5.4 Strategic informed trading: the Kyle model

Motivation

In Glosten–Milgrom the informed trader takes whatever size arrives. A real informed trader chooses size, knowing that trading more moves the price against her. Kyle (1985) solves this game. Its parameter $\lambda$ ("Kyle's lambda"), the price impact per unit of net order flow, is the most widely used measure of market depth, and the model gives a precise sense in which uninformed flow pays informed traders.

5.4.1The one-period game#

Setupone period

Liquidation value $v \sim N(p_0, \Sigma_0)$ , $\Sigma_0 = \sigma_v^2$ . One risk-neutral insider observes $v$ .
Noise traders submit net quantity $u \sim N(0, \sigma_u^2)$ , independent of $v$ .
The insider submits $x = X(v)$ . A competitive, risk-neutral market maker observes only total order flow $y = x + u$ and sets the price $p = P(y)$ at which she fills $y$ .
Equilibrium: (i) $X$ maximises the insider's expected profit $E[(v - p)x \mid v]$ given $P$ ; (ii) $P(y) = E[v \mid y]$ given $X$ (zero expected profit for the market maker).
We look for a linear equilibrium: $P(y) = \mu + \lambda y$ and $X(v) = \alpha + \beta v$ .

Proposition 5.3Kyle, 1985

The unique linear equilibrium has

\lambda = \frac{\sigma_v}{2\sigma_u}, \qquad \beta = \frac{\sigma_u}{\sigma_v}, \qquad \mu = p_0, \qquad \alpha = -\beta p_0, \tag{5.7}

so $x = \beta(v - p_0)$ and $p = p_0 + \lambda y$ . Moreover:

(a) The insider's expected profit is $E[(v - p)x] = \tfrac12\sigma_v\sigma_u$ . (b) Noise traders' expected loss equals the insider's expected profit: $E[(p - v)u] = \lambda\sigma_u^2 = \tfrac12\sigma_v\sigma_u$ . (c) Price informativeness: $\operatorname{Var}(v \mid y) = \tfrac12\Sigma_0$ , independent of $\sigma_u$ . (d) Order flow is half information, half noise: $\operatorname{Var}(x) = \operatorname{Var}(u) = \sigma_u^2$ .

Proof

Step 1: insider's best response to a linear price. Given $p = \mu + \lambda(x + u)$ with $\lambda > 0$ , and $u$ independent of $v$ with mean 0,

E[(v - p)x \mid v] = (v - \mu)x - \lambda x^2.

This is a concave quadratic in $x$ , maximised at

x = \frac{v - \mu}{2\lambda}, \qquad \text{i.e.}\quad \beta = \frac{1}{2\lambda},\quad \alpha = -\frac{\mu}{2\lambda}. \tag{5.8}

(If $\lambda \le 0$ the insider's problem is unbounded, so no equilibrium has $\lambda \le 0$ .)

Step 2: market maker's pricing given a linear insider. If $x = \alpha + \beta v$ , then $(v, y)$ is jointly Gaussian with $E y = \alpha + \beta p_0$ , $\operatorname{Var}(y) = \beta^2\Sigma_0 + \sigma_u^2$ and $\operatorname{Cov}(v, y) = \beta\Sigma_0$ . For jointly Gaussian variables the conditional expectation is linear (the projection theorem):

E[v \mid y] = p_0 + \frac{\beta\Sigma_0}{\beta^2\Sigma_0 + \sigma_u^2}\,\big(y - \alpha - \beta p_0\big). \tag{5.9}

So zero profit requires $\lambda = \dfrac{\beta\Sigma_0}{\beta^2\Sigma_0 + \sigma_u^2}$ and $\mu = p_0 - \lambda(\alpha + \beta p_0)$ .

Step 3: fixed point. From (5.8), $\alpha + \beta p_0 = (p_0 - \mu)/(2\lambda)$ . Substituting into the expression for $\mu$ : $\mu = p_0 - (p_0 - \mu)/2$ , so $\mu = p_0$ , and then $\alpha = -\beta p_0$ . Substituting $\beta = 1/(2\lambda)$ into the expression for $\lambda$ :

\lambda\Big(\frac{\Sigma_0}{4\lambda^2} + \sigma_u^2\Big) = \frac{\Sigma_0}{2\lambda} \;\Longrightarrow\; \frac{\Sigma_0}{4\lambda} + \lambda\sigma_u^2 = \frac{\Sigma_0}{2\lambda} \;\Longrightarrow\; \lambda^2 = \frac{\Sigma_0}{4\sigma_u^2}. \tag{5.10}

Taking the positive root (Step 1 requires $\lambda > 0$ ), $\lambda = \sigma_v/(2\sigma_u)$ and $\beta = 1/(2\lambda) = \sigma_u/\sigma_v$ . Uniqueness among linear equilibria follows because each step's solution was unique.

