Adverse selection, formally (Glosten–Milgrom)
M1 — Market microstructure
M0.4 gave you adverse selection as arithmetic: if 1 taker in 13 is informed, a 4p spread roughly breaks even. That was a consistency check on a number pulled from the air. Glosten and Milgrom (1985) turn it round and derive the number, and the result is clean enough to remember for life.
The trick is a single change of question. Don’t ask “what spread breaks even on average?” Ask:
What price would break even given that someone just chose to buy?
That conditioning is everything. The maker who ignores it goes broke.
The setup
Strip the world down as far as it goes:
- The asset is worth either V_H (high) or V_L (low). Nobody knows which yet; the prior is π on V_H. Start with π = ½.
- A fraction μ of traders are informed — they know V exactly. They buy if V = V_H, sell if V = V_L. Always.
- The remaining 1 − μ are uninformed. They buy or sell with probability ½ each, for reasons unrelated to V.
- The maker is risk-neutral, holds no inventory, has no costs, and faces perfect competition — so it earns exactly zero expected profit.
- Trades arrive one at a time, size fixed at one unit.
Everything M1.3 cares about is switched off here: no risk aversion, no inventory. This model is only about information.
Zero profit, conditionally
Competition drives expected profit to zero. But zero profit on each side separately, conditional on that side being hit — because the maker learns something from being hit:
ask = E[V | the next trade is a BUY]
bid = E[V | the next trade is a SELL]
If the maker set both to the unconditional E[V], it would sell to informed buyers whenever V = V_H and buy from informed sellers whenever V = V_L, losing on every informed trade and never recovering it. The spread exists because a buy and a sell carry different information.
The Bayes update
Two conditional probabilities do all the work. Given the true value is high, who buys?
P(buy | V_H) = μ·1 + (1−μ)·½ = (1 + μ) / 2
▲ ▲
│ └─ the uninformed, flipping a coin
└─ every informed trader buys
P(buy | V_L) = μ·0 + (1−μ)·½ = (1 − μ) / 2
Now Bayes, with prior π = ½ so the halves cancel:
P(buy | V_H)·π
P(V_H | buy) = ─────────────────────────────────────
P(buy | V_H)·π + P(buy | V_L)·(1−π)
½·(1+μ)/2 1 + μ
= ───────────────────────── = ─────
½·(1+μ)/2 + ½·(1−μ)/2 2
A pleasantly tidy result: after a buy, the probability the asset is worth V_H is (1 + μ)/2. By symmetry, after a sell it is (1 − μ)/2.
The spread falls out
Quotes are expectations under those posteriors:
ask = E[V | buy] = (1+μ)/2 · V_H + (1−μ)/2 · V_L
bid = E[V | sell] = (1−μ)/2 · V_H + (1+μ)/2 · V_L
────────────────────────────────────────────────────────
spread = ask − bid = μ · (V_H − V_L)
The spread is the fraction of informed flow times the range of possible value. Two limits are worth saying out loud:
- μ → 0 (nobody informed) → spread → 0. With no adverse selection, competitive makers quote a zero spread. Every penny of a real spread is paid for by someone’s information (or by inventory risk and costs, which this model deleted).
- μ → 1 (everybody informed) → spread → the whole range. The quotes reach V_H and V_L, no uninformed trader will touch them, and the market effectively shuts. Markets need noise traders to exist. That is not a cynical remark — it is a theorem.
The same numbers as M0.4
Put V_H = 110, V_L = 90 and μ = 0.2 into the machine:
P(V_H | buy) = (1 + 0.2)/2 = 0.60
ask = 0.60 × 110 + 0.40 × 90 = 102
bid = 0.40 × 110 + 0.60 × 90 = 98
spread = 4 = 0.2 × 20 ✓
A £4 spread on a £100 asset, and it isn’t a convention or a fee — it is what 20% informed flow costs, given the asset might be worth 90 or 110.
The book learns
Here is the part that makes this more than a spread formula. The posterior after a trade becomes the prior for the next one, so quotes walk. Continue the example — same μ = 0.2, same V_H/V_L — and feed it a buy, a buy, then a sell:
trade prior π P(V_H|·) new mid bid ask
─────────────────────────────────────────────────────────────
(start) 0.500 — 100.00 98.00 102.00
BUY 0.500 0.600 102.00 100.00 103.85
BUY 0.600 0.692 103.85 102.00 105.43
SELL 0.692 0.600 102.00 100.00 103.85
Read the third row: two buys in a row and the bid is now 102.00 — exactly the ask you could have lifted at the start. The maker will now happily buy from you at the price it was selling at two trades ago. Nobody announced anything. No news broke. The market revised its belief because buying is evidence, and the mechanism is nothing more than repeated Bayes.
Two consequences to carry forward:
- This is price discovery, mechanised. M0.1 said price “emerges from the collision of buy and sell intentions.” This is that sentence with the arithmetic filled in.
- This is where permanent price impact comes from. The move after a buy does not revert, because the belief genuinely changed. M1.3’s inventory effects do revert. Telling those two apart is the whole subject of M1.5.
(One honest caveat: the clean spread = μ·(V_H − V_L) holds at π = ½. Away from a 50/50 prior the spread is (V_H − V_L)·[P(V_H|buy) − P(V_H|sell)], which is what generated the rows above — it narrows as the market becomes confident. The intuition survives; the tidy formula is a special case.)
What the model buys you
Spreads widen before earnings because μ rises — more people plausibly know something. Makers aren’t being opportunistic; they are being solvent.
Retail flow is cheap to trade against because it is close to μ = 0. This is the entire economic basis of PFOF from M0.3: wholesalers pay for flow whose informedness is low and predictable, and can therefore quote inside the public spread.
Illiquid names have wide spreads partly because (V_H − V_L) is large — the range of defensible values for a thinly-covered small cap is simply wider.
A market can fail entirely. As μ → 1 there is no price at which a maker will trade. Every liquidity crisis has this shape: not an absence of buyers, but an inability to quote to a pool of counterparties who might all know something.
What the model doesn’t give you: any role for trade size. Every trade here is one unit, and the informed trader has no choice to make — he buys, or he sells. Let him choose how much, and you get Kyle (M1.4).
Source: Glosten & Milgrom (1985), “Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders” — the original is readable and the two-state version above is its first section. O’Hara, Market Microstructure Theory, ch.3 works it more slowly. Worth doing once by hand: redo the Bayes step with π ≠ ½ and watch the tidy formula turn into the general one.