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A VISUAL GUIDE TO MARKET MICROSTRUCTURE MODELS
NIHAD ALIYEV
ABSTRACT.In this note, I provide geometric interpretations of the classic asymmetric
information models of Kyle (1985) and Glosten and Milgrom (1985). This visual ap-
proach helps to better understand how market makers infer private information from
order flow and how prices adjust in response.
JEL Classification: G14, D4, D82.
Keywords: Market microstructure, Kyle, Glosten-Milgrom.
Current version: April 20, 2025.
Email: nihad.aliyev@uts.edu.au.
I have written this note during my PhD at the University of Technology Sydney (UTS) as a way to better understand
market microstructure models.
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1. INTRODUCTION
Classic asymmetric information models of market microstructure study how informa-
tion is revealed and prices are formed in financial markets. Two workhorse models in this
literature are Kyle (1985) and Glosten and Milgrom (1985). They differ in their assumptions
and mechanics, but both aim to explain how prices reflect private information in the presence of
noise and informed traders. In this note, I provide a geometric interpretation of each model and
translate their analytical insights into visual frameworks. I focus on how market makers infer
information from order flow and how equilibrium prices emerge as a function of traders’ beliefs
and behaviors.
2. KY LE (1985)
Consider a single-period economy and one risky asset with the value vdrawn from a
normal distribution N(µv, σ2
v), with σv>0. There are three types of agents in the market:
informed traders who know v, noise traders who trade randomly, and a competitive market
maker who sets the price pbased on total order flow y. The portion of the aggregate demand
that comes from the informed trader is yIand the remaining that comes from the noise trader is
yE∼N(0, σ2
ε), with σε>0, i.e., y=yI+yε. Suppose the price function of the risk-neutral
market maker who observes the aggregate demand yand breaks-even in expectation is given by
p=E[v|y] = µ+λy =µ+λ(yI+yε)(1)
With the price representation of Eq. (1), µis the price of the asset with zero net aggregate
demand, λis the price responsiveness to aggregate demand changes. The coefficients µand λ
will be determined in equilibrium. Suppose the informed demand is linear in the asset value v
yI=α+βv (2)
where the coefficients αand βwill also be determined in equilibrium. The expected profit of
informed traders is π=E[(v−p)yI], which is equal to
π=E[(v−µ−λyI−λyε)yI](3)
after substituting the price function in Eq. (1)
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The informed demand yIthat maximises the expected profit in Eq. (3) follows from the
first order condition ∂π
∂yI= 0 as
yI=v−µ
2λ(4)
The price set by the market maker who only observes the aggregate demand yand
breaks-even in equilibrium due to the perfect competition among market makers is
p=E[v|y] = E[v] + Cov(y, v)
V ar(y)(y−E[y]) (5)
where the second equality follows from the projection theorem for joint normal random vari-
ables. Substituting y=yI+yε=v−µ
2λ+yεto Eq. (5) obtains
p=µv+
1
2λσ2
v
(1
2λ)2σ2
v+σ2
ε
(µv−µ
2λ)
| {z }
µ
+1
2λσ2
v
(1
2λ)2σ2
v+σ2
ε
| {z }
λ
y(6)
Imposing the linear pricing function in Eq. (1) obtains µ=µvfrom the first part of Eq.
(6) and λ=σv
2σεfrom the second part. Substituting µ=µvand λ=σv
2σεto Eq. (1) obtains the
equilibrium price as
p=µv+σv
2σε
y(7)
and to Eq. (4) obtains the equilibrium informed demand as
yI= (v−µv)σε
σv
(8)
Eqs. (7) and (8) together constitute an equilibrium in which competitive market maker
sets the price and informed traders choose their demand that maximises their profits. It follows
from Eq. (7) that the market maker sets the price to µvwhen the aggregate demand is zero
(y= 0) and changes the prices more in response to aggregate demand when the asset volatility
is high ( ∂p
∂σv>0) or noise trader demand volatility is low ( ∂p
∂σϵ<0). It follows from Eq. (8)
that the informed buy when v > µvand sell v < µvwith the amount of buy and sell orders
increasing with noise trader demand volatility σεand decreasing with the asset volatility σv.
