6.4.1 Adverse Selection from Zero

Basic Prerequisites: Probability & Measure Theory.

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1. Intuition & Practical Objective

This page builds the why of the bid-ask spread with no prior microstructure knowledge needed. The objective is one unshakeable idea: even with zero exchange fees, zero inventory-carrying costs, and zero dealer monopoly power, a bid-ask spread must still exist - because the market maker risks trading against someone who knows more than they do.

Start with the dumbest question: why is a stock’s ask higher than its bid? Three candidate answers - (a) to pay exchange fees, (b) to compensate the maker for holding inventory overnight, (c) to exploit customers. All three fail to explain the spread in a liquid, near-frictionless, competitive market. The answer GM (1985) nailed is (d): the maker’s counterparty might be better informed than the maker. When that happens the maker is systematically on the wrong side - buying right before a fall, selling right before a rise. The spread is collected from uninformed traders to pay for those unavoidable informed losses.

Three intuitions in increasing sophistication:

1. The mid-price-only dealer goes bankrupt. Quote a single price and buy/sell at it. Informed traders hurt you every single time; uninformed traders just shuffle cash around at no profit to you. Your P&L is pure negative - the liquidity you “facilitate” is a net transfer out of your books to the informed.
2. The spread is a toll, not a fee. It bounces you out of what the information costs. Thinner than the information cost and the maker loses; thicker and a competitor quotes past you. Competition pins the spread down to exactly the adverse-selection cost.
3. The mid is not “the price.” The efficient price - what the maker believes the asset is worth given everything observed - is a conditional mean that moves with order flow. The quoted bid/ask straddle it asymmetrically (see 03 · The GM Model).

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2. Mathematical Ground Truth & Derivations

Set up a one-round adverse-selection game. There is a single “true” but as-yet-unrevealed terminal value; it is $+1$ above the initial mid $m=100$ with probability $\frac{1}{2}$ and $-1$ below it with probability $\frac{1}{2}$. A trader arrives:

• Informed (prob $\pi$): knows the sign of the move and always trades in its direction (buys if up, sells if down).
• Uninformed (prob $1-\pi$): buys or sells with equal probability for liquidity; carries no information.

Suppose the dealer trades at the mid with no spread. If the informed trader buys (value about to be $101$), the dealer sells at $100$ and immediately loses $1$ per share; symmetric loss on informed sells. Uninformed trades transact at fair value and break the dealer even. So the expected loss per trade at zero spread is $\pi$ (per unit of the move): the dealer who quotes no spread loses money in direct proportion to the informed fraction. Let the half-spread be $h$, i.e. ask $=100+h$, bid $=100-h$. Then:

• informed buy: dealer sells at $100+h$ but the share is worth $101$ → P&L $=h-1$;
• informed sell: dealer buys at $100-h$ but the share is worth $99$ → P&L $=h-1$;
• uninformed (either direction): dealer banks the half-spread $h$.

Expected P&L per trade:

$$\mathbb{E}[\text{P\&L}]=\pi (h-1)+(1-\pi )h=h-\pi .$$

Zero-expected-profit quote: set $\mathbb{E}[\text{P\&L}]=0⟹\boxed{h^{\star }=\pi }$ - the required half-spread equals the informed probability. This is the discrete, no-friction seed of the Glosten–Milgrom result. The full symmetric GM formula $A-B=\pi (V_H-V_L)$ (with $V_H-V_L=2$) is exactly this "$h^{\star }=\pi$" written against the value dispersion.

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3. Computational Implementation - the dealer who quotes the mid loses; the GM-dealer breaks even

Stdlib only. We simulate thousands of trades against a dealer who (a) quotes no spread (mid only) and (b) quotes the break-even half-spread $h=\pi$ of GM.

Two solid numbers from the run: quoting the mid costs the dealer $0.301$ per trade (matches $-\pi =-0.30$ to Monte Carlo error), and quoting the GM half-spread brings P&L to $\approx 0$ (the noise band around the theoretical $0$). The spread is not a fee for services rendered - it is insurance against information.

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4. Failure Modes & First-Principles Breakdowns

1. The “markets are efficient so orders carry no news” trap. Roll’s model and naive quoting assume trades are uninformative. GM’s whole point is they are not - the moment you quote the mid you assume $\pi =0$ and bleed. This is why normal-distribution, frictionless “mid execution” cost models misprice liquidity for illiquid/information-heavy names (see 05 · Failure Modes).
2. Confusing the spread with the price. The midpoint is not the efficient price except at maximal uncertainty. A dealer who thinks the mid is “fair value” is pricing the asset as if orders were noise, i.e. exactly the assumption that makes them a target.
3. The half-spread is not a free parameter. It is pinned by the informed fraction, not by what the dealer wants to earn. Competing makers who can quote the same value dispersion at the same information risk will undercut any thicker spread.

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5. References

• Glosten & Milgrom (1985), Bid, ask and transaction prices in a specialist market with heterogeneously informed traders, JFE 14(1), 71–100
• Copeland & Galai (1983), Information effects on the bid-ask spread, J. Finance 38(5), 1457–1469
• Hasbrouck (2007), Empirical Market Microstructure

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6. Connected Graph Bridges

• Base: Probability & Measure Theory · Bayesian Statistics
• Continue: 02 · Informed vs Uninformed · Index Hub
• Sibling: Limit Order Book Mechanics · Toxic Order Flow