2.10.2 Kyle (1985) and the Back Continuous-Time Limit

Basic Prerequisites: 01 - From Zero and basic probability (normal distributions, conditional variance, OLS).

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1. Intuition & Practical Objective

Kyle (1985) asked the sharpest possible version of the informed-trader question. Instead of a two-point value and a one-shot trade, suppose:

• the asset’s value is $v\sim \mathcal{N}(p_0,{\Sigma }_0)$ - a continuum of possible values, so the insider has how much better informed she is rather than which way;
• a single strategic insider observes $v$ and chooses how much to trade;
• noise traders submit $u\sim \mathcal{N}(0,{\sigma }_u^2)$ - random volume whose only job is camouflage;
• competitive market makers see only the total signed order flow $y=x+u$ and set a price.

The insider’s problem is a genuine strategic dilemma. Trade big and you move the price against yourself; trade small and you earn the impact-free edge on fewer shares. Kyle showed the equilibrium is startlingly clean: the insider submits a quantity linear in her edge, the makers use a linear price rule, and - the famous result - exactly half of the insider’s private information is impounded into the price, regardless of how much noise there is.

The second half of this page is the continuous-time question, and it contains the most useful trap in the literature. Everyone’s first instinct is: “run more and more auctions over a fixed horizon and take the limit”. That limit does not exist. Kyle’s multi-auction model releases information geometrically (half per auction), and if you shrink the interval between auctions while holding the total noise budget fixed, the first-period price impact blows up. The correct continuous-time object - derived by Back (1992) - has linear information release and a constant impact coefficient $\lambda =\sqrt{{\Sigma }_0/({\sigma }_u^{2}T)}$. Knowing which limit you are taking is the difference between a correct high-frequency model and nonsense.

The one-sentence essence. “A strategic insider trades proportionally to her edge, the makers price linearly, exactly half the private information gets revealed - and in continuous time the impact coefficient is constant and equal to $\sqrt{{\Sigma }_0/({\sigma }_u^{2}T)}$, which you only get from Back’s linear-release equilibrium, not from a naive limit of the discrete game.”

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2. Mathematical Ground Truth & Derivations

2.1 The single auction. $v\sim \mathcal{N}(p_0,{\Sigma }_0)$, $u\sim \mathcal{N}(0,{\sigma }_u^2)$, insider demand $x$, total flow $y=x+u$, linear price $P=p_0+\lambda y$. Makers are competitive, so they earn zero expected profit; by the projection theorem $P=\mathbb{E}[v\mid y]$, which for jointly normal variables is

$$P=p_0+\frac{\mathrm{Cov}(v,y)}{\mathrm{Var}(y)}y⟹\lambda =\frac{\beta {\Sigma }_0}{{\beta }^2{\Sigma }_0+{\sigma }_u^2}$$

if the insider uses $x=\beta (v-p_0)$. The insider maximises $\mathbb{E}[(v-P)x]=\beta {\Sigma }_0-\lambda {\beta }^2{\Sigma }_0$; the first-order condition ${\Sigma }_0=2\lambda \beta {\Sigma }_0$ gives $\lambda \beta =\frac{1}{2}$, and solving the two equations together:

$$\boxed{\beta =\sqrt{\frac{{\sigma }_u^2}{{\Sigma }_0}},\lambda =\frac{1}{2}\sqrt{\frac{{\Sigma }_0}{{\sigma }_u^2}},\frac{1}{\lambda }=2\sqrt{\frac{{\sigma }_u^2}{{\Sigma }_0}}}$$

Substituting back, the residual variance is

$$\mathrm{Var}[v\mid P]={\Sigma }_0-\frac{{\beta }^2{\Sigma }_0^2}{{\beta }^2{\Sigma }_0+{\sigma }_u^2}={\Sigma }_0-\frac{{\Sigma }_0}{{\beta }^2{\Sigma }_0/{\sigma }_u^2+1}={\Sigma }_0-\frac{{\Sigma }_0}{2}=\frac{{\Sigma }_0}{2}.$$

Half the information, whatever the noise. And $\mathbb{E}[\text{profit}]=\beta {\Sigma }_0-\lambda {\beta }^2{\Sigma }_0=\frac{1}{2}\sqrt{{\sigma }_u^2{\Sigma }_0}$.

2.2 The $N$-auction game. Repeat the auction $N$ times with equal noise variance ${\sigma }_u^2$ per round and prior variance ${\Sigma }_{n-1}$. Write $x_n={\beta }_n(v-p_{n-1})$, ${\Delta }_n=v-p_{n-1}\sim \mathcal{N}(0,{\Sigma }_{n-1})$. Since $y_n={\beta }_n{\Delta }_n+u_n$,

$${\lambda }_n=\frac{{\beta }_n{\Sigma }_{n-1}}{{\beta }_n^2{\Sigma }_{n-1}+{\sigma }_u^2},{\Sigma }_n={\Sigma }_{n-1}-\frac{{\beta }_n^2{\Sigma }_{n-1}^2}{{\beta }_n^2{\Sigma }_{n-1}+{\sigma }_u^2}=\frac{{\Sigma }_{n-1}{\sigma }_u^2}{{\beta }_n^2{\Sigma }_{n-1}+{\sigma }_u^2}.$$

