Glosten-Milgrom Model: Adverse-Selection Spreads (2026)

What the Glosten-Milgrom model explains

On 1 March 1985, Lawrence Glosten and Paul Milgrom published Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders in the Journal of Financial Economics. Their result is sharper than the familiar claim that dealers widen spreads when trading is risky.

A positive spread can exist even when the dealer is competitive, risk-neutral, carries no inventory penalty and has no processing cost. The dealer loses, on average, when the counterparty knows more about the asset. Bid and ask quotes are conditional expectations that offset that adverse selection.

The model is sequential. After every customer action, the specialist updates beliefs and posts new quotes. That makes it a useful complement to Kyle's informed-trading model, where a strategic insider hides inside aggregate noise flow. Our market microstructure guide places both models alongside inventory and order-processing explanations of the spread.

The calculation below uses a two-value, one-unit teaching version. It preserves the Bayesian mechanism, but it is simpler than the original paper's richer customer valuations and option not to trade. That distinction matters: this is a special case of the Glosten-Milgrom logic, not a verbatim reproduction of every assumption in the paper.

The Bayesian bid and ask

Let the liquidation value be either $V_H$ or $V_L$, with current public belief

$$\Pr(V=V_H)=\theta, \qquad \Pr(V=V_L)=1-\theta.$$

An arriving trader is informed with probability $\mu$. An informed trader buys one unit in the high state and sells one unit in the low state. With probability $1-\mu$, the trader is liquidity-motivated and buys or sells with equal probability. In this simplified version everyone trades.

The likelihood of a buy is therefore

$$\Pr(B\mid V_H)=\mu+\frac{1-\mu}{2} =\frac{1+\mu}{2},$$ $$\Pr(B\mid V_L)=\frac{1-\mu}{2}.$$

Bayes' rule gives the posterior high-state probability after a buy:

$$\theta_B =\Pr(V_H\mid B) =\frac{(1+\mu)\theta} {(1+\mu)\theta+(1-\mu)(1-\theta)}.$$

A competitive dealer's zero-profit ask is the expected value conditional on receiving that buy:

$$\boxed{ A=\mathbb{E}[V\mid B] =\theta_BV_H+(1-\theta_B)V_L }.$$

For a sell, the likelihoods reverse:

$$\theta_S =\Pr(V_H\mid S) =\frac{(1-\mu)\theta} {(1-\mu)\theta+(1+\mu)(1-\theta)},$$ $$\boxed{ B=\mathbb{E}[V\mid S] =\theta_SV_H+(1-\theta_S)V_L }.$$

These are break-even conditions conditional on trade direction. Posting the unconditional mean would make the dealer lose to informed buyers in the high state and informed sellers in the low state.

The symmetric-prior case makes the economics visible. If $\theta=1/2$, then

$$\theta_B=\frac{1+\mu}{2}, \qquad \theta_S=\frac{1-\mu}{2}.$$

Writing $m=(V_H+V_L)/2$ and $\Delta=V_H-V_L$,

$$A=m+\frac{\mu\Delta}{2}, \qquad B=m-\frac{\mu\Delta}{2},$$

so the full adverse-selection spread is

$$\boxed{A-B=\mu(V_H-V_L)}.$$

The spread rises with both informed-trader incidence and the value gap. If either $\mu=0$ or $V_H=V_L$, it collapses to zero in this stripped-down model.

Posterior odds make repeated updating easier to audit. Let $O=\theta/(1-\theta)$ be the odds of the high state. A buy multiplies those odds by the likelihood ratio

$$\frac{\Pr(B\mid V_H)}{\Pr(B\mid V_L)} =\frac{1+\mu}{1-\mu},$$

while a sell multiplies them by its reciprocal. With $\mu=0.2$, one buy multiplies high-state odds by $1.5$. Starting from even odds, the posterior odds are therefore $1.5$, which converts to $1.5/(1+1.5)=0.6$. This is the same answer as the probability calculation and a useful implementation test.

The no-trade decision complicates the original model. Uninformed customers have heterogeneous valuations and may decline both quotes; informed customers trade only when their information makes a quote attractive. The dealer's ask therefore affects who chooses to buy, and that selected set determines the conditional expectation used to set the ask. Quotes solve a fixed-point problem. Treating every arrival as a compulsory unit trade removes that selection and is why the binary example has such a compact spread formula.

That richer setup also clarifies terminology. $A-B$ is the quoted spread. A customer's realised effective spread depends on the transaction price relative to the pre-trade midpoint, while the dealer's adverse-selection loss depends on value after information is revealed. Those quantities can differ even when the model attributes the entire quoted spread to information.

Quotes after a trade

Take $V_H=105$ dollars, $V_L=95$ dollars, $\theta=0.5$ and $\mu=0.2$. Before the first trade, expected value is 100 dollars.

For a buy,

$$\Pr(B\mid V_H)=0.6, \qquad \Pr(B\mid V_L)=0.4,$$

and therefore

$$\theta_B=\frac{0.6\times0.5} {0.6\times0.5+0.4\times0.5} =0.6.$$

The ask is $0.6(105)+0.4(95)=101$ dollars. By symmetry, the bid is 99 dollars, giving a two-dollar spread. The closed form confirms it: $0.2(105-95)=2$ dollars.

