The Glosten–Milgrom model

"Adverse selection and informed trading" showed that a market maker, a dealer who continuously quotes a price to buy and a price to sell, loses money to traders who know more than the quote does, and that the spread has to be wide enough to pay for those losses. This lesson builds the simplest model in which that statement is derived instead of asserted: the Glosten–Milgrom model. It uses the conditional expectations and the martingale property from "Conditional expectation and martingales". The quotes are conditional expectations, the maker's belief is a martingale, and the spread, the learning and the speed of learning all follow from Bayes' rule.

Set-up

The asset is worth $V$, which is either $0$ or $1$ and is revealed only at the end. Everyone starts from the same prior belief $\mu_0 = P(V = 1)$. Traders arrive one at a time, and each trades one unit with the market maker, who quotes a bid $b$ (the price the maker pays) and an ask $a$ (the price the maker charges). Independently of everything else, each trader is:

• informed with probability $\pi$: knows $V$, buys if $V = 1$ and sells if $V = 0$;
• a noise trader with probability $1-\pi$: buys or sells with probability $\tfrac12$ each, whatever $V$ is.

The maker is risk-neutral and competes with other makers, so quotes are set to make the expected profit on each trade zero. The maker knows $\pi$ and the current belief $\mu$, but cannot tell an informed trader from a noise trader: all that is seen is the direction of the trade.

Quotes as conditional expectations

Take the ask. When a trader buys at $a$, the maker receives $a$ and owes an asset worth $V$, so the profit on the trade is $a - V$. Requiring zero expected profit given that a buy has happened, and the same for a sell, gives

Because $V$ is $0$ or $1$, its conditional expectation is the conditional probability that $V = 1$, and Bayes' rule computes it. First the chance of a buy in each state. If $V = 1$, an informed trader buys for sure and a noise trader buys half the time; if $V = 0$, only noise traders buy:

Selling is the mirror image: $P(\text{sell} \mid V=1) = (1-\pi)/2$ and $P(\text{sell} \mid V=0) = (1+\pi)/2$. Bayes' rule with prior $\mu$ then gives

A buy is more likely when the asset is worth 1, so a buy raises the belief and $a > \mu$; a sell lowers it and $b < \mu$. The ask lies above the prior and the bid below it.

The spread

At $\mu = \tfrac12$ the factors $\mu$ and $1-\mu$ cancel and

The spread equals the informed share. With no informed traders, $\pi = 0$, a trade says nothing about $V$, so both quotes equal the prior, the fair value $E[V]$, and the spread is zero. As $\pi$ grows the spread grows, and at $\pi = 1$ every trade reveals $V$, so the quotes are 0 and 1: the whole range. The spread is not a fee for effort. It is exactly the amount that makes the maker break even against the traders who know more.

Learning from the trades

After a buy the maker's belief is $P(V=1 \mid \text{buy})$, which by construction is the ask. A buy therefore moves the belief from $\mu$ to $a$, a sell moves it to $b$, and the next quotes are computed from the new belief. The belief is the only thing the maker has to remember.

In odds, Bayes' rule is a multiplication. A buy multiplies the odds $\mu/(1-\mu)$ by the likelihood ratio $P(\text{buy}\mid V=1)/P(\text{buy}\mid V=0) = (1+\pi)/(1-\pi)$, and a sell divides by the same factor. After $B$ buys and $S$ sells, in any order,

Only the difference between buys and sells matters. With $\pi = 0.2$ the factor is $1.5$, and a buy followed by a sell returns the belief exactly to where it was.

The belief is also a martingale from the maker's point of view. Let $q = P(\text{buy}) = \mu\,\tfrac{1+\pi}{2} + (1-\mu)\,\tfrac{1-\pi}{2}$. The denominator of $a$ is $2q$, so $q\,a = \mu\,(1+\pi)/2$, and in the same way $(1-q)\,b = \mu\,(1-\pi)/2$. Adding the two,

The expected next belief equals the current belief, so $\mu_t = E[V \mid \text{trades so far}]$ is a martingale, as in "Conditional expectation and martingales". Quotes never drift in a predictable direction; they move only when a trade surprises the maker. A bounded martingale converges, and since informed trades push towards the truth, the belief converges to $V$: the price discovers the value through the order flow.

A forty-trade example

The figure draws 40 trades from the model. The red line is the ask, the green line the bid and the dashed line the belief $\mu$; the dotted horizontal line marks the true value $V$. The tiles under the chart are the trades, B for a buy and S for a sell. Click a tile to flip that trade and every later quote changes, while the quotes before it stay put. At the defaults ($\pi = 20\%$, $V = 1$) the belief after 40 trades is 0.692 and the spread is 0.171. The belief first reached 90% after 30 trades and then fell back, because a run of noise sells outweighed the informed buys. Now move $\pi$ to 50%: the spread starts at 0.5 and the belief reaches 90% after four trades. Set $\pi$ to 0% and the three lines collapse onto 0.5, because nothing can be learned.

How fast the belief learns

Each trade moves the log-odds of $V=1$ by $\pm\ln\frac{1+\pi}{1-\pi}$. If $V=1$, the move is up with probability $(1+\pi)/2$ and down with probability $(1-\pi)/2$, so the expected gain per trade is

For small $\pi$, $\ln\frac{1+\pi}{1-\pi} = 2\pi + \tfrac23\pi^3 + \cdots$, so the gain is about $2\pi^2$. The information in a trade grows with the square of the informed share: doubling $\pi$ makes learning about four times faster. Reaching 90% means moving the log-odds from 0 to $\ln 9 \approx 2.197$, which takes about $2.197/(0.2\ln 1.5) \approx 27$ trades on average at $\pi = 0.2$ and about 6.5 at $\pi = 0.4$. One sequence, like the one in the figure, scatters around these averages.

What the model leaves out

Every trade is one unit, so size carries no information, and the maker's inventory plays no role. The informed trader is mechanical: always trading towards $V$, with no concern for what the trade does to the price. A strategic insider would spread orders over time, which is what Kyle's model of informed trading studies. The maker must also know $\pi$, which in practice has to be estimated. The two results that survive in richer models are that the spread compensates for adverse selection and that prices learn through the order flow.

Summary

• A competitive, risk-neutral maker sets the ask to $E[V \mid \text{buy}]$ and the bid to $E[V \mid \text{sell}]$; Bayes' rule gives both in closed form.
• At a prior of $\tfrac12$ the spread equals the informed share $\pi$; it is zero when nobody is informed.
• After a trade the belief moves to the quote that was hit. It is a martingale, depends only on the number of buys minus sells, and converges to the true value.
• The expected log-odds gain per trade is $\pi\ln\frac{1+\pi}{1-\pi} \approx 2\pi^2$, so doubling $\pi$ makes learning about four times faster.
• The model ignores size, inventory and strategic informed traders.