [v

E (P (z)jx)] denote the pro…ts that the insider expects from placing order x when

the observed value of the asset is v. The expression E (P (z)jx) denotes the price that the 3

Rochet and Vila (1994) do not explicitly specify a price rule for aggregate orders that are not consistent with the equilibrium play. This is done by Bias and Rochet (1997). At the end of this section, I will note that the price rule speci…ed in Biais and Rochet is indeed inadequate to support an equilibrium where the insider follows strategy (1).

insider expects to pay (or to receive) when he places order x. The actual price that the insider pays depends on the price rule P (z) (which is known to the insider as part of the equilibrium) and by the aggregate order z = x + u (which is unknown to the insider at the moment of placing order x). Since u takes on value E (P (z)jx) = 21 P (x + 1) + 12 P (x

1 and 1 with equal probability,

1). Part (i) of De…nition 1 is equivalent to say that given

price rule P (z), the following conditions are both satis…ed:

for all v 2 f 2; 1; 1; 2g and x 2

i,

for all v 2 f 2; 1; 1; 2g and x 2 R

where B(v; X(v)) = x [v

B(v; X(v)) i,

(3)

B(v; x)

B(v; X(v)) > B(v; x)

(4)

E (P (z)jX(v))] denotes the pro…ts that the insider expects from

following equilibrium strategy X(v). First, consider condition (3), which focuses on insider’s orders x 2

i

consistent with the equilibrium play). By construction, an order x 2 an aggregate trade z 2

a.

(i.e., insider’s orders i

always generates

We know that for this set of aggregate orders, the price rule is

de…ned by (2). By using (2), we can compute the values of E (P (z)jX(v)) and eventually those of B(v; X(v)) for every value of v and obtain:

B(v; X(v)) =

8 > > <

3 6w 2

> > : 3w (1

(1 + q) for v = 2; 2 (5) q)

for v = 1; 1

Likewise, we can use (2) to compute the values of E(P (z) jx) and B(v; x) corresponding to any order x 2

i.

By doing this for each x 2

i,

it is immediate to show that (3) is veri…ed

if and only if the two following conditions are veri…ed:

3

6w 2

(1 + q)

3w (1

q)

q)

(6)

(1 + q)

(7)

(1 + 3w) (1 1

6w 2

It is easy to show that conditions (6) and (7) are both satis…ed for Notice that

12w 1 5

is always smaller than 12w

two conditions are both satis…ed if and only if w 2 i

q

12w

1.

1 for any w 2 0; 21 . However, since q is

restricted to take values in (0; 1), we must require that: 12w

any order x 2

12w 1 5

1 1 ; 12 2

can be prevented if and only if w 2

1 > 0 and

12w 1 5

< 1. These

. In other words, a deviation to

1 1 ; 12 2

. In fact, an inspection of (6)

1 the insider always prefers to place an order equal and (7) reveals that when w 2 0; 12

1 + q irrespective of the value of the asset v that he has observed. The following remarks summarizes the previous …ndings. Remark 1 For any w 2

1 1 ; 12 2

, condition (3) is satis…ed for

12w 1 5

q

min(12w

1; 1).

1 For any w 2 0; 12 , there does not exist a q 2 (0; 1) such that the insider condition (3) is

satis…ed. Now, consider condition (4). Notice that an order x 2 R aggregate trade z 2 R

a,

i

always results in an

i.e. an aggregate trade that is inconsistent with the equilibrium

play. For such trades, Bayes rule cannot be used to compute E (vjz). Therefore, in order to compute B(v; x), a function that prices aggregate trades z 2 R

a

needs to be speci…ed.

