An asset is worth 110 or 90 with equal probability. A fraction 20% of the order flow is informed and always trades the right way; the rest buys or sells with equal probability. A competitive risk-neutral maker quotes to zero expected profit. What is the ask price?

An asset is worth 110 or 90 with equal probability. A fraction 20% of the order flow is informed and always trades the right way; the rest buys or sells with equal probability. A competitive risk-neutral maker quotes to zero expected profit. What is the ask price?

Approach: Set the ask equal to the expected value conditional on observing a buy, using Bayes rule on the arrival probability of a buy under each state of the world.

102. Conditional on the value being 110, a buy arrives with probability 0.2 + 0.8 * 0.5 = 0.6; conditional on 90 it arrives with probability 0.8 * 0.5 = 0.4. Bayes rule gives P(V = 110 | buy) = 0.6 * 0.5 / (0.6 * 0.5 + 0.4 * 0.5) = 0.6, so the ask is 0.6 * 110 + 0.4 * 90 = 102. The bid is 98 by symmetry, so the bid-ask spread is 4, which equals alpha * (V_H - V_L) = 0.2 * 20. That is the Glosten-Milgrom result: the whole spread is adverse selection, and the maker breaks even by losing to the informed 20% exactly what it earns from the uninformed 80%.

Follow-up: How wide must the quote be if the informed trader can also choose to trade two units when the edge is larger?

Key concepts: Glosten-Milgrom, adverse selection, Bayes rule, bid-ask spread.