Glosten-Milgrom Adverse Selection Model

Glosten-Milgrom (1985): Adverse Selection and the Spread

This is the most important model in market microstructure. Where ho-stoll-inventory-model explains the spread as compensation for inventory risk, Glosten-Milgrom explains it as a defense against informed traders. The mechanism is pure Bayesian inference: the market maker updates beliefs about the asset’s true value based on the direction of incoming orders.

The Core Insight

A market maker faces two types of counterparties:

• Informed traders who know something the maker does not
• Uninformed traders (noise/liquidity traders) who trade for exogenous reasons — rebalancing, redemptions, hedging

The maker cannot distinguish between them. But informed traders systematically buy when the asset is underpriced and sell when overpriced. This means:

• Buy orders are bad news for the maker (the buyer may know it is worth more)
• Sell orders are also bad news (the seller may know it is worth less)

The spread exists because the maker must protect against this adverse selection. Every trade carries an implicit information signal.

The math below formalizes this intuition using Bayes’ theorem — a method for updating beliefs given new evidence. The market maker starts with a guess about the asset’s value, then adjusts that guess each time someone buys or sells. A buy is evidence the asset is worth more (the buyer might be informed); a sell is evidence it’s worth less. The formulas compute exactly how much to adjust.

Probability prerequisites

The derivation below uses Bayes’ theorem and conditional probability. If these are unfamiliar, the Core Insight section above gives you the full conceptual understanding — the math just computes the exact bid and ask prices. See expectation-variance-and-mgfs and probability-measures-and-axioms for the probability foundations.

Setup

The asset has an unknown true value $V$ which takes one of two values:

$V=\{\begin{array}{ll}{V}_H& \text{with prior probability }p\\ V_L& \text{with prior probability }1-p\end{array}$

where $V_H>V_L$.

The population of traders:

• Fraction $\pi$ are informed: they know the true value of $V$
• Fraction $1-\pi$ are uninformed: they buy or sell with equal probability $\frac{1}{2}$, regardless of $V$

The market maker sets a bid $b$ and ask $a$, and must trade at those prices on demand. Competition among market makers drives expected profit to zero on each trade.

Derivation of the Ask Price

The ask is the price at which the maker is willing to sell. A buy order arrives. What is the expected value of $V$ given that someone wants to buy?

By Bayes’ theorem:

$a=E[V\mid \text{buy}]=V_H\cdot P(V=V_H\mid \text{buy})+V_L\cdot P(V=V_L\mid \text{buy})$

We need $P(V=V_H\mid \text{buy})$. The probability of observing a buy depends on the true state:

$P(\text{buy}\mid V=V_H)=\pi \cdot 1+(1-\pi )\cdot \frac{1}{2}=\frac{1+\pi }{2}$

Informed traders buy with certainty when $V=V_H$; uninformed buy with probability $\frac{1}{2}$.

$P(\text{buy}\mid V=V_L)=\pi \cdot 0+(1-\pi )\cdot \frac{1}{2}=\frac{1-\pi }{2}$

Informed traders never buy when $V=V_L$.

Applying Bayes:

$P(V=V_H\mid \text{buy})=\frac{P(\text{buy}\mid V_H)\cdot p}{P(\text{buy})}$

where:

$P(\text{buy})=p\cdot \frac{1+\pi }{2}+(1-p)\cdot \frac{1-\pi }{2}$

Therefore:

$\boxed{a=E[V\mid \text{buy}]=\frac{p(1+\pi )V_H+(1-p)(1-\pi )V_L}{p(1+\pi )+(1-p)(1-\pi )}}$

Derivation of the Bid Price

Symmetrically, the bid reflects the expected value given a sell:

$P(\text{sell}\mid V=V_H)=\frac{1-\pi }{2}$

$P(\text{sell}\mid V=V_L)=\frac{1+\pi }{2}$

$\boxed{b=E[V\mid \text{sell}]=\frac{p(1-\pi )V_H+(1-p)(1+\pi )V_L}{p(1-\pi )+(1-p)(1+\pi )}}$

Properties of the Spread

The spread widens with $\pi$

When $\pi =0$ (no informed traders), both buy and sell are equally likely under either state. The maker learns nothing from order direction, so $a=b=p{V}_H+(1-p)V_L$ — the spread is zero.

When $\pi =1$ (all traders informed), the ask jumps to $V_H$ and the bid drops to $V_L$. The maker knows that every buyer knows $V=V_H$ and every seller knows $V=V_L$, so the ask must be the high value and the bid the low value. The spread equals the full range of uncertainty.

For intermediate $\pi$, the spread is monotonically increasing in $\pi$. The more informed traders in the population, the wider the spread.

Zero-profit condition

The maker breaks even in expectation on each trade. Gains from trading with the uninformed exactly offset losses from trading with the informed. This is not generosity — it is the equilibrium outcome of competition among makers.

The spread is a tax on the uninformed

Uninformed traders pay the spread even though they carry no information. They subsidize the maker’s losses to the informed. This is the fundamental tension: liquidity provision is only viable if enough uninformed flow exists to absorb adverse selection costs.

Sequential Version: Bayesian Learning

In the full dynamic model, the maker updates beliefs after each trade. Let $p_t$ be the prior that $V=V_H$ before trade $t$.

After observing a buy:

$p_{t+1}=P(V_H\mid \text{buy})=\frac{p_t(1+\pi )}{p_t(1+\pi )+(1-p_t)(1-\pi )}$

After observing a sell:

$p_{t+1}=P(V_H\mid \text{sell})=\frac{p_t(1-\pi )}{p_t(1-\pi )+(1-p_t)(1+\pi )}$

This creates a martingale (a sequence where the best prediction of the next value is the current value — “today’s price is the best forecast of tomorrow’s price”): the sequence of mids $m_t=p_t{V}_H+(1-p_t)V_L$ converges to the true value as trades accumulate. The order flow is the price discovery mechanism.

Key dynamic properties:

• After a sequence of buys, $p_t\to 1$ and the spread narrows (less remaining uncertainty)
• After mixed flow, $p_t$ fluctuates and the spread reflects remaining uncertainty
• The speed of convergence depends on $\pi$: more informed traders means faster price discovery but wider spreads along the way

For how the Glosten-Milgrom adverse selection framework applies to token launches and bonding curves, see bonding-curves.

Connection to Other Models

• ho-stoll-inventory-model: inventory risk is complementary to adverse selection. A maker faces both simultaneously.
• kyle-lambda: Kyle models the strategic behavior of the informed trader — how they optimally trade to exploit their information without revealing it too quickly.
• spread-decomposition: Huang-Stoll (1997) empirically separates the adverse selection component from inventory and order-processing costs.

---

Companion notebook: notebook — simulate sequential Bayesian updating, visualize belief trajectories and spread dynamics as a function of $\pi$.

Questions to sit with:

1. In the zero-profit equilibrium, the maker breaks even. Who, then, benefits from the spread? Who loses?
2. If you could observe $\pi$ in real time (e.g., by classifying flow as informed vs. uninformed), how would you use that to set quotes?
3. In OTC bond markets, dealers see each trade individually (no consolidated tape). How does this affect the speed of Bayesian learning compared to an exchange-traded stock?