Let Poe ( ) denote a candidate function.4 Notice that if we want Poe ( ) to support our putative equilibrium, condition (4) mandates that the following inequalities must be satis…ed for any x2R

i:

(1 + q)

3

6w

3w(1 (1 + q)

> x (2

E (Poe ( )jx))

(8a)

q) > x (1

E (Poe ( )jx))

(8b)

2

3

6w

> x( 2

E (Poe ( )jx))

(8c)

q) > x ( 1

E (Poe ( )jx))

(8d)

2 3w(1

As we will explicitly show below, inequalities (8a) to (8d) de…ne a set of conditions that must be satis…ed by Poe ( ) to induce the insider to follow his equilibrium strategy and not to deviate to any order x 2 R

i.

Notice that the only restriction imposed by Bayesian

perfection is that Poe ( ) must lie in the support of v, that is:

Poe ( ) : R 4

a

! [ 2; 2]

(9)

Formally, for every z 2 R a , we should specify a set of market makers’ out-of-equilibrium beliefs (about the value of v conditional on z) from which Poe (z) = E (vjz) is derived. For the sake of exposition, we focus directly on de…ning Poe (z): In section 2.1, where we provide an example of how an equilibrium of the trade game under consideration can be constructed, we explicitly de…ne the set of market makers’ out-of-equilibrium beliefs from which Poe (z) is derived.

We will now show that a price function satisfying property (9) and conditions (8a) to (8d) exists if an only if the values of w are below a certain threshold. Let us …rst consider inequalities (8a) to (8d), and focus on orders x 2 R

i

such that x

2 or x

2.5

Such orders will always generate out-of-equilibrium aggregate trades z = x + u that are either greater than 1 or less than

1. Let us tentatively assume that for these set of out-of-

equilibrium aggregate trades, Poe ( ) satis…es:

Poe (z) =

8 > > < 2, for z > > :

2, for z

1, z 6= 2

q; 2 + q

1, z 6=

2 + q; 2

(10) q

It is immediate to notice that under property (10), the price expected from placing any order x2R

i

greater than 2 (lower than

2) is equal to 2 ( 2). This clearly guarantees that

conditions (8a) to (8d) are satis…ed for any order x 2 R

i

greater than 2 or lower than

2. Let us now consider orders x 2 R which 0

i

such that

2 < x < 2. Consider …rst the case in

x < 2. Notice that since x is positive, we can restrict our attention to the analysis

of conditions (8a) and (8b).6 Moreover, if we assume that Poe ( ) satis…es property (10), then 5

Recall that in steps (a), (b), and (c) we are restricting attention on insider’s and aggregate orders that belong to i and a respectively (i.e. on orders inconsistent with the equilibrium play). 6 It is immediate to notice that for positive x, (8c) is always satis…ed whenever (8a) is, as well as (8d) is always satis…ed whenever (8b) is.

we have that:

1 1 E (Poe (z)j0 < x < 2) = Poe (x + 1) + Poe (x 2 2

1 1) = 1 + Poe (x 2

1)

Notice that the last equality holds because z = x + 1 is always greater than or equal to 1 when 0

x < 2, and thus, by (10), Poe (x + 1) = 2. Using the previous result, after a bit of

algebra we can write inequalities (8a) and (8b) as follows:

Poe (x

1) > 2

Poe (x

1) >

(3

6w) (1 + q) x 6w(1 q) x

These two inequalities represent a …rst set of conditions that must be met by our candidate function Poe ( ). They guarantee that if out-of-equilibrium aggregate trades z = x

1 are

priced such that these conditions are met, the insider will never deviate to any out-ofequilibrium order 0

x < 2. Notice that since 0 < x < 2, any corresponding out-of-

equilibrium aggregate trade z = x

1 lies in ( 1; 1). Therefore, by changing variable from

x to z and considering property (9), we can equivalently write the previous conditions as follows: Poe (z) > max ( 2; Gmax (z)) , for where Gmax (z; w; q)

sup 2 z

Let us now consider orders x 2 R

i

1 < z < 1; z 6=

(3 6w)(1+q) ; z+1

such that

2
q; q (11)

6w(1 q) z+1

0. By following the same line

of reasoning above, we can derive a second set of conditions that must be satis…ed by Poe (z) in order to avoid that the insider deviates to any order

