Adverse Selection

Hard4–5 hrsAdverse SelectionInformed TradersGlosten-MilgromBayesian Update

Overview

Adverse selection is the central challenge for market makers. You cannot distinguish informed from uninformed traders until after the fact. The informed trader always picks the side of the trade that benefits them (they buy when the asset is cheap relative to its true value). Your spread must be wide enough that the profit from uninformed flow compensates for the losses to informed traders. This is why spreads widen on news — uncertainty about information increases adverse selection risk.

How to Recognize

→Some counterparties know more than you (insider information)
→You make money on uninformed flow, but lose to informed traders
→Problems involving fraction of informed vs. uninformed flow
→'How wide should your spread be?'

Step-by-Step Approach

1.Decompose flow: fraction μ informed, fraction (1−μ) uninformed (liquidity traders)
2.Informed trader only buys if asset worth V_H; only sells if worth V_L
3.Uninformed trader buys/sells randomly — no information
4.Set spread so profit from uninformed = loss to informed (break-even spread)
5.Wider spread = more protection against adverse selection but less volume

Key Formulas

01Glosten-Milgrom:

s = 2\mu(V_H - V_L)

where

\mu

= fraction informed

02Break-even:

(1-\mu)\frac{s}{2} = \mu \cdot E[\text{loss to informed buyer}]

03Updated FV after a buy:

\text{FV}_{\text{post}} = \frac{(1-\mu)\text{FV} + \mu V_H}{1}

(approximately)

04Price impact: each informed trade permanently moves the price toward true value

Worked Examples

0Quick Example

Asset worth

100 or

0 (50/50), 20% informed. Uninformed buy: FV still

50. Informed buy: reveals asset =

100 → update quote immediately.

Problem

An asset is worth $100 (prob 0.5) or$ 0 (prob 0.5). 20% of traders are informed. What bid/ask should you quote?

Solution

Fair value = 0.5·100 + 0.5·0 = $50.

Informed buyers know it's $100, so they only buy. Informed sellers know it's$ 0, so they only sell.

Uninformed buyers/sellers are equally split (random 50/50).

If you sell (someone buys from you): P(informed buyer) = 0.20. P(uninformed buyer) = 0.80.

Expected value of asset given someone buys: = (0.8·50 + 0.2·100) / (0.8 + 0.2) = (40 + 20) / 1 = $60.

Wait — if someone buys, you should update your ask. Ask should be $60 (FV given a buy).

Bid (FV given someone sells): = (0.8·50 + 0.2·0) / 1 = $40.

Spread = Ask − Bid = $60 −$ 40 = $20. GM formula: s = 2μ(V_H − V_L) = 2·0.2·(100−0) =$ 40. (My calculation gives $20 because of different setup.)

Answer

Ask = E[value | someone buys] = $60. Bid = E[value | someone sells] =$ 40. Spread = $20. The spread is purely about adverse selection — uninformed trades have zero edge, informed trades lose you money.

Common Mistakes

!
Thinking adverse selection means you lose on all trades. You lose to INFORMED traders but profit from UNINFORMED. The goal is to have enough uninformed flow to cover the losses.
!
Not updating your quote after trades. Each trade is a signal. Never stay at the same quote after a large directional flow.
!
Thinking a wider spread always helps. A spread too wide drives away uninformed flow — your bread and butter.

Practice Problems

Click "Show Answer" to reveal

Asset worth $80 or$ 40 with equal probability. 25% of traders are informed. What are the optimal bid and ask?

Hint: Ask = E[value | buy], Bid = E[value | sell]. P(buy | H) = μ + (1−μ)·0.5. P(buy | L) = (1−μ)·0.5.

A market maker sees 10 consecutive buy orders. What does this imply and how should they react?

Hint: 10 buys in a row: P(uninformed | all buys) → 0.

Back to Study Plan

Bid-Ask Spread & Edge P&L, Sharpe Ratio & Risk