G. Trigilia, February 2015
Consider a trading model similar to that in Glosten and Milgrom’s paper, except that suppose regulation forces the specialist to set one price for buy and sell orders. That is, the specialist is not allowed to set a bid-ask spread he can only specify one price at which he will buy or sell one unit. Suppose there is only one trading period before the liquidation of the risky asset. The liquidation value can be either 2 or 4 with equal probabilities. An arriving agent is a liquidity trader with probability δ or is perfectly informed with probability 1 − δ . If the trader is a liquidity trader then he is a buyer with probability α and a seller with probability 1 − α .
1. Consider all possible values of δ and α . Does there exist a price P such that the specialist makes zero expected profits if he posts the price P ?
Give conditions under which such a price exists, and calculate its value when it exists.
2. Suppose that there are many potential market makers. Each one of them announces a price at which he is willing to buy or sell one unit of the risky asset. A market maker can also refuse to trade. The arriving agent can then choose which the market maker to trade with. A price
P is called an equilibrium price if, when P is quoted by some market maker (s), there does not exist another price P 1
P
1 such that by offering another market maker can earn positive expected profits. Consider again all possible values of α and δ . Give conditions (if any) under which an equilibrium price exits, and calculate its value when it exists.
Solution to part (A): Given that the value of the asset is binary – v ∈ { 2 , 4 } , we need to consider three cases in the analysis:
1. First, we might have P > 4: in this case, whenever a trader is informed he wants to sell the asset. As a result, the market maker faces an expected loss of 3 − P < 0 against (i) any informed trader (there are
1 − δ of them), and (ii) any uninformed traders who want to sell (there are δ (1 − α ) of them). In contrast, he expects to gain P − 3 > 0 against uninformed traders that want to buy (there are δα of them). Summing up, the expected profits of the market maker are:
(1 − δ )(3 − P ) + δ [(1 − α )(3 − P ) − α (3 − P )] = ( P − 3)[2 δα − 1]
1

Since P = 3 such a price can lead to zero expected profits for the market maker if only if 2 δα = 1;
2. Second, we might have P < 2. In this case, the logic of the previous case is reverse: the informed traders all want to buy and the market maker gains only if he faces an uninformed trader who sells. Formally, the expected profits of the market maker are:
(1 − δ )( P − 3) + δ [ α (3 − P ) + (1 − α )( P − 3)] = (3 − P )[1 − 2 αδ ]
Once again, P = 3 implies that zero profits only occur at such price when 2 δα = 1;
3. Finally, the more interesting case: 2 ≤ P ≤ 4. Now, the informed trader buy or sell depending on whether the value of the asset is high or low, respectively. As a result, the expected profits read:
(1 − δ )[0 .
5(2 − P ) + 0 .
5( P − 4)] + δ [ α ( P − 3) + (1 − α )(3 − P )]
Setting the expression equal to zero, and doing some manipulations, we get:
1 − δ
P = 3 +
δ (2 α − 1)
It remains to check for what values of α and δ such a price would indeed belong to the range 2 ≤ P ≤ 4. Starting with P ≤ 4, we get that:
1 − δ
4 ≥ 3 +
δ (2 α − 1)
I conjecture that the solution requires α < 0 .
5, therefore we can rewrite
it as: 1
2 δα ≥ 1 ⇐⇒ α ≥ (2 δ )
− 1 and because δ ∈ (0 , 1) we confirm that indeed α < 0 .
5.
Now, consider P ≥ 2. We have:
1 − δ
2 ≤ 3 +
δ (2 α − 1)
I conjecture that the solution requires α > 0 .
5, therefore we can rewrite it as:
0 ≥ 2 δα + 1 − 2 δ ⇐⇒ α ≤ 1 − (2 δ )
− 1 and because δ ∈ (0 , 1) we confirm that indeed α > 0 .
5.
1 I mean, we do not flip the inequality direction.
2