Figure 1 visualises the market maker’s rationale when setting the price as in Eq. (7).
Figure 1A plots the price pagainst the aggregate demand y. This illustrates the equilibrium
price rule in Eq. (7).
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Figure 1B plots what happens in the background. By observing the aggregate demand
y, the market maker guesses the informed demand. From the market maker’s perspective, the
best guess for the informed traders’ demand follows from Eq. (8) as
E[yI|y] = E[(v−µv)σε
σv
|y] = E[v|y]σε
σv
−µv
σε
σv
= (µv+σv
2σε
y)σε
σv
−µv
σε
σv
=y
2
(9)
where the second line follows from the equilibrium price in Eq. (7). With no additional infor-
mation, the market maker simply guesses that half of the aggregate demand is from the informed
traders.
Figure 1C shows the equilibrium price with respect to the best guess on the informed
demand using
p=µv+σv
σε
E[yI|y](10)
Eq. (10) seems slightly more intuitive than Eq. (7) and its illustration in Figure 1C
is an interesting way of thinking about the equilibrium concept. In this equilibrium, from the
market maker’s perspective, the informed demand E[yI|y]maps to price plinearly with slope
σv
σε. However, it is also the case that from the informed trader’s perspective, the actual informed
demand yImaps to the actual asset value vlinearly with the same slope σv
σε. To see this, re-write
Eq. (8) as
v=µv+σv
σε
yI(11)
Eqs. (10) and (11) show that the mapping of E[yI|y]to pfrom the market maker’s
perspective is similar to the mapping of yIto vfrom the informed trader’s perspective. In
other words, the price function of the market maker aims to replicate the value function of
the asset using the best guess for informed demand. Therefore, Figure 1C can be interpreted
as (E[yI|y],p) plane from the market maker’s perspective and (yI,v) plane from the informed
traders’ perspective. Rather than looking at Figure 1A to understand just the equilibrium pricing
rule of the market maker, it seems more intuitive to look at Figure 1C to understand both the
equilibrium pricing rule and the informed trader’s demand, which together constitute the overall
equilibrium.
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Figure 1. Kyle (1985) model
Price ()
Aggregate
demand ()
Best guess for informed
demand ()
1A
1B
1C
3. GL OS TE N AN D MIL GRO M (1985)
Consider one risky asset with value vdrawn from the set v∈0,1. A competitive market
maker has an initial belief Pr(v= 1) = p1=1
2. The market maker sets the bid (Bt) and
ask (At) prices at time tbefore trade. Trading takes place over periods t= 1, . . . ,T , and the
risky security pays off in period T+ 1. In each period, a trader arrives sequentially and trades
one unit with a competitive market maker. The trader may be informed or uninformed. The
informed trader knows the true asset value vand chooses the order that maximizes their profit.
The uninformed trader buys or sells at random with equal probability. The probability of an
informed arrival is µ, and of an uninformed arrival is 1−µ.
Let Dtdenote the trade direction at time t, where Dt=−1for a sell and Dt= +1 for
a buy. Let Ptdenote the transaction price. Public information at time tconsists of the sequence
of past trades and prices, denoted by ht= (Dτ, Pτ)t−1
τ=1. The market maker observes ht, updates
her belief pt= Pr(v= 1 |ht)via Bayes’ rule, and sets quotes (Btand At) to break even in
expectation.
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In equilibrium, the informed trader with a signal v= 1 will always buy and with a
signal v= 0 will always sell. This is because in equilibrium v= 1 ≥At(buying at Atis
always profitable when v= 1) and v= 0 ≤Bt(selling at Btis always profitable when v= 0).
The market maker’s conditional probabilities follow from Bayes rule as
Pr(Dt= +1|v= 1, ht) = 1 + µ
2(12)
Pr(Dt=−1|v= 1, ht) = 1−µ
2(13)
Pr(Dt= +1|ht) = 1 + µ(2pt−1)
2(14)
Pr(Dt=−1|ht) = 1 + µ(1 −2pt)
2(15)
The market maker’s zero expected profit (break-even) condition requires that the quoted price
equals the expected asset value conditional on the trade direction.