Poe (z) < min (2; Gmin (z)) , for Gmin (z)

inf z

2

(3 6w)(1+q) ; z 1

2
0:

1 < z < 1; z 6=

q; q (12)

6w(1 q) z 1

The previous analysis suggests that if we de…ne Poe ( ) so that conditions (10), (11) and (12) are met, then condition (4) is satis…ed too. It is immediate to notice that a necessary condition for (11) and (12) to be simultaneously satis…ed is that:

for all

1 < z < 1, z 6=

q; q, max ( 2; Gmax (z)) < min (2; Gmin (z))

(13)

Notice that both Gmax (z) and Gmin (z) depend on w and q, which we restrict to take values in 0; 21

and (0; 1) respectively. Therefore, condition (13) has to be satis…ed under the

constraint that w 2 0; 12 and q 2 (0; 1). To get an intuition of the impact of this constraint on condition (13), suppose that there did not exist any pair (w; q) 2 0; 21

(0; 1) for which

condition (13) can be satis…ed. This would imply that it is not possible to specify any price function satisfying conditions (11) and (12), and thus it is not possible to support our putative equilibrium. On the contrary, if a pair (w0 ; q 0 ) 2 0; 21

(0; 1) existed such that condition

(13) can be satis…ed, then we could conclude that when w = w0 , we can de…ne a price function that induces the insider to follow the equilibrium strategy X(2) =

X( 2) = 1+q 0 ,

Figure 1: Condition (13) when w = 14 and q = 12 . Notice that in this case there exists an in…nity of functions satisfying conditions conditions (11) and (12), for example any continuous function that goes through the shaded area.

X(1) =

X( 1) = 1

q0.

Figure 1 provides a graphical illustration of condition (13) for the speci…c case of w =

1 4

and q = 21 . Intuitively, condition (13) ensures that the shaded area in …gure 1 is a connected set, which in turn guarantees that it is possible to specify a functional form for Poe ( ) such that property (9) and conditions (11) and (12) are met. In other words, when w =

1 4

it is always

possible to de…ne a price rule for out-of-equilibrium trades that supports an equilibrium where the insider’s strategy reads: X(2) = In general, for any pair (w; q) 2 0; 12

X( 2) = 12 , X(1) =

X( 1) = 23 .7

(0; 1) satisfying (13), it is possible to de…ne

Notice that when w = 14 and q = 21 there exists an in…nity of functions satisfying conditions (11) and (12), for example any continuous function that goes through the shaded area. However, notice that we do not require that Poe ( ) be continuous. 7

a price function Poe ( ) : R

a

! [ 2; 2] that meets conditions (10), (11) and (12), and

therefore satis…es condition (4) as well. We formally analyze condition (13) in the appendix where, for any values of w 2 0 ; 21 , we …nd the values of q 2 (0 ; 1) that satisfy condition (13). The following remark summarizes the results: Remark 2 For any w 2 0; 16 , condition (13) is satis…ed for all max (q1 (w); 0) < q < 1. For any w 2

1 ;w 6

, condition (13) is satis…ed for all 0 < q2 (w) < q < q3 (w) < 1. For any

w 2 w; 12 , there does not exist a q 2 (0; 1) such that condition (13) is satis…ed. (Proof in the Appendix). Essentially, Remark 2 suggests that a price rule that satis…es condition (4) can be de…ned if and only if w 2 (0; w). Notice that this condition is not su¢ cient to guarantee that an equilibrium where the insider follows strategy (1) exists. Indeed, also the incentive compatibility constraint (3) must me met. That is, also the conditions stated in remark 1 must be satis…ed. The following proposition summarizes the results that we obtain by jointly examining the conditions stated in Remark 1 and Remark 2. Proposition 1 An equilibrium where the insider follows strategy (1) can be constructed if and only if w 2

1 ;w 12

.(Proof in the Appendix).

