Structural Market Microstructure

EKOP PIN (1996) Probability of Informed Trading Lab

Estimate the Probability of Informed Trading ($\text{PIN} = \frac{\alpha\mu}{\alpha\mu + 2\epsilon}$) from daily buy and sell order imbalances using numerical Maximum Likelihood Estimation.

Market Liquidity Presets:
EKOP Model Parameters
$\alpha$
Probability that a private information event occurs on a given day
$\delta$
Probability that news is unfavorable (informed traders sell)
Daily Poisson intensity of informed trading volume
Daily Poisson intensity of uninformed buy and sell orders
True Benchmark PIN

33.1%

Theoretical Asymmetry
MLE Estimated PIN

32.8%

Fitted Model Value
Fitted Event Prob ($\hat{\alpha}$)

36.5% (True: 38%)

Likelihood Maximizer
Fitted Informed Rate ($\hat{\mu}$)

68 / day (True: 65)

Informed Trade Intensity
Fitted Noise Rate ($\hat{\epsilon}$)

26 / day (True: 25)

Uninformed Noise Intensity
EKOP Microstructure Visualizer
Sensitivity: Event Probability ($\alpha$) vs Informed Arrival ($\mu$) on PIN (%)

Theoretical Probability of Informed Trading with fixed noise rate $\epsilon = 100$.

Sensitivity: Noise Rate ($\epsilon$) vs Informed Arrival ($\mu$) on PIN (%)

Impact of liquidity trading volume on information dilution with fixed $\alpha = 35\%$.

Mathematical Derivation: EKOP (1996) Structural Likelihood

1. Information Event Tree and Daily Trade Arrival

In the classic Easley, Kiefer, O'Hara, and Paperman (1996) framework, securities trade over discrete calendar days $d \in \{1, \dots, D\}$. Prior to the market open each day, an information event occurs with probability $\alpha$. If an event occurs, the news is bad ($L$) with probability $\delta$, or good ($H$) with probability $1-\delta$.

Throughout trading day $d$, orders arrive according to independent Poisson processes:

  • Uninformed Liquidity Traders: Exogenously motivated buyers arrive at Poisson rate $\epsilon$, and sellers arrive at Poisson rate $\epsilon$.
  • Informed Traders: If an information event occurs, privately informed traders enter the market at Poisson rate $\mu$. On good news days ($H$), they buy; on bad news days ($L$), they sell.
2. Daily Joint Probability Distribution

Conditional on the unobserved information state, the joint distribution of observed daily buys $B_d$ and sells $S_d$ is the mixture of three Poisson processes:

$$L(B_d, S_d \mid \theta) = (1-\alpha) e^{-2\epsilon} \frac{\epsilon^{B_d + S_d}}{B_d! S_d!} + \alpha (1-\delta) e^{-(2\epsilon+\mu)} \frac{(\mu+\epsilon)^{B_d} \epsilon^{S_d}}{B_d! S_d!} + \alpha \delta e^{-(2\epsilon+\mu)} \frac{\epsilon^{B_d} (\mu+\epsilon)^{S_d}}{B_d! S_d!}$$
3. The Probability of Informed Trading (PIN)

The Probability of Informed Trading is defined as the expected fraction of daily orders originating from informed traders:

$$\text{PIN} = \frac{\mathbb{E}[\text{Informed Trades}]}{\mathbb{E}[\text{Total Trades}]} = \frac{\alpha \mu}{\alpha \mu + 2\epsilon}$$

Higher values of PIN denote severe information asymmetry, wider dealer bid-ask spreads, and elevated adverse selection risks for passive market participants.

4. Numerically Stable Maximum Likelihood Estimation

Given $D$ independent trading days, the joint sample log-likelihood is:

$$\ln L(\theta) = \sum_{d=1}^D \ln L(B_d, S_d \mid \theta)$$

To prevent floating-point overflow for large order counts ($B_d, S_d > 100$), the factorial terms $\ln(B_d!) + \ln(S_d!)$ are dropped from the objective function, and log-sum-exp factorization is applied:

$$\ell_d(\theta) = \text{LogSumExp}\left(x_{1,d}, x_{2,d}, x_{3,d}\right)$$ $$x_{1,d} = \ln(1-\alpha) - 2\epsilon + (B_d+S_d)\ln \epsilon$$ $$x_{2,d} = \ln\alpha + \ln(1-\delta) - (2\epsilon+\mu) + B_d\ln(\mu+\epsilon) + S_d\ln \epsilon$$ $$x_{3,d} = \ln\alpha + \ln\delta - (2\epsilon+\mu) + B_d\ln \epsilon + S_d\ln(\mu+\epsilon)$$

Microstructure Concept Mastery Quiz

1. What is the closed-form definition of PIN in the EKOP (1996) model with symmetric noise?

2. How does the EKOP structural likelihood identify whether a given day was a good news day vs. bad news day?

3. What is the primary technical barrier in numerically estimating the original EKOP likelihood in high-volume stocks?

4. Why does PIN drop toward zero in heavily traded mega-cap stocks?

Related Quantitative Microstructure Labs

VPIN Order Flow Toxicity Lab

Model David Easley, Marcos López de Prado, and Maureen O'Hara's (2012) Volume-Synchronized Probability of Toxicity.

Glosten-Milgrom Microstructure Lab

Simulate sequential trades, Bayesian quote updating, and adverse selection spread decay.

Kyle's Lambda & Microstructure Lab

Model Pete Kyle's (1985) continuous informed trading equilibrium, market depth, and price impact.

Corwin-Schultz Spread Lab

Estimate effective bid-ask transaction spreads and liquidity using daily high and low prices.

Frequently Asked Questions

The EKOP (1996) model, formulated by David Easley, Nicholas Kiefer, Maureen O'Hara, and Joseph Paperman, is a structural market microstructure model that estimates the probability that an incoming order is submitted by a privately informed trader. Using daily counts of buyer-initiated and seller-initiated trades, it estimates parameters (alpha, delta, mu, epsilon) via numerical maximum likelihood estimation to quantify information asymmetry in securities.

Uninformed liquidity traders arrive symmetrically throughout every trading day following independent Poisson processes with intensity rate epsilon for buys and epsilon for sells. When private information events occur (with probability alpha), informed traders enter the market on the profitable side (with arrival intensity mu): buying if good news (delta = 0), or selling if bad news (delta = 1). The resulting daily order imbalance (B - S) provides identifying variation for the likelihood function.

In liquid stocks with high trading volume, calculating Poisson probabilities involves large factorials (B!, S!) and exponentiations (mu^B, eps^S) that cause arithmetic overflow or underflow in standard computers. Modern implementations overcome this by dropping constant factorial terms from the objective function, reparameterizing into log-odds space, and using stable Log-Sum-Exp transformations.

In mega-cap stocks, the volume of uninformed noise trading (epsilon) is enormous, generated by institutional indexers, mutual funds, retail rebalancing, and algorithmic market makers. Because PIN = (alpha * mu) / (alpha * mu + 2 * epsilon), massive noise volume dwarfs informed trade intensity, causing PIN to drop to 1-3%, whereas illiquid micro-caps exhibit PIN values exceeding 30-40%.

EKOP PIN (1996) is a static, parametric model estimated over historical daily trade count aggregates using numerical maximum likelihood optimization. In contrast, VPIN (Volume-Synchronized Probability of Toxicity, Easley, Lopez de Prado, and O'Hara 2012) is a non-parametric, continuous metric measured in volume time using rolling trade buckets and bulk volume classification without numerical optimization.