Glosten-Milgrom Zero-Profit Bid-Ask Quotes

Worked Solution

How to Think About It: You are a market maker posting quotes against a mix of informed and uninformed flow. The informed traders pick you off -- they only buy when the asset is worth 1 and only sell when it is worth 0. Noise traders are your profit center since they trade randomly. The zero-profit condition means you set each quote equal to the expected value of the asset conditional on being hit on that side. This is just Bayes' theorem: a buy tells you it is more likely the counterparty is informed and $V = 1$, while a sell tells you $V = 0$ is more likely. The ask will be above $p$ (adverse selection on the buy side) and the bid will be below $p$ (adverse selection on the sell side), with the gap driven by how much informed flow there is.

Quick Estimate: Take $p = 0.5$, $q = 0.5$ (half informed, half noise, fair coin prior). If someone buys, the informed component pushes $P(V=1)$ up. By symmetry the ask should be above 0.5 and the bid below by the same amount. Rough guess: $a \approx 0.7$, $b \approx 0.3$, spread $\approx 0.4$. As we will see, the exact answer gives $a = 3/4$, $b = 1/4$, spread $= 1/2$.

Approach: Apply Bayes' theorem separately to the buy event and the sell event, then set each quote equal to the posterior expectation of $V$.

Formal Solution:

*Step 1: Ask quote (zero-profit on the sell side of the book).*

When someone buys from you, compute the conditional probabilities:

• $P(\text{buy} \mid V = 1) = (1 - q) \cdot 1 + q \cdot \tfrac{1}{2} = 1 - \tfrac{q}{2}$
• $P(\text{buy} \mid V = 0) = (1 - q) \cdot 0 + q \cdot \tfrac{1}{2} = \tfrac{q}{2}$

By Bayes' theorem:

$$P(V = 1 \mid \text{buy}) = \frac{p\,(1 - q/2)}{p\,(1 - q/2) + (1 - p)\,q/2} = \frac{p(2 - q)}{2p + q - 2pq}$$

The zero-profit ask is $a = E[V \mid \text{buy}] = P(V = 1 \mid \text{buy})$:

$$\boxed{a(p, q) = \frac{p(2 - q)}{2p + q - 2pq}}$$

With $p = q = \tfrac{1}{2}$: $P(\text{buy}\mid V=1) = 1 - \tfrac{1}{4} = \tfrac{3}{4}$ and $P(\text{buy}\mid V=0) = \tfrac{1}{4}$, so

$$a = \frac{\tfrac{1}{2}\cdot\tfrac{3}{4}}{\tfrac{1}{2}\cdot\tfrac{3}{4} + \tfrac{1}{2}\cdot\tfrac{1}{4}} = \frac{3/8}{3/8 + 1/8} = \frac{3/8}{1/2} = \frac{3}{4}.$$

Equivalently from the formula, $a = \frac{(1/2)(3/2)}{1 + 1/2 - 1/2} = \frac{3/4}{1} = \frac{3}{4}$.

*Step 2: Bid quote (zero-profit on the buy side of the book).*

When someone sells to you:

• $P(\text{sell} \mid V = 1) = (1 - q) \cdot 0 + q \cdot \tfrac{1}{2} = \tfrac{q}{2}$
• $P(\text{sell} \mid V = 0) = (1 - q) \cdot 1 + q \cdot \tfrac{1}{2} = 1 - \tfrac{q}{2}$

By Bayes' theorem:

$$P(V = 1 \mid \text{sell}) = \frac{p\,q/2}{p\,q/2 + (1 - p)(1 - q/2)} = \frac{pq}{pq + (1 - p)(2 - q)}$$

The zero-profit bid:

$$\boxed{b(p, q) = \frac{pq}{pq + (1 - p)(2 - q)}}$$

With $p = q = \tfrac{1}{2}$: $b = \frac{(1/2)(1/2)}{(1/2)(1/2) + (1/2)(3/2)} = \frac{1/4}{1/4 + 3/4} = \frac{1/4}{1} = \frac{1}{4}$.

*Step 3: Implied spread.*

Let $D_a = 2p + q - 2pq$ and $D_b = pq + (1 - p)(2 - q) = 2 - 2p - q + 2pq$. Note $D_a + D_b = 2$.

$$a - b = \frac{p(2 - q)}{D_a} - \frac{pq}{D_b}$$

Putting over the common denominator $D_a D_b$ and expanding the numerator:

$$p(2 - q) D_b - pq \, D_a = p\bigl[(2 - q)(2 - 2p - q + 2pq) - q(2p + q - 2pq)\bigr]$$

Expanding and collecting terms, the bracket simplifies to $4(1 - p)(1 - q)$, so:

$$\boxed{a - b = \frac{4p(1 - p)(1 - q)}{(2p + q - 2pq)(2 - 2p - q + 2pq)}}$$

With $p = q = \tfrac{1}{2}$: $a - b = \frac{4(1/2)(1/2)(1/2)}{(1)(1)} = \frac{1/2}{1} = \frac{1}{2}$, matching $a - b = \tfrac{3}{4} - \tfrac{1}{4} = \tfrac{1}{2}$.

*Step 4: Limiting cases.*

$q \to 0$ (all informed): $a \to \frac{2p}{2p} = 1$ and $b \to \frac{0}{(1-p) \cdot 2} = 0$. The spread goes to 1, the maximum possible. Every counterparty is informed, so the market maker must quote the widest possible spread to avoid being picked off. Trading effectively shuts down.

$q \to 1$ (all noise): $a \to \frac{p \cdot 1}{0 + 1} = p$ and $b \to \frac{p}{p + 0} = p$. The spread collapses to zero. With no informed traders, a buy or sell carries no information about $V$, so both quotes equal the prior $p$. The market is perfectly liquid.

Answer: The zero-profit quotes are $a(p, q) = \frac{p(2 - q)}{2p + q - 2pq}$ and $b(p, q) = \frac{pq}{pq + (1-p)(2-q)}$. The spread is $\frac{4p(1-p)(1-q)}{(2p+q-2pq)(2-2p-q+2pq)}$. For $p = q = \tfrac{1}{2}$ this gives $a = \tfrac{3}{4}$, $b = \tfrac{1}{4}$, spread $= \tfrac{1}{2}$. The spread ranges from 0 (pure noise) to 1 (pure informed), and is maximized in the prior at $p = 1/2$ for any fixed $q$.