In other words, for any given value of w 2

1 ;w 12

, say w0 , we can …nd a value of q 2 (0; 1),

say q 0 , such that for w = w0 and q = q 0 the following is true: (i) The incentive compatibility

constraint (3) is satis…ed; (ii) a price rule can be speci…ed so that condition (4) is satis…ed. In other words, when w = w0 an equilibrium can be constructed where the insider’s strategy reads: X(2) =

X( 2) = 1 + q 0 , X(1) =

X( 1) = 1

In general, it is true that given a value of w 2 Poe (z), there exist a set of value of q, say q; q

q0.

1 ;w 12

and a speci…ed price function

(0; 1), for which the incentive compatibility

constraint (3) and condition (4) are satis…ed. In other words, given a value of w 2

1 ;w 12

,

there generally exists a non-countable family of equilibria, indexed with q 2 q; q , where the insider follows strategy (3). The values of q and q will depend on the value of w and on the speci…c functional form chosen for Poe (z). In the next section, we will provide an example by constructing an equilibrium for the case discussed in Biais and Rochet (1997), i.e. the case in which w = 41 .

2.1

An Example

Let w = 14 . Proposition 1 ensures that an equilibrium can be constructed in this case. Let us assume that Poe ( ) satis…es:

Poe (z) =

8 > > > 2 > > > <

z

1; z 6= 2

q; 2 + q

2z 1 < z < 1; z = 6 q; q > > > > > > : 2 z 1; z 6= 2 q; 2 + q

(14)

Poe(z)

Figure 2: Price fuction (14) satis…es property (9) and meets conditions (10), (11) and (12), thereby guaranteeing that the insider does not deviate to any out-of-equilibrium orders.

Figure 2 represents price function (14). It is immediate to verify that given (14), condition (4) is satis…ed for all q 2 for all q 2

2 ;1 5

1 2 ; 3 3

. Since the incentive compatibility constraint (3) is satis…ed

, we can conclude that when w =

of equilibria indexed by q 2

2 2 ; 5 3

1 4

there exists a non-countable family

, such that: (i) The insider trades according to (1); (ii)

Market makers price equilibrium aggregate trades using (2) and out-of-equilibrium aggregate trades using (14). It is worth noticing that price function (14) is derived from the following

set of market makers’out-of-equilibrium beliefs:

- F or z

1; z 6= 2

q; 2 + q :

Pr(v = 2jz) = 1; Pr(v = 1jz) = Pr(v = - F or 0

z+1 ; Pr(v = 1jz) = Pr(v = 2 0; z 6=

1
Pr(v = 2) = - F or z

2jz) = 0:

z < 1; z 6= q :

Pr(v = 2) = - F or

1jz) = Pr(v =

1

z 2

1; z 6=

2) =

1jz) = 0; Pr(v =

2) =

1

z 2

:

q:

; Pr(v = 1jz) = Pr(v = 2

1jz) = 0; Pr(v =

z+1 : 2

q; 2 + q :

Pr(v = 2jz) = Pr(v = 1jz) = Pr(v =

1jz) = 0; Pr(v =

2jz) = 1:

In fact, it is this set of out-of-equilibrium beliefs that supports the family of equilibria that we have characterized above. Indeed, the previous set of beliefs allows us to compute E (vjz) for every out-of-equilibrium order z 2 R

a.

Price function (14) is derived from

this set of beliefs as Poe ( ) = E (vjz). Clearly, if we had assumed a functional form for Poe ( ) di¤erent from (14), the family of equilibria would have been identi…ed by a di¤erent set of values of q, essentially because condition (4) would be now satis…ed by a set of values of q di¤erent from

1 2 ; 3 3

.8 The only

general conclusion that can be drawn from Remarks 1 and 2 for the case of w = 8

1 4

is that it

Assuming a di¤erent functional form for Poe implicitly implies that we are assuming a di¤erent set of market makers’out-of-equilibrium beliefs that support the equilibrium under consideration.