(a) $E[(v-p)x \mid v] = (v - p_0)x - \lambda x^2$ at $x = (v - p_0)/(2\lambda)$ equals $(v - p_0)^2/(4\lambda)$ . Taking expectations,

E[(v - p)x] = \frac{\Sigma_0}{4\lambda} = \frac{\sigma_v^2 \cdot 2\sigma_u}{4\sigma_v} = \frac{\sigma_v\sigma_u}{2}. \tag{5.11}

(b) The market maker's expected profit is $E[(p - v)y] = 0$ by construction ( $p = E[v \mid y]$ implies $E[(p - v)g(y)] = 0$ for any $g$ ). Since $y = x + u$ , $E[(p - v)u] = -E[(p - v)x] = E[(v - p)x] = \sigma_v\sigma_u/2$ . Directly: $E[(p - v)u] = E[(p_0 + \lambda x + \lambda u - v)u] = \lambda\sigma_u^2$ , using independence of $u$ from $v$ and $x$ ; and $\lambda\sigma_u^2 = \sigma_v\sigma_u/2$ .

(c) For jointly Gaussian variables the conditional variance is non-random:

\operatorname{Var}(v \mid y) = \Sigma_0 - \frac{\operatorname{Cov}(v,y)^2}{\operatorname{Var}(y)} = \Sigma_0 - \frac{\beta^2\Sigma_0^2}{\beta^2\Sigma_0 + \sigma_u^2} = \Sigma_0 - \frac{\Sigma_0}{2} = \frac{\Sigma_0}{2}, \tag{5.12}

using $\beta^2\Sigma_0 = \sigma_u^2$ .

(d) $\operatorname{Var}(x) = \beta^2\Sigma_0 = \sigma_u^2$ .∎

5.4.2Who pays the insider#

Interpretation

The insider hides behind noise: she trades exactly so that her order flow has the same variance as the noise. More noise ( $\sigma_u$ up) makes the market deeper ( $\lambda$ down), and she trades more aggressively ( $\beta$ up), so her profit $\sigma_v\sigma_u/2$ rises, but the price reveals the same fraction of her information (c).
The insider's profit is paid entirely by noise traders (b). If retail flow is noise, the model says retail flow funds informed and faster traders, and more retail flow means more profit for them. SEBI's finding that individuals' gross losses were mirrored by algorithmic prop and FPI gains (5.1) is consistent with this, though it does not prove that the winners were "informed" in Kyle's sense rather than faster.
$1/\lambda$ is depth: the order flow needed to move the price by ₹1.

5.4.3Kyle's lambda in data#

Worked example

For an NSE stock at ₹1,500, suppose the private information resolved by the close has standard deviation $\sigma_v = \text{₹}30$ (2%) and the standard deviation of net noise flow over the same period is $\sigma_u = 2{,}00{,}000$ shares. Then $\lambda = 30/(2 \times 2{,}00{,}000) = \text{₹}7.5\times10^{-5}$ per share: a net buy imbalance of 1 lakh shares moves the price by ₹7.50 (50 bp). The insider's equilibrium trading intensity is $\beta = 2{,}00{,}000/30 \approx 6{,}667$ shares per rupee of mispricing, and her expected profit is $30 \times 2{,}00{,}000/2 = \text{₹}30$ lakh, which is also the expected loss of the noise traders.

Estimating $\lambda$ In data, regress the price change over an interval on signed order flow (buyer-initiated minus seller-initiated volume) over the same interval. In the model, the slope of the later value $v$ on $y$ is exactly $\lambda$ , by (5.9). Exercise m05_kyle simulates 400,000 one-period games, regresses $v$ on $y$ , and recovers $\lambda$ within 1%. A further test fixes $\lambda$ and shows by simulation that the insider's mean profit over a grid of $\beta$ is maximised at $\beta = 1/(2\lambda)$ , which is Step 1 checked numerically.

Practitioner notes

The multi-period version (the sequential and continuous auctions in Kyle 1985; K. Back, "Insider trading in continuous time", Review of Financial Studies 5(3), 1992) has the insider trade gradually, so information enters prices smoothly and order flow is persistently signed. That persistence is what (A2) of the Roll model ignores.
Linear impact is an equilibrium property of this Gaussian model. Empirically, the impact of a large order executed over time is concave in its size (5.5). Kyle's $\lambda$ is best read as the local slope for small imbalances.
Kyle-type reasoning also shows why "following the smart money" from order flow is hard: the informed trader's flow is, by design, indistinguishable from noise trade by trade.