Glosten–Milgrom Bid–Ask Spread Model

Glosten–Milgrom Bid–Ask Spread Model is a foundational sequential trade framework in financial market microstructure, adverse selection, and business and economics theory, formulated by American financial economists Lawrence R. Glosten and Paul R. Milgrom in their landmark 1985 Journal of Financial Economics paper "Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders." Whereas earlier theories of dealership markets explained the bid–ask spread through fixed order-processing costs (Demsetz, 1968; Roll, 1984) or dealer risk aversion and inventory holding costs (Stoll, 1978; Ho and Stoll, 1981), Glosten and Milgrom proved that a positive bid–ask spread S_t = A_t − B_t > 0 emerges even when market makers are risk-neutral, incur zero inventory or administrative costs, and earn zero expected profits under Bertrand competition. Because an anonymous trader arriving at the market maker's quote desk may be an informed trader possessing superior private information about the asset's liquidation value V, every incoming buy order conveys upward news about V to the dealer and every incoming sell order conveys downward news. To break even in expectation across informed and uninformed counterparties, competitive dealers set an ask price A_t = E[V | Buy_t, ℋ_t] equal to the conditional expectation of value given a buy order and a bid price B_t = E[V | Sell_t, ℋ_t] equal to the conditional expectation of value given a sell order, recouping their expected adverse-selection losses to informed traders from the spread paid by inelastic liquidity (noise) traders.

Order-Processing, Inventory-Control, and Adverse-Selection Theories

In quote-driven securities markets—such as foreign exchange dealerships, corporate bond markets, and specialist-assisted stock exchanges—liquidity providers post two prices: a bid price (B) at which the dealer stands ready to buy one unit of the security from an incoming seller, and a higher ask or offer price (A > B) at which the dealer stands ready to sell one unit to an incoming buyer. The difference S = A − B is the quoted bid–ask spread.

Financial economists attribute the bid–ask spread to three distinct mechanisms:

1. Order-Processing and Immediacy Costs (Demsetz, 1968; Roll, 1984): Harold Demsetz (1968) modeled the spread as the competitive price of immediacy, compensating specialists for clearing, settlement, and fixed overhead. Richard Roll (1984) showed that with a constant order-processing half-spread c, transaction prices bounce between bid and ask quotes, generating negative first-order serial covariance in price changes: Cov(ΔP_t, ΔP_t−1) = −c².
2. Inventory-Holding Risk (Stoll, 1978; Ho & Stoll, 1981): Because order flow arrives asynchronously, a risk-averse dealer who absorbs customer orders is forced to hold a suboptimal portfolio inventory away from target benchmarks, charging a spread for bearing idiosyncratic return variance and shading quotes to mean-revert inventory.
3. Informational Asymmetry and Adverse Selection (Bagehot, 1971; Copeland & Galai, 1983; Glosten & Milgrom, 1985): Writing under the pseudonym Walter Bagehot in 1971, Jack Treynor observed that a market maker faces both liquidity-motivated traders (who trade for exogenous cash needs) and informed traders (who trade only when the dealer's quote is mispriced relative to private information). Trading with informed insiders is a losing proposition for the dealer—analogous to writing naked call and put options (Copeland and Galai, 1983). Glosten and Milgrom (1985) formalized Bagehot's insight into a dynamic sequential Bayesian game in which dealers update beliefs trade by trade.

Asset Payoff, Trader Population, and Arrival Process

Consider a market for a single risky asset whose terminal liquidation value V takes one of two values:

At date t = 0, nature draws V = V_H with prior probability π₀ ∈ (0, 1) and V = V_L with probability 1 − π₀, so the unconditional prior expected value is E[V | ℋ₀] = π₀V_H + (1 − π₀)V_L.

Trading occurs sequentially at discrete arrival times t = 1, 2, 3, …. At each step t, a single trader is drawn at random from an infinite population to trade at most one unit of the asset (buy +1 unit at the ask A_t, sell −1 unit at the bid B_t, or abstain). The trader population consists of two groups:

• Informed Traders (Fraction μ ∈ (0, 1)): With probability μ, the arriving trader at step t is an informed insider who privately knows the true realization of V ∈ {V_L, V_H}. Because V_L < B_t ≤ A_t < V_H for any interior belief π_t ∈ (0, 1), a risk-neutral informed trader always buys one unit at A_t when V = V_H (earning V_H − A_t > 0) and sells one unit at B_t when V = V_L (earning B_t − V_L > 0).
• Uninformed Liquidity / Noise Traders (Fraction 1 − μ): With probability 1 − μ, the arriving trader is an uninformed liquidity trader who knows only the public history ℋ_t and trades for exogenous liquidity or hedging needs. An arriving liquidity trader is equally likely to be a buyer (probability 1/2) or a seller (probability 1/2), with price-inelastic demand within (V_L, V_H).