is always possible to …nd a family of equilibria characterized by strategies (1),(2), and (14) h p i with q taking values in some subset of 25 ; 2 3 2 .9

2.2

A note on Bias and Rochet (1997)

Biais and Rochet (1997) analyze the previous trade game for the case w = 41 . For the set of out-of-equilibrium aggregate trades z 2 R

P (z) =

a

they specify the following price rule:

8 > > < 2, for z > 0, z 6= q; 2 > > :

2, for z < 0, z 6=

q; 2 + q (15)

q; 2 + q; 2

q

It is immediate to show that (15) does not support an equilibrium where the insider trades according to strategy (1). Using (2) and setting w = rule corresponding to orders z 2

we obtain the market maker’s price

a:

P (z) = E(vjz) =

9

1 4

8 > > > > > > <

1 2

z=q

1 z = 2 q ; P ( z) = > > > > > > : 2 z =2+q

P (z)

(16)

Notice that di¤erently from what it is concluded in Rochet and Vila (1994) (see their paper at p. 151), it is never possible to characterize a family of equilibria indexed by values of q 2 25 ; 1 .where the insider follows (1) and market makers follow (2).

Using (16), it is easy to show that:

B(v; X(v)) =

8 > > < > > :

3 4

(1 + q) for v = 2; 2

3 4

(1

q) for v = 1; 1

Now let x0 denote an out-of-equilibrium order. Let 0 < x0 < 1. Given (15), E (P (z)jx0 ) = 1 2 2

+ 12 ( 2) = 0. This implies that B(2; x0 ) = 2x0 and B(1; x0 ) = x0 . A necessary condi-

tion for the existence of the equilibrium is that the following two conditions are satis…ed simultaneously:

3 (1 + q) > 2x0 4 3 (1 q) > x0 4

However, since q is restricted to take values in (0; 1), it is immediate to notice that there always exist some orders x0 2 (0; 1) for which the insider can make a pro…table deviation from his prescribed strategy. Indeed, the …rst inequality is satis…ed for any x0 2 (0; 1) if and only if q > 53 , while the second inequality is satis…ed for every x0 2 (0; 1) if and only if q<

1 . 3

Therefore, there does not exist any q that simultaneously satis…es both conditions.

Appendix Proof of Remark 2. Let me write condition (13) as follows:

max ( 2; Gmax (z)) < min (2; Gmin (z)) , for all

1 < z < 1, z 6=

q; q

where

Gmax (z)

sup 2

(3

z

Gmin (z)

inf z

2

6w) (1 + q) ; z+1

(3

6w) (1 + q) ; z 1

6w(1 q) z+1 6w(1 q) z 1

It is immediate to verify that since q 2 (0; 1) and z 0; 21 , both the inequalities Gmax (z) < 2 and Gmin (z) >

2 are satis…ed for all

1 < z < 1. Let us then analyze when the following

condition holds: Gmax (z) < Gmin (z); for

1 < z < 1; z 6=

q; q

This condition holds if and only if, for all

1 < z < 1; z 6=

q; q, the following conditions

are veri…ed:

6w(1 q) 6w(1 q) < z+1 z 1 6w(1 q) (3 6w) (1 + q) < 2 z+1 z 1 6w(1 q) (3 6w) (1 + q) < 2 z+1 z 1 (3 6w) (1 + q) 6w(1 q) 2 < z+1 z 1

(17) (18) (19) (20)

It is immediate to verify that inequality (17) is always satis…ed for all

1 < z < 1; z 6=

q; q.

