Glosten-Milgrom Zero-Profit Bid-Ask Quotes
A binary asset has fundamental value $V \in \{0, 1\}$ with prior $P(V = 1) = p$, where $p \in (0, 1)$.
A single counterparty arrives to trade one unit. With probability
- q$ the counterparty is perfectly informed (buys if $V = 1$, sells if $V = 0$). With probability $q$ the counterparty is a noise trader who buys or sells each with probability $\frac{1}{2}$.
You are the market maker. You must post a bid $b$ and an ask $a$ (unit size) such that your expected profit is exactly zero conditional on being hit on that side.
1. Derive the zero-profit ask $a(p, q)$ and bid $b(p, q)$.
2. Compute the implied spread $a - b$ and simplify.
3. Interpret the limiting cases $q \to 0$ (all informed) and $q \to 1$ (all noise).
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