Sequential Trade & Information Asymmetry

Glosten-Milgrom (1985) Microstructure Lab

Simulate sequential order flow, Bayesian quote adjustments ($A_t, B_t$), adverse selection pricing, and spread decay toward zero as private information impounds into prices.

Market Information Regimes:
Model Configuration
prob
Initial dealer belief that asset has high valuation
$\mu$
Probability that arriving trader possesses private info
$\gamma$
True Value State

High ($115.00)

Fundamental Target
Initial Spread $S_0$

$7.50 (7.5%)

Opening Adverse Selection
Final Spread $S_N$

$0.32

Post-Revelation Friction
Ending Market Maker Belief

98.4%

Posterior $p_N = \mathbb{P}(V = V_H)$
Convergence Threshold

Trade #34

Trade where $|p_t - \mathbf{1}| \le 0.05$
Sequential Microstructure Visualizer
Sensitivity: Informed Fraction ($\mu$) vs Prior ($p_0$) on Initial Spread ($)

Opening adverse selection spread ($A_0 - B_0$) under varying insider presence and market priors.

Sensitivity: Informed Fraction ($\mu$) vs Valuation Gap on Convergence Speed

Average sequential trades required to reach 95% Bayesian belief certainty.

Mathematical Derivation: Glosten-Milgrom (1985) Bayesian Equilibrium

1. Economic Setup and Information Structure

Let the true terminal liquidation value of an asset be a binary random variable $V \in \{V_L, V_H\}$ with $V_H > V_L$. The public prior belief held by competitive, risk-neutral market makers is:

$$p_0 = \mathbb{P}(V = V_H), \quad 1 - p_0 = \mathbb{P}(V = V_L)$$

At each discrete trade time $t = 1, 2, \dots$, a single anonymous trader arrives from a pool of potential market participants:

  • Informed Traders (fraction $\mu$): Possess perfect private knowledge of the true state $V$. If $V = V_H$, they always buy ($\text{trade} = +1$); if $V = V_L$, they always sell ($\text{trade} = -1$).
  • Uninformed Noise Traders (fraction $1-\mu$): Trade for exogenous liquidity/rebalancing reasons. They buy with probability $\gamma$ and sell with probability $1-\gamma$ (standardly $\gamma = 0.50$).
2. Bayesian Quote Setting (Bertrand Competition)

Because dealers operate in a perfectly competitive, zero-profit environment, Bertrand competition forces quotes to equal the conditional expected value of the asset given the trade direction:

$$A_t = \mathbb{E}[V \mid \text{trade}_t = \text{Buy}], \quad B_t = \mathbb{E}[V \mid \text{trade}_t = \text{Sell}]$$

Using Bayes' Law, the market maker updates their subjective probability belief $p_{t-1} = \mathbb{P}(V = V_H)$ upon observing an order:

$$\mathbb{P}(\text{Buy} \mid V_H) = \mu + (1-\mu)\gamma, \quad \mathbb{P}(\text{Buy} \mid V_L) = (1-\mu)\gamma$$ $$\mathbb{P}(V_H \mid \text{Buy}) = \frac{p_{t-1}[\mu + (1-\mu)\gamma]}{p_{t-1}[\mu + (1-\mu)\gamma] + (1-p_{t-1})(1-\mu)\gamma}$$

Symmetrically, for incoming sell orders:

$$\mathbb{P}(\text{Sell} \mid V_H) = (1-\mu)(1-\gamma), \quad \mathbb{P}(\text{Sell} \mid V_L) = \mu + (1-\mu)(1-\gamma)$$ $$\mathbb{P}(V_H \mid \text{Sell}) = \frac{p_{t-1}(1-\mu)(1-\gamma)}{p_{t-1}(1-\mu)(1-\gamma) + (1-p_{t-1})[\mu + (1-\mu)(1-\gamma)]}$$
3. Bid-Ask Spread and Information Impounding

The quotes are affine transformations of the updated posterior probabilities:

$$A_t = V_L + (V_H - V_L) \cdot \mathbb{P}(V_H \mid \text{Buy})$$ $$B_t = V_L + (V_H - V_L) \cdot \mathbb{P}(V_H \mid \text{Sell})$$

