Adverse Selection — Glosten-Milgrom and Kyle’s Lambda

Companion notebook to the Adverse Selection — Glosten-Milgrom and Kyle’s Lambda article (Market Microstructure, Module 3). All computations use the Python standard library (math, random, fractions) only — no NumPy or SciPy.

• §0 — Imports and constants
• §1 — Glosten-Milgrom spread: exact rational arithmetic for varying α
• §2 — Bayesian posterior updating after sequential buy orders
• §3 — Kyle’s λ and β table: 3×3 grid over Σ₀ and σₐ
• §4 — Kyle model simulation: 50 auction rounds, price convergence to v
• §5 — Validation summary: ✓/✗ checks for all quantitative claims

§0 — Imports and Model Parameters

We import only the Python standard library. fractions.Fraction is used for exact rational arithmetic in the Glosten-Milgrom section, avoiding floating-point rounding when computing the spread as a ratio. random.gauss drives the Kyle simulation.

§1 — Glosten-Milgrom Spread: Exact Arithmetic

We compute the ask price, bid price, and spread for α ranging from 0 to 0.9 using fractions.Fraction for exact rational results. The formula under π₀ = 1/2 is:

s = (alpha / (2 - alpha)) * (V_H - V_L)

At α = 0 the spread is exactly zero; as α → 1 the spread approaches V₁ − V₂ = 4.

§2 — Bayesian Posterior Updating

Starting from prior π₀ = 1/2 and α = 0.3, we simulate 10 consecutive buy orders and update the belief π = P(v = V₁) after each. Under all-buy signals the posterior should increase monotonically toward 1.

Update rule (Bayes’ theorem): after observing a buy,

pi_new = P(buy | V_H) * pi / P(buy)

§3 — Kyle’s Lambda: 3×3 Parameter Grid

For Σ₀ ∈ {0.25, 1.0, 4.0} and σₐ ∈ {0.5, 1.0, 2.0}, we compute:

• λ = sqrt(Σ₀) / (2σₐ) — price impact coefficient
• β = σₐ / sqrt(Σ₀) — informed trader aggressiveness
• Product λ · β = 1/2 (always, in equilibrium)

§4 — Kyle Model Simulation: Price Convergence

We draw the true value v once (seed 42) and simulate 50 auction rounds. In each round:

• The informed trader submits x = beta * (v - p_prev)
• Noise trader submits u ~ N(0, sigma_u^2)
• Market maker updates: p_new = p_prev + lambda * (x + u)

Kyle’s theory predicts the price converges toward v, with residual uncertainty decreasing each round. The initial mid-price is 100; true value is drawn from N(100, 1).

§5 — Validation Summary

Collect pass/fail checks for all quantitative claims made in the article. Each check compares computed results against theoretically predicted values.