Information Sets and Regret-Free Bertrand Quote Competition

Let ℋ_t = (Q₁, Q₂, …, Q_t−1) denote the public history of all past transaction types and prices up to the start of period t, where Q_k ∈ {Buy_k, Sell_k}. Let π_t = P(V = V_H | ℋ_t) denote the market makers' common posterior probability that V = V_H entering period t, and let p_t = E[V | ℋ_t] = π_tV_H + (1 − π_t)V_L denote the pre-trade public valuation.

Because multiple risk-neutral market makers compete in Bertrand fashion for each unit order and incur zero inventory or processing costs, competition drives expected dealer profit conditional on every trade type to zero. When setting the ask price A_t, a dealer knows that A_t will be hit if and only if the incoming customer chooses to buy (Q_t = Buy_t). Similarly, B_t will be hit if and only if the customer chooses to sell (Q_t = Sell_t). Zero expected profit conditional on execution requires the regret-free quoting conditions:

Even after observing that the customer chose to buy at A_t, the dealer has zero ex-post regret over having offered A_t because A_t already incorporates the Bayesian upward revision triggered by a buy order.

Conditional Order-Arrival Probabilities and Bayes' Rule

Entering period t with belief π_t = P(V = V_H | ℋ_t), the conditional probabilities of observing a buy or sell order in each state are:

• If V = V_H: Every informed trader (μ) and half of liquidity traders ((1 − μ)/2) buy:

• If V = V_L: Every informed trader (μ) and half of liquidity traders ((1 − μ)/2) sell:

By the law of total probability, the unconditional probabilities of a buy or sell at step t are:

Applying Bayes' Rule, a buy order updates the posterior probability that V = V_H upward to π_t+1⁺, whereas a sell order updates it downward to π_t+1⁻:

Consequently, the equilibrium ask price A_t and bid price B_t are given in closed form by:

Closed-Form Expression for the Adverse-Selection Bid–Ask Spread

Subtracting B_t from A_t yields the closed-form Glosten–Milgrom bid–ask spread S_t = A_t − B_t:

At a symmetric prior (π_t = 1/2, so p_t = (V_H + V_L)/2), the quotes and spread simplify to:

Four comparative-statics results follow from this equation:

1. Positivity Under Asymmetric Information: Whenever μ > 0 and 0 < π_t < 1, the spread is positive (S_t > 0). If no traders have private information (μ = 0), S_t = 0.
2. Monotonicity in the Fraction of Informed Traders (∂S_t / ∂μ > 0): A higher share of informed traders μ widens the bid–ask spread monotonically.
3. Proportionality to Fundamental Uncertainty: The numerator 4π_t(1 − π_t)(V_H − V_L) is proportional to conditional variance Var(V | ℋ_t). The spread is widest at π_t = 1/2 and shrinks toward zero as π_t → 0 or π_t → 1.
4. Zero-Sum Redistribution: At π_t = 1/2, an informed trader earns (1 − μ)(V_H − V_L)/2 per trade while a liquidity trader loses the half-spread μ(V_H − V_L)/2. Weighting by population shares μ and 1 − μ confirms that expected dealer profit is zero: μ[−(1 − μ)(V_H − V_L)/2] + (1 − μ)[+μ(V_H − V_L)/2] = 0.

Transaction Prices Form a Martingale (Proposition 2 of Glosten–Milgrom)

Let P_t denote the realized transaction price of the t-th trade (P_t = A_t on a buy and P_t = B_t on a sell). Because A_t = E[V | ℋ_t, Buy_t] and B_t = E[V | ℋ_t, Sell_t], the transaction price equals the updated public expectation entering period t + 1: P_t = E[V | ℋ_t+1] = p_t+1. By the Law of Iterated Expectations:

Proposition 2 (Martingale Property of Transaction Prices): In the Glosten–Milgrom pure adverse-selection model, transaction prices {P_t}_t≥1 form a martingale with respect to public history {ℋ_t+1}, implying zero serial correlation of price changes: Cov(ΔP_t+1, ΔP_t) = 0.

Unlike Roll (1984)—where quotes remain fixed and bid–ask bounce produces negative serial covariance −c²—in Glosten–Milgrom a buy at A_t causes the dealer to revise the next period's midpoint p_t+1 all the way up to A_t. Because every price impact is 100% permanent, bid–ask bounce creates zero negative serial correlation.

Log-Likelihood Ratio Convergence and Market Breakdown

Writing the dealer's Bayesian update in terms of the posterior odds ratio Λ_t = π_t / (1 − π_t) shows that the log-odds ratio follows a biased random walk:

where Z_k = +1 for a buy and −1 for a sell. When V = V_H, E[Z_k | V = V_H] = +μ > 0, so by the Strong Law of Large Numbers π_t → 1, P_t → V_H, and the spread S_t → 0 geometrically as t → ∞.

