2.10.3 Glosten–Milgrom

Basic Prerequisites: 01 - From Zero and Bayesian Statistics (Bayes’ rule, conditional expectation).

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1. Intuition & Practical Objective

The two-point, one-shot model of page 01 is a snapshot. The real game is a sequence: trades arrive one at a time, the maker updates her belief after every one of them, and re-quotes. Glosten & Milgrom (1985) showed that this sequential structure - not the size of the trade, not inventory, not hedging - is by itself enough to produce a spread that behaves like a real one: it is positive at maximum uncertainty, it narrows as information is revealed, and the price it generates is a martingale.

The essential insight is a change of what “the price” means. There is no single price. There is a bid and an ask, and each is the expected value of the asset conditional on the direction of the trade that is about to occur. The maker is not setting a price she believes; she is setting two prices, each conditioned on being adversely selected in a known direction.

Three structural results carry the rest of the folder:

• The quotes are the maker’s own posteriors. $A=\mathbb{E}[V\mid \text{buy}]$ and $B=\mathbb{E}[V\mid \text{sell}]$ are, mechanically, the two possible Bayesian updates of the current belief. The quote is the update, pre-computed.
• The price is a martingale. Because the quotes are conditional expectations, the expected next price equals the current price exactly - no drift, no “adverse selection drag” term, nothing.
• Adverse selection is an equilibrium, not an assumption. Nobody in the model is assuming a spread. The spread is the unique price at which a competitive maker does not lose money to informed flow. If you enter the game with a different spread, you either bleed (too tight) or are picked off by order-flow competition (too wide).

The one-sentence essence. “The maker’s bid and ask are her two possible Bayesian posteriors of the asset’s value after seeing a sell and after seeing a buy - so the spread is the change in belief the two directions imply, and the resulting price process is a martingale by construction.”

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2. Mathematical Ground Truth & Derivations

Setup. $V\in \{V_L,V_H\}$, prior belief ${\theta }_t=\mathbb{P}(V=V_H\mid {\mathcal{F}}_t)$, informed share $\pi$. Arrivals:

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

Zero-profit quotes (the equilibrium). Competition forces expected profit to zero conditional on the direction of the trade - a maker cannot cross-subsidise the ask with the bid, because any maker who tried would be picked off on one side. Hence

$$A_t=\mathbb{E}[V\mid B,{\mathcal{F}}_t],B_t=\mathbb{E}[V\mid S,{\mathcal{F}}_t].$$

Writing $n_A=(1+\pi ){\theta }_t+(1-\pi )(1-{\theta }_t)$ and $n_B=(1-\pi ){\theta }_t+(1+\pi )(1-{\theta }_t)$ (these are twice the arrival probabilities), Bayes’ rule gives

$$\boxed{A_t=\frac{V_H(1+\pi ){\theta }_t+V_L(1-\pi )(1-{\theta }_t)}{(1+\pi ){\theta }_t+(1-\pi )(1-{\theta }_t)},B_t=\frac{V_H(1-\pi ){\theta }_t+V_L(1+\pi )(1-{\theta }_t)}{(1-\pi ){\theta }_t+(1+\pi )(1-{\theta }_t)}}$$

The belief recursion. $A_t$ is the posterior after a buy, so

$$\boxed{{\theta }_t^{+}=\frac{\mathbb{P}(B\mid V_H){\theta }_t}{\mathbb{P}(B\mid V_H){\theta }_t+\mathbb{P}(B\mid V_L)(1-{\theta }_t)}=\frac{(1+\pi ){\theta }_t}{(1+\pi ){\theta }_t+(1-\pi )(1-{\theta }_t)}=\frac{(1+\pi ){\theta }_t}{n_A}}$$

and symmetrically ${\theta }_t^{-}=(1-\pi ){\theta }_t/n_B$. At $\theta =\frac{1}{2}$: ${\theta }^{+}=(1+\pi )/2$ and ${\theta }^{-}=(1-\pi )/2$, so the spread at maximum uncertainty is

$$A-B=\pi (V_H-V_L).$$

The martingale property (exact). Arrival probabilities at the current belief are $\mathbb{P}(B)=\frac{1}{2}{n}_A$ and $\mathbb{P}(S)=\frac{1}{2}{n}_B$. Therefore, using the expressions above,

$$\mathbb{E}[P_{t+1}\mid {\mathcal{F}}_t]=\frac{1}{2}{n}_A\cdot \frac{V_H(1+\pi )\theta +V_L(1-\pi )(1-\theta )}{n_A}+\frac{1}{2}{n}_B\cdot \frac{V_H(1-\pi )\theta +V_L(1+\pi )(1-\theta )}{n_B}$$ $$=\frac{1}{2}\left[V_H\theta \right((1+\pi )+(1-\pi ))+V_L(1-\theta )((1-\pi )+(1+\pi )\left)\right]=V_H\theta +V_L(1-\theta )=P_t.$$

Zero drift, exactly, at every belief. Verified to machine precision in §3.