Bt=E[v= 1|ht, Dt=−1] = pt·Pr(Dt=−1|v= 1, ht)
Pr(Dt=−1|ht)(16)
At=E[v= 1|ht, Dt= +1] = pt·Pr(Dt= +1|v= 1, ht)
Pr(Dt= +1|ht)(17)
Substituting conditional probabilities into Eqs. (16)-(17) obtains:
Bt=pt
pt+δ·(1 −pt)(18)
At=pt
pt+δ−1·(1 −pt)(19)
where
δ=1 + µ
1−µ(20)
and the market maker’s belief ptbased on the order imbalance Nt(number of buys minus sells
up to time t) is
pt=δNt
1 + δNt(21)
Substituting ptinto Eqs. (18) and (19), and finding the their difference obtains the bid-ask
spread at time tas
St=At−Bt=δNt·(δ−δ−1)
(δNt+δ)·(δNt+δ−1)(22)
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The model admits a natural geometric interpretation in which the pricing mechanism
can be understood as a transformation of the order imbalance Nt. This imbalance serves as a
sufficient statistic for the market maker’s belief about the asset’s value (pt), and is the time-
varying determinant of the bid-ask spread St. The other key parameter that affects the spread
is δ=1+µ
1−µ, which captures the informativeness of order flow. Since δincreases with the
probability of informed trading µ, a higher δimplies that trades are more informative, causing
the market maker to adjust beliefs more aggressively in response to order flow.
Figure 2A plots the bid-ask spread Stagainst the order imbalance Ntbased on Eq. (22)
for a given value of δ. When Nt= 0, the market maker observes a perfectly balanced sequence
of buys and sells and therefore does not learn anything about the asset’s value (pt=p1= 0.5).
The market maker remains maximally uncertain about the asset value, which leads to the widest
possible spread. This is the point where informational asymmetry is highest, and the spread
serves as protection against trading with an informed agent. As Ntgrows in either direction
(more buys or more sells) the spread narrows. The market maker gradually leans quotes in the
direction of the order flow and becomes more confident about the asset’s true value. In the
limit, as Nt→+∞(resp. Nt→ −∞), the spread converges to zero, as both bid and ask prices
approach 1 (resp. 0).
Figure 2B shows how the market maker updates the belief about the asset’s value based
on observed imbalance. Specifically, it plots the mapping from Ntto pt, the probability that
the asset value is 1, as given in Eq. (21). This belief function is smooth, strictly increasing,
and S-shaped. When Nt= 0, the market maker assigns equal probability to either state, i.e.,
pt= 0.5. As buy orders dominate (Nt>0), the market maker’s belief ptapproaches 1. As
sell orders dominate (Nt<0), the market maker’s belief ptapproaches 0. This belief forms the
basis for the bid and ask quotes posted in each round as per Eqs. (18) and (19).
Figure 2C plots the spread Stdirectly against the belief pt. The relationship is symmetric
around pt= 0.5, where the spread is maximum. As the belief moves closer to either extreme
(toward 0 or 1) the spread declines. Intuitively, when the market maker is most uncertain about
the asset’s value (pt= 0.5), they set the maximum spread to protect against adverse selection.
As belief becomes more precise in either direction, this informational risk decreases, and the
market maker tightens the bid-ask spread accordingly.
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Figure 2. Glosten and Milgrom (1985) model
Imbalance (
Belief (
Bid-ask
spread ()
0
2A
2B
2C
4. CONCLUSION
The models of Kyle (1985) and Glosten and Milgrom (1985) offer foundational insights
into how prices incorporate private information in financial markets. By providing geometric
interpretations, this note highlights how market makers use order flow to infer asset values and
manage adverse selection risk. These visual representations make it easier to understand the
mechanisms through which equilibrium prices and spreads emerge, and how beliefs evolve in
response to trading activity.
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REFERENCES
Glosten, L. R. and Milgrom, P. R. (1985), ‘Bid, ask and transaction prices in a specialist market with heteroge-
neously informed traders’, Journal of Financial Economics 14(1), 71–100.
Kyle, A. S. (1985), ‘Continuous auctions and insider trading’, Econometrica 15(35), 1315–1335.