Suppose the next observed action is a buy. The public prior for the following round becomes $\theta=0.6$, not $0.5$. Substituting that posterior into the same quote equations gives

$$A_{\mathrm{next}} =\frac{0.6(0.6)(105)+0.4(0.4)(95)} {0.6(0.6)+0.4(0.4)} \approx101.92\text{ dollars},$$ $$B_{\mathrm{next}} =\frac{0.4(0.6)(105)+0.6(0.4)(95)} {0.4(0.6)+0.6(0.4)} =100\text{ dollars}.$$

Both quotes shift upwards because a buy is evidence for the high state. They are no longer symmetric around the original 100-dollar prior.

The calculation also shows why "a buy widens the spread" is not a general implication. Here the next ask is about 101.92 dollars and the next bid is 100 dollars, so the spread narrows slightly from two dollars to about 1.92 dollars as beliefs move away from maximum uncertainty at $\theta=0.5$. A trade shifts the level of both quotes; whether it widens or narrows their distance depends on the posterior distribution and customer-demand model.

The transaction price at each trade is the relevant conditional expectation - ask after a buy, bid after a sell. Glosten and Milgrom prove that this transaction-price sequence is a martingale with respect to public and specialist information. That does not mean consecutive trade-price changes are independent. Bid-ask bounce and changing posterior beliefs can still create serial dependence.

The model's comparative statics are conditional, not slogans. Holding customer demand fixed, better insider information and a higher informed-to-uninformed arrival ratio increase the adverse-selection component of quotes. But changing liquidity-trader demand can alter both trade probabilities and selection at the quote. A production calibration should fit the arrival process and quote response together rather than infer informed-trader prevalence from spread alone.

The same point applies around announcements. A wider spread before earnings may reflect a broader value distribution, a larger chance of informed arrival, thinner displayed depth or dealer inventory policy. Glosten-Milgrom supplies a disciplined information channel; it does not identify that channel from one observed price difference.

For research, report the assumed state values, prior, arrival probabilities and no-trade treatment. Without them, a quoted informed-trader fraction cannot be mapped back to the spread calculation or reproduced in another sample.

For a market maker, the practical lesson is about conditioning. A quote model should ask how trade sign, size and the no-trade decision alter the posterior, not merely add a fixed margin to a stale midpoint. That logic survives in modern market-making systems, even though production models use far more state.

A practical diagnostic is to compare quote revision with half-spread revenue. For each aggressive trade, record the midpoint immediately before the trade, the execution price and the midpoint after a fixed horizon. The pre-trade execution concession estimates gross spread capture; the later midpoint move against the dealer estimates adverse selection. The horizon is part of the definition. At 100 milliseconds the measure may mostly capture latency; at 30 minutes it may include unrelated news.

What the model leaves out

The binary example makes informed traders mechanical: buy high, sell low. Real informed traders choose timing, size, venue and whether to trade. Their orders arrive in runs because parent orders are split. A dealer that assumes independent one-unit arrivals will underestimate the information in persistent flow.

The original framework also abstracts from dealer inventory costs, exchange fees, latency, tick size and queue position. Those forces can produce or widen a spread even without private information. Observing a wide spread does not identify $\mu$ unless the other components are modelled or measured.

Empirical PIN models add latent information-event and order-arrival processes. Easley, Kiefer, O'Hara and Paperman's 1996 paper is the canonical reference, but PIN is an estimated model parameter, not the literal fraction of all traders who possess private information. Later VPIN work uses a different volume-clock construction and remains contested as a real-time toxicity measure.

Finally, beliefs do not always converge neatly to the true state. In the model they can under regularity conditions and continued informative arrivals. In a live market the liquidation value moves, public news arrives and the signal distribution changes while the dealer learns. A high-frequency trading system therefore treats state estimates as perishable.

Glosten-Milgrom is best read as a mechanism, not a spread calculator. Trade direction changes beliefs; competitive quotes reflect those conditional beliefs; the distance between them pays for being picked off.

Frequently Asked Questions

What is the Glosten-Milgrom model?

It is a sequential-trade model in which a specialist quotes against customers who may possess superior information. Competitive bid and ask prices equal expected asset value conditional on a sell or buy respectively.

Why is the bid below the ask?

A buy is evidence that value may be high, while a sell is evidence that value may be low. The dealer conditions on those different events, so $\mathbb{E}[V\mid B]>\mathbb{E}[V\mid S]$ when informed trading is possible.

Is the spread always $\mu(V_H-V_L)$?

Only in the symmetric two-state special case with equal buy and sell probabilities for liquidity traders. General priors, customer valuations, abstention and varying order sizes change the formula.

What happens after each trade?

The posterior after the trade becomes the prior for the next quote. A buy generally shifts both quotes upwards; a sell shifts them downwards.

How does Glosten-Milgrom differ from Kyle?

Glosten-Milgrom studies sequential customer trades and bid-ask quotes. Kyle studies a strategic insider whose order is pooled with noise flow and derives a price-impact slope. Both price the information in order flow.

Which paper should I cite?

Cite Glosten and Milgrom (1985), Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders, Journal of Financial Economics 14, 71 to 100.