However, if liquidity traders have price-elastic valuations—refusing to buy if the half-spread A_t − p_t exceeds their liquidity benefit b > 0—then at π_t = 1/2 the market can operate if and only if:

Whenever the informed share μ exceeds μ̄, the break-even spread drives liquidity traders out of the market, leaving only informed traders and triggering a complete market breakdown (liquidity freeze) analogous to George Akerlof's (1970) market for "lemons."

Structural Spread Decomposition and Probability of Informed Trading (PIN)

In empirical market microstructure, Glosten and Harris (1988) and Madhavan, Richardson, and Roomans (1997) combined Glosten–Milgrom with Roll (1984) to decompose intraday price changes ΔP_t using trade-direction indicators Q_t ∈ {−1, +1}:

where θ > 0 measures the Glosten–Milgrom permanent adverse-selection component (~35%–50% of NYSE spreads) and ϕ > 0 measures the Roll transitory order-processing component. Extending Glosten–Milgrom to continuous-time Poisson arrival rates of information days (α), informed trades (μ), and uninformed buy/sell trades (ε), David Easley, Nicholas Kiefer, Maureen O'Hara, and Joseph Paperman (1996) derived the Probability of Informed Trading:

which directly determines the opening proportional Glosten–Milgrom spread and serves as a benchmark empirical metric of informational asymmetry across equities.

How can a risk-neutral market maker earn zero expected profit while charging a positive bid–ask spread?

A risk-neutral dealer in Bertrand competition faces two groups of anonymous counterparties. Whenever the counterparty is an informed trader (probability μ), the trader knows V and trades only when the quote is unfavorable to the dealer (buying at A_t < V_H or selling at B_t > V_L), inflicting an expected loss on the dealer. Whenever the counterparty is an uninformed liquidity trader (probability 1 − μ), the trader buys at A_t or sells at B_t around the public mean p_t, paying the expected half-spread S_t/2 to the dealer. The equilibrium spread S_t is set where expected gains from liquidity traders offset expected adverse-selection losses to informed traders dollar-for-dollar.

Why do transaction prices follow a martingale in the Glosten–Milgrom model despite bouncing between bid and ask quotes?

In a pure order-processing model (Roll, 1984), quotes B and A remain fixed when trades occur, so a buy at A followed by a sell at B creates negative serial correlation. In the Glosten–Milgrom model, when a buy order executes at A_t, the dealer updates the public mean p_t+1 = E[V | ℋ_t, Buy_t] all the way to A_t, centering the next period's quotes (B_t+1, A_t+1) around p_t+1 = A_t. Because quote midpoints shift by the full half-spread after every trade, E[P_t+1 | ℋ_t+1] = P_t.

What happens in the Glosten–Milgrom model if traders can choose between small and large trade sizes?

Easley and O'Hara (1987) showed that when traders can choose between a small order (q₁) and a large block order (q₂ > q₁), risk-neutral informed traders prefer to trade the large size q₂ to maximize profit. Competitive market makers therefore infer a higher conditional probability of informed trading when hit with a large order and quote a wider bid–ask spread for block trades than for small orders (A_t(q₂) > A_t(q₁) and B_t(q₂) < B_t(q₁)).

Why can a monopolist specialist prevent market breakdown when competitive Glosten–Milgrom dealers shut down?

Glosten (1989) showed that whereas competitive Bertrand dealers must break even trade-by-trade—forcing them to shut down the market when the informed share μ exceeds 2b/(V_H − V_L)—a monopolist specialist can cross-subsidize across time and across order sizes, keeping the market open during high-asymmetry states and recouping losses when uncertainty subsides.

1. Glosten, L. R., & Milgrom, P. R. (1985). "Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders." Journal of Financial Economics, 14(1), 71–100. https://doi.org/10.1016/0304-405X(85)90044-390044-3)
2. Roll, R. (1984). "A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market." Journal of Finance, 39(4), 1127–1139. https://doi.org/10.1111/j.1540-6261.1984.tb03897.x
3. Easley, D., & O'Hara, M. (1987). "Price, Trade Size, and Information in Securities Markets." Journal of Financial Economics, 19(1), 69–90. https://doi.org/10.1016/0304-405X(87)90029-890029-8)
4. Glosten, L. R., & Harris, L. E. (1988). "Estimating the Components of the Bid/Ask Spread." Journal of Financial Economics, 21(1), 123–142. https://doi.org/10.1016/0304-405X(88)90034-790034-7)
5. Easley, D., Kiefer, N. M., O'Hara, M., & Paperman, J. B. (1996). "Liquidity, Information, and Infrequently Traded Stocks." Journal of Finance, 51(4), 1405–1436. https://doi.org/10.1111/j.1540-6261.1996.tb04074.x