Glosten-Milgrom Dynamic Informed Trading and Spread Calibration

Scenario

In the Glosten-Milgrom (1985) sequential trading market microstructure model, an asset has unknown terminal value $V \in \{V_L, V_H\}$ with equal initial prior $\pi_0 = \mathbb{P}(V = V_H) = 0.5$. Traders arrive sequentially. With probability $\mu$, a trader is informed (knowing the true state $V$); with probability $1 - \mu$, the trader is an uninformed noise trader who buys with probability $0.5$ and sells with probability $0.5$.

Your Task

1. Given prior belief $\pi_t = \mathbb{P}(V = V_H \mid \mathcal{H}_t)$, derive the Bayesian posterior beliefs following a Buy order ($\pi_{t+1}^B$) and following a Sell order ($\pi_{t+1}^S$).
2. Under Bertrand zero-profit competition, derive the market maker's quoted Ask $A_t$ and Bid $B_t$.
3. Prove that the quoted bid-ask spread $S_t = A_t - B_t$ is strictly positive and proportional to the adverse selection parameter $\mu$.
4. Simulate a sequence of 5 consecutive buy orders for $V_L = 0, V_H = 100, \mu = 0.30$, and demonstrate how beliefs and prices converge toward $V_H$.

Constraints

• Risk-neutral market maker, competitive Bertrand market.

Examples If $\mu = 0$ (no informed traders), Bid = Ask = Mid-price, and spread is zero. When $\mu > 0$, adverse selection generates a positive spread.