The Kyle Model

Where The Glosten-Milgrom Model has many small, non-strategic informed traders, Kyle (1985) asks the sharper question: what if there is one informed trader who knows they will move the price, and trades strategically to hide their information and maximize profit? The answer is a clean linear equilibrium whose central object, Kyle's lambda, the price-impact coefficient, has become the standard measure of market illiquidity and the theoretical anchor for every Market Impact model.

Setup: one insider, noise traders, a competitive maker

Three players in a single-period batch auction:

• An informed trader (insider) observes the asset's liquidation value $v \sim \mathcal{N}(p_0, \Sigma_0)$ and submits a market order of size $x$.
• Noise traders submit a total order $u \sim \mathcal{N}(0, \sigma_u^2)$, independent of $v$, the crowd the insider hides inside.
• A competitive, risk-neutral market maker observes only the aggregate order flow $y = x + u$ (not its components) and sets a single clearing price $p$ equal to the expected value given that flow: $p = \mathbb{E}[v \mid y]$.

The insider's dilemma is that trading more aggressively both increases exposure to their edge and moves the price against themselves, revealing information. The equilibrium is where the insider's optimal size and the maker's pricing rule are mutually consistent.

The linear equilibrium

Guess a linear form and verify it is a fixed point:

$x = \beta\,(v - p_0), \qquad p = p_0 + \lambda\, y.$

Here $\beta$ is the insider's trading intensity and $\lambda$ is the maker's price-impact coefficient (Kyle's lambda).

Insider's problem. Given the pricing rule, the insider's profit is $\pi = x\,(v - p) = x\big(v - p_0 - \lambda(x + u)\big)$. Taking expectations over the noise ($\mathbb{E}[u]=0$),

$\mathbb{E}[\pi \mid v] = x\,(v - p_0) - \lambda x^2.$

This is concave in $x$; the first-order condition $\;(v - p_0) - 2\lambda x = 0\;$ gives the optimal order

$x = \frac{v - p_0}{2\lambda} \quad\Longrightarrow\quad \beta = \frac{1}{2\lambda}.$

The insider trades half as aggressively as a price-taker would, deliberately holding back to limit self-impact.

Market maker's problem. With $v, u$ jointly Gaussian, $\mathbb{E}[v \mid y]$ is the linear projection, so

$\lambda = \frac{\operatorname{Cov}(v, y)}{\operatorname{Var}(y)} = \frac{\beta\,\Sigma_0}{\beta^2 \Sigma_0 + \sigma_u^2},$

using $y = \beta(v - p_0) + u$, $\operatorname{Cov}(v,y) = \beta\Sigma_0$, and $\operatorname{Var}(y) = \beta^2\Sigma_0 + \sigma_u^2$.

Solve the two equations. Substitute $\beta = 1/(2\lambda)$ into the maker's equation. After simplifying (multiply through by $4\lambda^2$) the $\lambda$'s cancel to give $\;\Sigma_0 + 4\lambda^2\sigma_u^2 = 2\Sigma_0$, hence $4\lambda^2\sigma_u^2 = \Sigma_0$ and, writing $\sigma_v = \sqrt{\Sigma_0}$,

$\boxed{\lambda = \frac{1}{2}\frac{\sigma_v}{\sigma_u}, \qquad \beta = \frac{1}{2\lambda} = \frac{\sigma_u}{\sigma_v}.}$

Reading Kyle's lambda

This tiny formula carries the whole model:

• Lambda is price impact per unit of order flow. The price moves $\Delta p = \lambda\,\Delta y$; the maker's depth, order flow needed to move price one unit, is $1/\lambda = 2\sigma_u/\sigma_v$. More noise trading ($\sigma_u$ large) makes the market deeper and impact smaller, because the insider is easier to hide; more fundamental uncertainty ($\sigma_v$ large) makes it shallower, because each trade is more informative.
• Impact is linear in size here, a direct consequence of the Gaussian/linear structure. Empirically impact is concave (see The Square-Root Impact Law); reconciling Kyle's linear per-trade impact with the empirical square-root aggregate law is a central theme of modern microstructure.
• Exactly half the information is revealed. Compute the maker's residual uncertainty: $\beta^2\Sigma_0 = (\sigma_u/\sigma_v)^2\sigma_v^2 = \sigma_u^2$, so $\operatorname{Var}(y) = 2\sigma_u^2$ and the posterior variance is $\operatorname{Var}(v \mid y) = \Sigma_0 - \dfrac{\operatorname{Cov}(v,y)^2}{\operatorname{Var}(y)} = \sigma_v^2 - \dfrac{(\sigma_u\sigma_v)^2}{2\sigma_u^2} = \tfrac12\sigma_v^2$. The insider's strategic restraint leaks exactly half of their private variance into the price, a beautiful, clean result. In Kyle's multi-period continuous version, information is revealed smoothly over the session and the price is a martingale that ends at $v$.

Worked example

An insider knows a stock is worth $v$, drawn around $p_0 = 100$ with value volatility \sigma_v = \2$. Daily noise-trader volume has standard deviation$\sigma_u = 10{,}000$ shares. Then

$\lambda = \frac{1}{2}\cdot\frac{2}{10{,}000} = \$0.0001 \text{ per share}, \qquad \beta = \frac{10{,}000}{2} = 5{,}000 \text{ shares per \$}.$

Suppose the insider learns $v = 103$ (three dollars of edge). They submit $x = \beta(v - p_0) = 5{,}000 \times 3 = 15{,}000$ shares. If net noise that period is $u = -5{,}000$, aggregate flow is $y = 10{,}000$, and the price prints at p = 100 + \lambda y = 100 + 0.0001(10{,}000) = \101$. The insider bought 15,000 shares at an average impacted price and captures roughly$x(v - p_0) - \lambda x^2 = 15{,}000(3) - 0.0001(15{,}000)^2 = 45{,}000 - 22{,}500 = $22{,}500$, half the naive edge, the rest lost to their own impact and shared with the market via price discovery.

Failure modes and caveats

• Linear impact is an artifact of Gaussianity. The real aggregate impact of large "meta-orders" is concave (square-root); Kyle's linearity holds per-auction with normal priors but does not describe splitting one big order over a day.
• Single insider, known $\sigma_u$. With multiple competing informed traders, they trade more aggressively, more information is revealed, and the insider's profit shrinks (competition erodes the informational rent). The model also assumes the maker knows the noise-trading variance.
• Static value. The basic model has one liquidation value; extensions with a flowing signal or with risk-averse makers change the dynamics.
• No spread, one price. Kyle produces a price-impact coefficient, not a bid-ask spread, the complementary object to Glosten–Milgrom's spread. Real markets have both.

In interviews

Kyle is the model to know cold if you interview for market-making or execution. Be ready to: posit the linear forms $x = \beta(v - p_0)$ and $p = p_0 + \lambda y$; derive the insider's optimum $\beta = 1/(2\lambda)$ from maximizing $x(v-p_0) - \lambda x^2$; derive the maker's $\lambda = \beta\Sigma_0/(\beta^2\Sigma_0 + \sigma_u^2)$ from the Gaussian projection; and solve the pair to land on $\lambda = \tfrac12\sigma_v/\sigma_u$. The insight interviewers want stated aloud: lambda measures illiquidity (impact per share), rises with information $\sigma_v$, falls with noise-trading liquidity $\sigma_u$, and the insider optimally reveals exactly half their edge. The standard contrast is with The Glosten-Milgrom Model (spread from many small informed traders) versus Kyle (impact from one strategic insider).