Price impact describes how the order flow generated by a trade moves the market price against the trader. In "Continuous Auctions and Insider Trading," Kyle (1985) built a dynamic model with one risk-neutral insider, random noise traders, and competitive market makers. The unique linear equilibrium captures the insider's price impact in a single parameter λ. In this single-auction version, one-half of the insider's private information enters prices (Σ₁ = ½Σ₀). The insider earns an expected profit of ½(Σ₀σ_u²)^½.
What the Study Found
In the single auction, the price error variance falls to Σ₁ = ½Σ₀, impounding half of the insider's private information. The insider's expected unconditional profit equals ½(Σ₀σ_u²)^½, proportional to the standard deviations of both value and noise trading. In the continuous-auction limit, the price error variance follows Σ(t) = (1 − t)Σ₀. It reaches Σ(1) = 0, so all private information is revealed by the close. The continuous-limit insider earns Σ₀^½·σ_u, exactly double the single-auction profit. Doubling noise-trading volatility σ_u halves the price-impact parameter λ while doubling insider profits and leaving price informativeness unchanged.
Methodology
The paper is a theoretical model, not an empirical study, so it uses no dataset, sample, or time period. It assumes the liquidation value v is normal with mean p₀ and variance Σ₀, and noise-trader volume u is independent normal with variance σ_u². Trading is structured as a sequence of N auctions, with market makers setting a semi-strong efficient price p = E{v | order flow}. The continuous-time model is obtained as the interval between auctions goes to zero.
Key Statistics
| Metric | Finding | Context |
|---|---|---|
| Information impounded (single auction) | Σ₁ = ½Σ₀ | Half of private information in price after one auction |
| Insider profit (single auction) | ½(Σ₀σ_u²)^½ | Expected unconditional profit |
| Insider profit (continuous) | Σ₀^½·σ_u | Exactly double the single-auction profit |
| Price error variance (continuous) | Σ(t) = (1 − t)Σ₀ | Σ(1) = 0; all information revealed by close |
| Market depth parameter | λ(t) = (Σ₀/σ_u²)^½ | Constant over time in the continuous limit |
| Single-auction equilibrium | X(v) = β(v − p₀), P = p₀ + λ(x + u) | β = (σ_u²/Σ₀)^½, λ = ½(σ_u²/Σ₀)^(−½) |
Why This Matters
The model gives traders and exchanges a precise language for liquidity. Depth is the order flow needed to move price one dollar, and it scales with how much uninformed volume hides informed flow. It shows that an insider optimally bleeds private information into the market gradually rather than all at once, trading slowly to avoid revealing his hand. The λ parameter became the standard measure of price impact used to size large orders and design execution algorithms. Modern transaction-cost models still estimate λ empirically to forecast the slippage that large institutional orders incur.