Kyle / Glosten-Milgrom: how informed traders force market makers to widen spreads
Bid-ask spreads aren't fees. They are the price a market maker charges to insure against trading with someone smarter than them. Kyle (1985) and Glosten-Milgrom (1985) derive the exact width.
Why do bid-ask spreads exist even when there is no execution cost? The answer, formalised by Albert Kyle and Lawrence Glosten-Paul Milgrom in back-to-back 1985 papers, is adverse selection. Some traders arrive with private information about the true asset value; others are noise traders with no informational edge. Because the market maker cannot tell them apart, every incoming buy shifts her belief upward and every sell shifts it downward. The zero-expected-profit condition then pins the spread to exactly the width that compensates her for losses on informed fills. This tutorial walks through the Glosten-Milgrom Bayesian update on a discrete trade, then connects it to Kyle's continuous-quantity formulation.
β Intro Β· expand
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β Try first Β· expand
Worked example
An asset will be revealed to be worth either $V_L = 50$ or $V_H = 100$ with equal prior probability $1/2$. Each arriving trader is *informed* with probability $\pi = 0.20$ — informed traders buy at $V_H$ and sell at $V_L$ — and a *noise trader* with probability $0.80$, equally likely to buy or sell regardless of $V$. A risk-neutral market maker quotes an ask $A$ and a bid $B$ such that her expected profit on each fill is zero. Compute the ask, the bid, and the bid-ask spread.
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
Reflection
Glosten-Milgrom uses a discrete buy/sell signal; Kyle uses a continuous order quantity. Where in the derivation did that choice actually change the math, and where did it not matter?