Consider inequality (18). With a bit of algebra we can write it as follows:

2z 2 + (3 + 3q

Notice that 1

z 2 < 0 for

strictly positive for to z1 =

3 3q+12w

p

12w)z + (1 + 3q z2 1

12qw)

<0

1 < z < 1. Thus, we need to impose that the numerator is

1 < z < 1. The numerator is strictly convex in z, with roots equal p 1 6q+9q 2 72w+24qw+144w2 3 3q+12w+ 1 6q+9q 2 72w+24qw+144w2 and z2 = . 4 4

When z1 ; z2 2 = R, the numerator is strictly positive for every z. This occurs when 1

6q +

9q 2

72w + 24qw + 144w2 < 0, which is satis…ed under the following condition:

w2 where q1 (w) =

0;

1 2

1 (1 3

and q 2 (q1 (w); q3 (w)) 4w)

8p w 3

2w2 and q3 (w) =

(A.1) 1 (1 3

4w) +

8p w 3

2w2

Condition (A.1) guarantees that the numerator is always positive. However this condition might be too strict. Indeed, there may be values of q for which z1 ; z2 2 R and the numerator is still positive for every

1 < z < 1. In particular, when q 2 = (q1 (w); q3 (w)), z1 ; z2 2 R

and the numerator is positive for z < z1 or z > z2 . It is easy to verify that the z1 is always smaller than 1. Thus, for the numerator to be strictly positive for every z 2 ( 1; 1), we must impose that z2 <

1 This is satis…ed when:

w2

0;

1 6

and q 2 (q3 (w); 1)

(A.2)

Therefore, the numerator of (18) is positive for z 2 ( 1; 1) when either condition (A.1) or condition (A.2) is satis…ed, or equivalently when the following condition is met:

(i) w 2 0; 61

and q 2 (q1 (w); 1) (A.3)

or (ii) w 2

1 1 ; 6 2

and q 2 (q1 (w); q3 (w))

Consider inequality (19). We can equivalently write it as:

4z 2 + 2q(3

Notice that 1

z 2 < 0 for

6w) 1 z2

12w + 2

<0

1 < z < 1. Thus, we need to impose that the numerator is

strictly positive for z 2 ( 1; 1). The numerator is strictly convex in z, with roots equal q q 1 3q+6w+6qw 1 3q+6w+6qw and z = . When z1 ; z2 2 R, the numerator is to: z1 = 2 2 2

positive for z 2 = (z1 ; z2 ). Since z1 < 0 < z2 , there always exist values of z 2 ( 1; 1) where the

numerator is non-positive. Since we require the numerator to be strictly positive for every z 2 ( 1; 1), we need to impose that z1 ; z2 2 = R, which is veri…ed when

1 3q +6w+6qw < 0.

This is satis…ed when:

w2

0;

1 2

and q > q2 (w)

where q2 (w) =

(A.4)

1 6w 3(2w 1)

Finally, consider inequality (20) and write it as follows:

2z 2 + (12w

Notice that 1

z 2 < 0 for

strictly positive for

3q

3)z 1 z2

12qw + 3q + 1

<0

1 < z < 1. Thus, we need to impose that the numerator is

1 < z < 1. The numerator is strictly convex in z, with roots equal

to z1 =

p

3+3q 12w

1 6q+9q 2 72w+24qw+144w2 4

and z2 =

3+3q 12w+

p

1 6q+9q 2 72w+24qw+144w2 . 4

It is

immediate to notice that z1 ; z2 2 = R and the numerator is positive when condition (A.1) is met. When z1 ; z2 2 R the numerator is positive for z 2 = (z1 ; z2 ). It is easy to verify that the highest root, z2 , is always greater than

1. Thus, for the numerator to be strictly positive

for every z 2 ( 1; 1) we must impose that z1 > 1 One can verify that this is satis…ed when condition (A.2) is met. Thus inequality (20) is satis…ed under condition (A.3). Joint consideration of (A.3), (A.4) together with the constraint q 2 (0; 1) implies that inequalities (17) through (20) are satis…ed simultaneously in our parameter region as long as: (i) w 2 0; 61

and max (q1 (w); q2 (w); 0) < q < 1 (A.5)

or (ii) w 2

1 1 ; 6 2

and max (q1 (w); q2 (w); 0) < q < min (q3 (w); 1)

