6.4.3 The Glosten–Milgrom Sequential-Trade Model

Basic Prerequisites: Bayesian Statistics and 02 · Informed vs Uninformed.

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1. Intuition & Practical Objective

Glosten–Milgrom (1985) built the smallest market where the spread is endogenous: one risky asset worth either $V_L$ or $V_H$; a risk-neutral, competitive market maker; and a random stream of traders who are either informed (know the state) or uninformed (trade noise). The maker’s only tool is a price rule. The model’s striking conclusions, all verified against Hasbrouck Ch 5 and Foucault Ch 3:

1. The spread exists purely because of potential informed trading. With zero informed traders ($\pi =0$) the spread collapses to zero even though the maker still does all the mechanics.
2. Quotes are “regret-free” conditional means. The maker sets ask = expected value given a buy just hit it and bid = expected value given a sell just hit it. Because a buy is mildly informative, the maker marks the ask up and the bid down - and the gap between the two is the spread.
3. Transaction prices are a martingale with respect to the maker’s information: prices adapt as fast as the maker learns, so the trade-price series cannot be predicted. Yet the observable tick-by-tick moves convey information - that is price discovery.

The practical objective: implement the market maker’s inference + quoting policy, run it, and watch the belief and the spread evolve through hundreds of trades - a faithful, working rendition of the primary paper.

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2. Mathematical Ground Truth & Derivations

Let $V\in \{V_L,V_H\}$, prior ${\theta }_{t-1}=\mathbb{P}(V=V_H)$, and $\pi$ = probability a given trader is informed. Uninformed buy/sell with probability $\frac{1}{2}$; informed buy if $V=V_H$, sell if $V=V_L$.

Arrival law of a Buy order:

$$\mathbb{P}(B\mid V_H)=\pi +(1-\pi )\frac{1}{2}=\frac{1+\pi }{2},\mathbb{P}(B\mid V_L)=(1-\pi )\frac{1}{2}=\frac{1-\pi }{2}.$$

Bayes update (Hasbrouck eq. 5.1/5.5; Foucault eq. 3.16–3.17):

$${\theta }_t^{+}=\frac{\frac{1+\pi }{2}{\theta }_{t-1}}{\frac{1+\pi }{2}{\theta }_{t-1}+\frac{1-\pi }{2}(1-{\theta }_{t-1})},{\theta }_t^{-}=\frac{\frac{1-\pi }{2}{\theta }_{t-1}}{\frac{1-\pi }{2}{\theta }_{t-1}+\frac{1+\pi }{2}(1-{\theta }_{t-1})}.$$

Zero-profit (competitive, regret-free) quotes (Hasbrouck eq. 5.2/5.6; Foucault eq. 3.5/3.7):

$$A_t=\mathbb{E}[V\mid B_t]=V_L+{\theta }_t^{+}(V_H-V_L),B_t=\mathbb{E}[V\mid S_t]=V_L+{\theta }_t^{-}(V_H-V_L).$$

Vanishing spread as learning converges. With ${\theta }_0=\frac{1}{2}$ and a value dispersion of $V_H-V_L=2$:

$$S_t=A_t-B_t=\frac{\pi {\theta }_{t-1}(1-{\theta }_{t-1})}{\pi {\theta }_{t-1}+\frac{1-\pi }{2}}(2)+\frac{\pi {\theta }_{t-1}(1-{\theta }_{t-1})}{\pi (1-{\theta }_{t-1})+\frac{1-\pi }{2}}(2),$$

which at ${\theta }_{t-1}=\frac{1}{2}$ gives the celebrated first-trade spread

$$\boxed{S_{\theta =\frac{1}{2}}=\pi (V_H-V_L)}.$$

The spread is widest at maximal uncertainty ($\theta =\frac{1}{2}$), shrinks toward $0$ as $\theta \to 1$ or $0$ (the maker learns the value), and is zero for all $\theta$ when $\pi =0$ - adverse selection alone generates the entire spread. Hasbrouck’s equivalent formula (with $\delta =\mathbb{P}(V=V_L)$) is $A-B=\frac{4(1-\delta )\delta \mu (V_H-V_L)}{1-(1-2\delta {)}^2{\mu }^2}$, identical at $\delta =\frac{1}{2}$ to $\pi (V_H-V_L)$.

Net wealth-transfer identity (Hasbrouck eq. 5.4): expected gains from uninformed exactly balance expected losses to informed,

$$(A-\mathbb{E}[V\mid U,B])\mathrm{Pr}(U\mid B)=-(A-\mathbb{E}[V\mid I,B])\mathrm{Pr}(I\mid B).$$

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3. Computational Implementation - the sequential Bayesian market maker (stdlib only)

Simulate the maker learning. With true value $V=V_H=12$, $\pi =0.25$: the maker starts at $\theta =0.5$, quotes ask $11.25$/bid $10.75$, and after thousands of informed-biased buys drives $\theta \to 1$ and the spread to $0$. Ran and verified.

The output is exactly the model’s three claims: (1) spread grows linearly in $\pi$ (and is $0$ at $\pi =0$), (2) the quoted values track $\theta$, and (3) over time the competing maker learns the truth - price converges to $V_H$ and the spread vanishes. This is price discovery: transaction prices are martingales for the maker, yet they converge to the true value.

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4. Failure Modes & First-Principles Breakdowns

1. Missing why the spread is asymmetric. In general $\theta \ne \frac{1}{2}$, so $A$ and $B$ are not symmetric about the unconditional mean. Quoting mid $=\mathbb{E}[V]$ is wrong once flow is one-sided - the mid should equal ${\mu }_t={\theta }_t{V}_H+(1-{\theta }_t)V_L$ only when $\theta =\frac{1}{2}$ (Hasbrouck Ch 5; Foucault Ch 3).
2. Forgetting the quotes are conditional, not point estimates. $A=\mathbb{E}[V\mid B_t]$ is an expected value given a specific order. A maker who treats the ask as “value + constant fee” has thrown away the information content of the order that just filled them.
3. The martingale does not mean “no information.” Transaction prices being a martingale w.r.t. the maker’s info does not imply trades are uninformative - it is precisely because they are informative that the maker marks them into the quotes. Conflating the two is the classic misread of GM.

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5. References

• Glosten & Milgrom (1985), JFE 14(1), 71–100
• Hasbrouck (2007), Empirical Market Microstructure
• Foucault, Pagano & Röell (2013), Market Liquidity
• Bagehot (1971) (= Walter Bagehot pseudonym), The only game in town, Financial Analysts Journal 27(2), 12–14

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6. Connected Graph Bridges

• Back: 02 · Informed vs Uninformed · 01 · From Zero
• Forward: 04 · Spread Decomposition (add order-processing & inventory on top of this information spread)
• Sibling: Avellaneda–Stoikov Optimal Quoting · Spread Decomposition & the Roll Model