Evidently, there are parameters for which an equilibrium does not exist. For example, for δ < 1 and α = 0 .
5.
Solution to part (B): First, observe that if δ = 1 there exists an equilibrium price P = 3. Now, consider δ < 1. For the market maker to make zero profits, it needs to be the case that the gains from one side of the market exactly compensate the losses from the other side. In other words, the proceeds from the buy side (B) plus those from the sell side (S) must be such that B + S = 0 at price P . Without loss of generality, suppose that the sell side is profitable. Then, there exists a price P
0
> P that would attract only the sellers, making a profit. Therefore, there cannot be a price equilibrium.
Consider a Kyle model with multiple informed agents. There are N informed agents each who knows the value v. Apart from this, we assume the same structure as in the original Kyle’s model.
1. Characterize a linear equilibrium.
2. How does the aggregate profit of the informed traders vary with their number, N? How does the conditional variance vary with N?
3. What happens as we increase the number of the informed traders to infinity?
Solution: Recall that the original Kyle’s model assumes the following:
1. The asset value is normally distributed as V :
N
[ ρ
0
, Σ
0
];
2. The noise traders submit order µ :
N
[0 , σ
2
µ
];
3. The order submitted by the informed trader i is denoted by x i
;
4. The market maker observes the aggregate order flow P
N j =1 x j
+ µ and it is common knowledge that he sets a price according to the following function:
P
X
N j =1 x j
+ µ =
E
V
X
N j =1 x j
+ µ
5. We focus on symmetric strategies x i i = 1 , ..., N .
= X ( v ) for every informed trader
3

Define k ≡ P
N j =1 x j
+ µ .
The optimal strategy maximises the payoff of the informed trader, anticipating the behaviour of the market maker. Formally:
X ( v ) = arg max
E
[ π | v ] =
E
[ v − P | v ] x
(1) An equilibrium is linear if the strategy takes the following form X ( v ) =
α + βv where the greek letters are constant parameters. Because of the normality assumption, we know that the conditional expectations are linear and hence:
E
[ V | k ] = γ + λk = ρ
0
+ cov( V, k ) var k k −
E
[ k ]
In particular, ( V, k ) is jointly normal with mean ( ρ
0
, N ( α + βρ
0
)]) and covariance matrix:
 Σ
0
βN Σ
0


βN Σ
0
β
2
N
2
Σ
0
+ σ
2
µ
 and hence we get:
E
[ V | k ] = ρ
0
+
βN Σ
0
β 2 N 2 Σ
0
+ σ 2
µ k − N ( α + βρ
0
) = P
And substituting in the objective function:
X ( v ) = arg max
E
[ v − P | v ] x
= v −
E
[ P ] x
= [ v − γ + λk ] x
= vx − γx − λx
2 − λ ( N − 1)( α + βv ) x
The FOC yields: x = v −
2 λ
γ
−
N − 1
( α + βv )
2 and because also x = α + βv by symmetry, we get:
λα ( N + 1) + λβ ( N + 1) v = v − γ or:
(
λα ( N + 1) = − γ
λβ ( N + 1) v = v
We know from the formula for the conditional expectation
E
[ V | k ] given above that:
γ = ρ
0
−
βN Σ
0
β 2 N 2 Σ
0
+ σ 2
µ
N ( α + βρ
0
)
4

and
λ =
βN Σ
0
β 2 N 2 Σ
0
+ σ 2
µ and hence we get:







−
β 2 N
βN Σ
0
2 Σ
βN Σ
0
0
+ σ 2
µ
α ( N + 1) = ρ
0
=
1
β 2 N 2 Σ
0
+ σ 2
µ
β ( N + 1)
−
β 2 N
βN Σ
0
2 Σ
0
+ σ 2
µ
N ( α + βρ
0
) that are two equations in the two unknowns α and β . Solving the system yields:
β = √
1
N s
σ 2
µ
Σ
0 and α = − βρ
0 which also gives us:
λ =
√
N
N + 1 s
Σ
0
σ 2
µ and γ = ρ
0
Finally, plugging back we obtain:
X ( v ) = β ( v − ρ
0
) = √
1
N s
σ
2
µ
Σ
0
( v − ρ
0
)
P ( k ) = ρ
0
+
√
N
N + 1 s
Σ
0
σ 2
µ k
(2) The aggregate order that comes from informed traders is
N ( α + βv ) =
√
N s
σ
2
µ
Σ
0
( v − ρ
0
)
(3) Notice that it is strictly increasing in the number of informed traders
N . As a result, their profits are strictly decreasing in N and the conditional variance of the price is ( N + 1)
− 1 Σ
0
, which converges to zero for large N.
5