Notice that for w 2 0; 16 , q2 (w) = 1 3

(1

w2

4w) 1 1 ; 6 2

8 3

p

w

1 6w 3(2w 1)

< 0. Then, consistency requires that q1 (w) =

2w2 < 1, which is always satis…ed for w 2 0; 61 . Now notice that for

, q1 (w) < 0 < q2 (w) and q3 (w) =

1 3

(1

4w) +

8 3

p

w

2w2 < 1. Thus, (A.5) can

be written as: (i) w 2 0; 16

and max (q1 (w); 0) < q < 1

or (ii) w 2 Finally, consistency requires that

1 1 ; 6 2

and q2 (w) < q < q3 (w)

1 6w 3(2w 1)

<

1 3

(1

4w) +

8 3

p

w

2w2 . This is satis…ed for

0
1 12

p p p 2 2+ 3 + 4 2 < 12 . Thus our …nal condition can be written as:

5

(i) w 2 0; 61

and max (0; q1 (w)) < q < 1

or 1 ;w 6

(ii) w 2 1 12

where w =

5

and q3 (w) = 31 (1

and q2 (w) < q < q3 (w)

p p p 2 2+ 3 + 4 2 , q1 (w) = 4w) +

8 3

p

1 3

(1

8 3

4w)

p

2w2 , q2 (w) =

w

1 6w 3(2w 1)

2w2 .

w

Proof of Proposition 1. Conditions in Remark 1 and Remark 2 are jointly satisfy if and only if:

(i) w 2

1 1 ; 12 6

and max

1 3

1 ;w 6

and max

1 6w ; 12w5 1 3(2w 1)

(1

4w)

8 3

p

w

1 6w 2w2 ; 3(2w ;0 1)

q

min(12w

1; 1)

or (ii) w 2

< q < min

1 3

(1

4w) +

8 3

p

w

2w2 ; 12w

Focus on (i) and notice that we have that the following holds true for all w 2 (a) max

1 3

(1

(b) min(12w (c) 0 <

12w 1 5

4w)

8 3

p

1; 1) = 12w < 12w

2w2 ; 12w5 1 ; 0 =

w

:

12w 1 ; 5

1;

1 < 1.

This means that for any w 2 can be constructed.

1 1 ; 12 6

1

1 1 ; 12 6

there exist a q 2 (0; 1) for which the equilibrium

Now focus on (ii) and notice that the following holds true for all w 2 (d) min

1 3

(1

(e) 0 < max

4w) +

8 3

p

w

1 6w ; 12w5 1 3(2w 1)

2w2 ; 12w < 31 (1

1 = 31 (1

4w) +

Again, this implies that also for any w 2

8 3

p

w

1 1 ; 12 6

4w) +

8 3

p

w

1 ;w 6

:

2w2

2w2 < 1 there exist a q 2 (0; 1) for which the

equilibrium can be constructed.

References [1] B. Biais and J. C. Rochet, Risk Sharing, Adverse Selection and Market Structure, in B. Biais, T. Biork, J. Cvitanic, N. El Karoui, E. Jouini and J.C. Rochet, Financial Mathematics (Springer-Verlag, Berlin, 1997) 1-51. [2] M. Bagnoli, S. Viswanathan and C. Holden, On the Existence of Linear Equilibria in Models of Market Making, Mathematical Finance 11 (2001) 1-31. [3] A. Boulatov, A. S. Kyle and D. Livdan, Uniqueness of Equilibrium in the Single-Period Kyle ’85 Model, Working Paper (2005). [4] M. K. Brunnermeier, Asset Pricing under Asymmetric Information - Bubbles, Crashes, Technical Analysis and Herding, (Oxford University Press, 2001). [5] A. S. Kyle, Continuous Auctions and Insider Trading, Econometrica 53 (1985) 1335-1355.

[6] J. C. Rochet and J. C. Vila, Insider Trading without Normality, Review of Economic Studies 61 (1994) 131-152