When uninformed flow is symmetric ($\gamma = 0.50$), the bid-ask spread simplifies to:

$$S_t = A_t - B_t = (V_H - V_L) \frac{4\mu p_{t-1}(1-p_{t-1})}{1 - \mu^2(2p_{t-1} - 1)^2}$$

Notice that the spread is strictly positive for $\mu > 0$ and $p_0 \in (0, 1)$, even with zero order processing costs and zero inventory risk. This is the pure adverse selection component of the bid-ask spread.

4. Martingale Property and Asymptotic Learning

By the law of iterated expectations, the belief process $\{p_t\}$ is a bounded martingale:

$$\mathbb{E}[p_t \mid \mathcal{F}_{t-1}] = p_{t-1}$$

By Doob's Martingale Convergence Theorem, $p_t$ converges almost surely to the true state $\mathbf{1}_{\{V = V_H\}}$. As uncertainty resolves ($p_t \to 1$ or $p_t \to 0$), the spread $S_t \to 0$, demonstrating how order flow acts as an information aggregation mechanism.

Microstructure Concept Mastery Quiz

1. In the Glosten-Milgrom (1985) model, what economic principle determines the dealer's Ask quote $A_t$?

2. Why does the bid-ask spread exist even when dealer order processing and clearing costs are zero?

3. What happens to the bid-ask spread as the number of sequential trades $N \to \infty$?

4. What occurs if the fraction of informed traders $\mu \to 1.0$ (complete adverse selection)?

Related Quantitative Microstructure Labs

Kyle's Lambda & Microstructure Lab

Model Pete Kyle's (1985) continuous informed trading equilibrium, market depth, and price impact.

Corwin-Schultz Spread Lab

Estimate effective bid-ask transaction spreads and liquidity using daily high and low prices.

Roll Effective Spread Lab

Estimate effective spreads from transaction price serial covariance using Richard Roll's (1984) model.

Huang-Stoll Spread Decomposition Lab

Decompose spreads into adverse selection, inventory holding, and order processing friction.

Frequently Asked Questions

The Glosten-Milgrom (1985) model is a foundational economic framework of sequential trading with asymmetric information. In the model, competitive risk-neutral market makers face informed traders who possess private knowledge of the asset's liquidation value and uninformed liquidity traders who trade randomly. Quotes (Bid and Ask) are set using Bayesian probability updates conditional on the direction of incoming order flow.

The market maker faces adverse selection risk: trades initiated by informed insiders systematically cause losses for the dealer (buying when undervalued, selling when overvalued). To break even overall under Bertrand competition, the dealer sets an ask price above expected value and a bid price below expected value so that profits earned from uninformed noise traders exactly offset expected losses to informed traders.

Before each trade, the market maker holds a prior belief $p_{t-1} = \mathbb{P}(V = V_H)$. When a buy order arrives, the dealer applies Bayes' Rule to increase the belief: $\mathbb{P}(V_H \mid \text{Buy}) = \frac{p_{t-1}[\mu + (1-\mu)\gamma]}{(1-\mu)\gamma + \mu p_{t-1}}$. When a sell order arrives, the belief drops accordingly. Quotes update dynamically with each trade.

As the sequence of trades progresses, each trade conveys information about the unobserved true liquidation value. By the Martingale Convergence Theorem, the market maker's subjective probability $p_t$ converges almost surely to 1 (if true value is $V_H$) or 0 (if true value is $V_L$). As uncertainty vanishes ($p_t \to 0$ or $1$), adverse selection risk disappears and the bid-ask spread shrinks to zero.

If the fraction of informed traders $\mu$ approaches 1, or uninformed noise trading liquidity dries up, the market maker can no longer cross-subsidize informed trader losses. The required bid-ask spread widens until it reaches the entire valuation gap $V_H - V_L$, causing trade to halt entirely—an illustration of George Akerlof's 'Lemons' market failure in financial microstructure.