Information eventually gets revealed. The belief is a bounded martingale, so it converges; the only absorbing points of the recursion are $\theta \in \{0,1\}$. Since informed arrivals always push in the direction of the true value, ${\theta }_t\to 1\{V=V_H\}$ almost surely. This is the sequential-trade analogue of Kyle’s $\Sigma (t)\to 0$ and the reason a market “learns”.

Toxicity of a fill. The probability that a trade you just received came from an informed counterparty is

$$\mathbb{P}(\text{informed}\mid B)=\frac{\pi {\theta }_t}{\mathbb{P}(B)}=\frac{2\pi {\theta }_t}{n_A}(=\pi \text{at }\theta =\frac{1}{2}).$$

This is the number a market maker actually cares about, and - crucially - it is not constant: it moves with $\theta$, so the value of a fill depends on where in the learning process you are.

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3. Computational Implementation - the sequential Bayesian game, simulated

Stdlib only. First the exact recursion and the martingale check at machine precision; then the dynamics over long paths.

Four readings:

1. The drift is exactly 0.00e+00. Not “small”, not “within sampling error” - the martingale property is an algebraic identity of the recursion, and the simulation confirms it numerically to all displayed digits.
2. ${\theta }^{+}=0.600000$ from a $0.500000$ prior. A single buy moves the maker’s belief by 10 percentage points at $\pi =0.20$. That belief move is the ask: the maker sells at 100.20 because that is her expected value given the buyer showed up.
3. Learning is fast and asymmetric. Over 400 independent 300-trade days, the mean terminal belief is $0.999971$ when $V=V_H$ and $0.000434$ when $V=V_L$. The engine does not merely “tend toward” the truth; it is essentially certain by trade 300. This is the sequential-trade engine behind “prices impound information”.
4. Toxicity equals the informed share at maximum uncertainty ($0.2000$) and moves away from it as the belief departs from $\frac{1}{2}$ - which is exactly the mechanism behind the failure in page 05.

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4. Failure Modes & First-Principles Breakdowns

1. The maker is assumed to price the average trade. Glosten–Milgrom’s maker cannot condition on size, venue, or timing. Real makers can and do - which is why the equilibrium spread in practice is a function of order attributes, not a scalar. The model is a lower bound on the sophistication of real liquidity provision.
2. $\pi$ is assumed constant and common knowledge. Both the informed share and the two-point value structure are equilibrium inputs, not outputs. If $\pi$ changes and the maker does not re-estimate, page 05 shows the loss is first-order in the error.
3. Trade direction must be observable. The whole recursion depends on labelling each trade buy or sell. With electronic order books the classifier (the Lee–Ready rule, tick tests) is imperfect; classification error feeds directly into the belief recursion and biases $\theta$.
4. Information is short-lived and single-shot. The informed trader knows $V$ once and trades, and then the game ends at $\theta \in \{0,1\}$. Real informed flow is long-lived - an informed trader may want to trade into several days of arrivals, which changes the equilibrium entirely (Holden–Subrahmanyam 1992; page 06).
5. The informed trader is assumed to be non-strategic in the GM version. In Glosten–Milgrom the informed trader trades every time in the correct direction - no attempt to disguise or time. Kyle’s insider is strategic about size; a full game would be strategic about both size and timing across many rounds.
6. Competition is modelled as the absence of profit, not as entry. Zero-profit quotes are an assumption about how many makers there are and how fast they can enter. With finite capital the equilibrium can sit above zero-profit indefinitely.

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5. References

• Glosten, Lawrence R.; Milgrom, Paul R. - “Bid, ask and transaction prices in a specialist market with heterogeneously informed traders,” Journal of Financial Economics 14(1), 71–100 (1985). *Sections 2–3 give the recursion and the zero-profit quotes;
• Easley, David; O’Hara, Maureen - “Price, trade size, and information in securities markets,” Journal of Financial Economics 19(1), 69–90 (1987). *Adds trade size to the sequential game
• Glosten, Lawrence R. - “Is the electronic open limit order book inevitable?” Journal of Finance 49(4), 1127–1161 (1994). The sequential-trade logic taken to the limit order book, where the “quotes” become the whole schedule.
• Hasbrouck, Joel - Empirical Market Microstructure (2007)
• Foucault, Thierry; Pagano, Marco; Röell, Ailsa - Market Liquidity: Theory, Evidence, and Policy (2013)
• Biais, Bruno; Glosten, Lawrence; Spatt, Chester - “Market microstructure: A survey of microfoundations, empirical results, and policy implications,” Journal of Financial Markets 8(2), 217–264 (2005). Where this model sits in the full taxonomy.

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6. Connected Graph Bridges

• Back: 01 - From Zero · 02 - Kyle & the Back Limit (the continuous-value counterpart) · Index Hub
• Forward: 04 - Predatory Trading & Execution Games
• The game in full detail: Adverse Selection & the Glosten–Milgrom Model
• Measuring the inputs: Toxic Order Flow & VPIN (estimating $\pi$) · Spread Decomposition & the Roll Model (how much of a quoted spread is adverse selection)
• Formal machinery: Bayesian Statistics · Probability & Measure Theory (martingale convergence - why ${\theta }_t\to 1\{V=V_H\}$)