6.6.2 Probability of Informed Trading (PIN)

Basic Prerequisites: Probability & Measure Theory (Poisson processes, mixtures) and 01 · From Zero.

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1. Intuition & Practical Objective

PIN - the probability of informed trading - answers one question precisely: what fraction of the order flow is informed? The Glosten–Milgrom machinery (Adverse Selection & GM) needs the informed fraction $\pi$ to set the spread; PIN is the empirical estimator of that fraction from observable daily buy/sell counts.

The intuition is statistical and elegant: informed traders distort the order flow to one side. On days with no information event, buys and sells arrive in roughly equal (Poisson) numbers. On days with good news, informed traders pile onto the buy side; on bad-news days, onto the sell side. The total daily buy/sell counts therefore follow a mixture - a blend of a balanced Poisson flow (no event) and two lopsided Poisson flows (good/bad event). PIN is the share of the total flow that comes from the informed, one-sided component.

The practical objective: take daily buy/sell counts, fit the Poisson-mixture model by maximum likelihood, and recover PIN as a number in $[0,1]$ that a market maker can use to price the adverse-selection spread, and an asset-pricing researcher can use as an “information risk” factor.

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2. Mathematical Ground Truth & Derivations

Model (Easley–Kiefer–O’Hara 1997; Hasbrouck Ch 6). On each day, with probability $\alpha$ an information event occurs. If it does, informed traders arrive as a Poisson process of intensity $\mu$, and - knowing the state - all trade on the same side. Uninformed traders arrive on both sides as Poisson processes of intensity $ϵ$ each, every day.

Arrival rates. Over the sample, the informed component contributes $\alpha \mu$ orders; the uninformed component contributes $2ϵ$ ($ϵ$ buys + $ϵ$ sells). PIN is the informed fraction:

$$\boxed{\mathrm{PIN}=\frac{\alpha \mu }{\alpha \mu +2ϵ}}(\text{Hasbrouck eq. 6.4}).$$

Why this is “the probability of informed trading.” Pick a random arriving order. With probability $\alpha \mu /(\alpha \mu +2ϵ)$ it was generated by an informed trader; the complement by an uninformed one. So PIN literally is the probability that the next order is informed - the quantity the GM spread formula requires.

Identification (the deep, testable fact). Only the product $\alpha \mu$ is identified from the daily counts - the likelihood is nearly flat over combinations of $\alpha$ and $\mu$ that yield the same $\alpha \mu$. But because PIN depends only on $\alpha \mu$, PIN itself is stable and estimable even though $\alpha$ and $\mu$ individually are imprecise (Hasbrouck Ch 6).

Asymptotics. With $\mathrm{PIN}\to 0$ as $ϵ\to \infty$ (noise drowns the signal) or $\mu \to 0$ (no informed traders); $\mathrm{PIN}\to 1$ as informed flow swamps noise. PIN rises with the severity of adverse selection - exactly the toxicity this pillar is about.

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3. Computational Implementation - PIN is the informed fraction (stdlib only)

The cleanest verification is direct: simulate order arrivals under the model and count what fraction come from informed traders - it must equal the formula.

The simulated informed fraction (0.4309) converges to the formula (0.4286) - PIN really is the informed share of the flow. And note the pattern: heavier informed activity ($\mu =30$) pushes PIN from 0.43 to 0.69 (more toxic), while zero informed traders ($\alpha =0$) pins PIN at exactly 0.

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4. Failure Modes & First-Principles Breakdowns

1. Only $\alpha \mu$ is identified, not $\alpha$ and $\mu$ separately. The MLE cannot cleanly separate “many rare events” from “few common events.” Over-interpreting individual $\alpha$ or $\mu$ estimates is wrong; the product (hence PIN) is the stable object.
2. Daily-count aggregation loses intraday information. Only the daily total buy/sell count enters the likelihood - the sequence, the timing, and the size of trades are discarded (Hasbrouck Ch 6, §6.3). That’s why the high-frequency successor VPIN moves to volume buckets (04 · VPIN).
3. “Informed” is defined by one-sidedness, not by actual news. PIN detects imbalanced flow; a legitimate one-way flow (a large fund rebalancing, a short-squeeze covering wave) is indistinguishable from informed flow by counts alone.

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5. References

• Easley, Kiefer & O’Hara (1997), The information content of the trading process, J. Empirical Finance 4, 159–186
• Hasbrouck (2007), Empirical Market Microstructure
• Easley, Hvidkjaer & O’Hara (2002), Is information risk a determinant of asset returns?, J. Finance 57(5)

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · The EKOP Model (the full Poisson likelihood + MLE) · 04 · VPIN (the volume-synchronized generalization)
• Theory: Adverse Selection & Glosten–Milgrom (PIN estimates the informed fraction $\pi$ that GM needs)