Summary Glosten and Milgrom model explanation

The Glosten and Milgrom (1985) Model:

Basic idea:

 If the insiders in the market are buyers, then the market maker will experience a fall in her stock
inventory and will increase the ask price to avoid large losses. They will increase this ask price to
V H as the ask price was too low. The stock inventory will decrease because buyers will buy more
and more stocks due to them having the knowledge, meaning the stock inventory decreases.
 The reverse happens if the insiders are sellers.
 In this model, the insiders perform a useful function: they push prices in line with the true value
b
P.
 This gives a favourable picture of the activities of insiders, unlike the King and Roell model where
they found that insider trading was harmful.

First trade:

 We call the initial expectation μ the prior expectation and 0.5 the prior probability of a high
value outcome, V H . Denote these initial values as μ1 and p1 to indicate that they are only valid
for the first trade.
 Recall that the market maker sets the initial ask price Pa (if she thinks the trade will happen)
such that she breaks even in expectation if a sale takes place.
 So Pa is the posterior expectation conditional upon finding a buyer at the first trade.
 By the same argument, Pb is the posterior expectation conditional upon meeting a seller.

Second trade:

 Once the first trade is completed, a rational market maker will reset μ2 and the trading process
will move into the next round.
 The only complication is that the probability a high value has changed, from prior probability of
0.5 to posterior probability, p2. NOT P1 because it is the second trade.

Then because the expectation is a probability weighted average, we have:

μ2= p2 V H +(1− p2 )V LWhere p2 V H is the probability of a high value and (1− p 2)V L is the
probability of a low value.

This can be rewritten as:
μ2−V L = p2 (V H −V L )
μ2−V L
OR: p2= (1) => this is what the posterior probability will be
V H −V L

Note that q, V H , V L are fixed.

 Now suppose that the market maker happened to meet a buyer
on the first trade. If the first client buys then μ2=P a
We have already seen that in the one-shot case: (previous lecture)

Pa=μ+ 0.5 q(V H −V L )
a
Hence, by using μ2=P and μ=μ 1

, μ2=μ 1+0.5 q (V H −V L ) (2)

Substituting (2) into (1), we get:

μ2−V L
p2 = and now sub in μ2=μ 1+0.5 q (V H −V L )
V H −V L

μ1 +0.5 q( V H −V L )−V L
p2 =
V H −V L
μ1 +0.5 q V H −0.5 qV L−V L
p2 =
V H −V L
 Separate out the fraction

μ1−V L 0.5 q V H −0.5 qV L
p2 = +
V H −V L V H −V L
 Second fraction can be simplified

μ1−V L 0.5 q (V H −V L )
p2 = +
V H −V L (V ¿ ¿ H −V L )¿

μ1−V L
p2 = +0.5 q
V H −V L
μ1−V L q
p2= +
V H −V L 2
μ2−V L μ1−V L
Above, we found that p2= so p1=
V H −V L V H −V L
Therefore:
q
p2= p1 + > p1 (3)
2
The dealer revises up the probability of a high value.

 If the first client that the MM meets is a seller then μ2=P b and
going through the same algebra, the probability of a high value is:
Remember in the one-shot case: Pb=μ−0.5 q (V H −V L )

By using μ2=P b and μ=μ 1

μ2=μ 1−0.5 q(V H −V L )
μ2−V L
Like we did before, sub this equation in p2=
V H −V L

μ1−0.5 q (V H −V L )−V L
p2 =
V H −V L