5.3 Sequential trade: Glosten–Milgrom

5.3.1Zero-profit quotes#

At $\theta = \tfrac12$:

where $\bar V = (V_L + V_H)/2$. The spread is increasing in $\mu$ and in $\Delta V$, zero when $\mu = 0$, and equal to $\Delta V$ when $\mu = 1$. The dealer's expected profit from uninformed traders exactly offsets the expected loss to informed traders.

Proof

$P(\text{buy} \mid V_H) = \mu + (1-\mu)/2 = (1+\mu)/2$ and $P(\text{buy} \mid V_L) = (1-\mu)/2$. By Bayes' rule,

$P(V_H \mid \text{buy}) = \dfrac{\theta(1+\mu)/2}{\theta(1+\mu)/2 + (1-\theta)(1-\mu)/2}$,

and $E[V \mid \text{buy}] = V_L + \Delta V \cdot P(V_H \mid \text{buy})$, which is the formula for $a$. The bid follows in the same way from $P(\text{sell} \mid V_H) = (1-\mu)/2$ and $P(\text{sell} \mid V_L) = (1+\mu)/2$. At $\theta = \tfrac12$ the denominators equal 1, so $P(V_H \mid \text{buy}) = (1+\mu)/2$ and $P(V_H \mid \text{sell}) = (1-\mu)/2$, giving (5.6).

Zero profit at $\theta = \tfrac12$: an informed buyer arrives when $V = V_H$ and gains $V_H - a = \Delta V(1-\mu)/2$; an informed seller symmetrically gains $b - V_L = \Delta V(1-\mu)/2$. An uninformed trader pays, in expectation, $a - \bar V = \mu\Delta V/2$ on a buy or $\bar V - b = \mu\Delta V/2$ on a sell. The dealer's expected profit per arrival is $(1-\mu)\cdot\frac{\mu\Delta V}{2} - \mu\cdot\frac{(1-\mu)\Delta V}{2} = 0$.∎

5.3.2Learning from order flow#

Dynamics Repeat the game: after each trade the dealer updates $\theta$ by Bayes' rule and requotes. The sequence of transaction prices $p_k = E[V \mid \text{trades } 1..k]$ is a martingale with respect to public information (tower property), so trade-price changes are unpredictable, unlike in the Roll model, where bounce creates negative autocorrelation. As trades accumulate, $\theta$ converges to 0 or 1, the posterior concentrates on the true $V$, and the spread shrinks to zero (by (5.5), $a - b \to 0$ as $\theta \to 0$ or $1$). Information gets into prices through order flow, and the spread is widest when the market knows least, which is one reason spreads are wide at the open and around news.

5.3.3Spreads around news#

Code Exercise m05_kyle (in the app: Exercises → m05_kyle) implements (5.5) and simulates 400,000 rounds to check that the dealer's mean P&L is statistically indistinguishable from zero at three $(\theta, \mu)$ pairs.