Each round the insider again faces ${\lambda }_n{\beta }_n=\frac{1}{2}$, so ${\beta }_n{\Sigma }_{n-1}=2{\lambda }_n{\beta }_n^2{\Sigma }_{n-1}$ and the recursion collapses to

$$\boxed{{\beta }_n=\sqrt{\frac{{\sigma }_u^2}{{\Sigma }_{n-1}}},{\lambda }_n=\frac{1}{2}\sqrt{\frac{{\Sigma }_{n-1}}{{\sigma }_u^2}},{\Sigma }_n=\frac{{\Sigma }_{n-1}}{2}={\Sigma }_0{2}^{-n},\mathbb{E}[{\text{profit}}_n]=\frac{1}{2}\sqrt{{\sigma }_u^2{\Sigma }_{n-1}}}$$

so information is released geometrically - half per auction - and ${\lambda }_n$ decays geometrically while ${\beta }_n$ grows geometrically. The insider’s total expected profit converges to a finite limit:

$$\sum _{n\ge 1}\frac{1}{2}\sqrt{{\sigma }_u^2{\Sigma }_0}{2}^{-(n-1)/2}=\frac{1}{2}\sqrt{{\sigma }_u^2{\Sigma }_0}\cdot \frac{1}{1-2^{-1/2}}=\boxed{\frac{2+\sqrt{2}}{2}\sqrt{{\sigma }_u^2{\Sigma }_0}}.$$

More auctions $\Rightarrow$ more profit, but bounded: $1.707\sqrt{{\sigma }_u^2{\Sigma }_0}$ versus the single auction’s $0.5\sqrt{{\sigma }_u^2{\Sigma }_0}$.

2.3 Why the naive continuous-time limit is ill-posed. Fix a horizon $T$, put $N$ auctions of length $\tau =T/N$, and give each round the “correct” noise variance ${\sigma }_u^2\tau$ (proportional to the clock, so the total noise budget over $[0,T]$ is ${\sigma }_u^{2}T$). Then

$${\lambda }_1=\frac{1}{2}\sqrt{\frac{{\Sigma }_0}{{\sigma }_u^2\tau }}=\frac{1}{2}\sqrt{\frac{{\Sigma }_{0}N}{{\sigma }_u^{2}T}}\underset{N\to \infty }{\to }\infty .$$

The discrete game does not converge: shrinking the auction interval while the insider’s information advantage stays fixed makes the first period infinitesimally informative and hence infinitely costly to trade into. Something must give - and what gives, in the correctly posed model, is the shape of information release.

2.4 Back (1992): the correct continuous-time limit. Let $d{U}_t={\sigma }_{u}d{B}_t$ and let the price be $P_t=p_0+\lambda (X_t+U_t)$. The correct equilibrium satisfies $P_t=\mathbb{E}[V\mid {\mathcal{F}}_t]$ with

$$\boxed{\Sigma (t)={\Sigma }_0(1-\frac{t}{T}),\lambda =\sqrt{\frac{{\Sigma }_0}{{\sigma }_u^{2}T}},\frac{1}{\lambda }={\sigma }_u\sqrt{\frac{T}{{\Sigma }_0}}}$$

The derivation is a one-line filtering identity, and it is worth internalising. If $P_t=\mathbb{E}[V\mid {\mathcal{F}}_t]$ and the innovation representation is $d{P}_t={\lambda }_{t}d{Y}_t$, then

$$d{\Sigma }_t=-\mathrm{Var}(d{P}_t\mid {\mathcal{F}}_{t^{-}})=-{\lambda }_t^2\mathrm{Var}(d{Y}_t\mid {\mathcal{F}}_{t^{-}})=-{\lambda }_t^2{\sigma }_u^{2}dt,$$

because the predictable part of $d{Y}_t$ is known and only the noise contributes variance. With $\Sigma (t)={\Sigma }_0(1-t/T)$ we have $d{\Sigma }_t=-\frac{{\Sigma }_0}{T}dt$, so ${\lambda }^2{\sigma }_u^2={\Sigma }_0/T$ - $\lambda$ is constant in $t$, and equal to $\sqrt{{\Sigma }_0/({\sigma }_u^{2}T)}$.

Two consequences worth memorising:

• ${\int }_0^T{\lambda }^2{\sigma }_u^{2}dt={\Sigma }_0$: all the information is released by $T$. Half of it is released by $t=T/2$ - the continuous analogue of the auction’s ${\Sigma }_0/2$.
• $\lambda \propto T^{-1/2}$: a longer horizon means a deeper market (more time for the noise to accumulate, so the flow-to-price map is flatter). This is the exact opposite of the trading-desk intuition “longer = more risk”, and both statements are true - they are about different objects (information per unit flow vs inventory risk).

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3. Computational Implementation - the Kyle engine and the Back limit, verified

Stdlib only. Three verified blocks of work: (a) the discrete $N$-auction recursion with its profit limit, (b) the divergence of the naive continuum, (c) Back’s continuous-time information accounting.

Three things to take away:

1. ${\lambda }_n$ halves every round ($1.0000\to 0.7071\to 0.5000\to \cdots$) while ${\beta }_n$ doubles, because ${\Sigma }_n$ halves and the two are reciprocal square roots of the same quantity. This geometric pattern is the discrete model.
2. The total profit saturates: $N=12$ already gives $3.3609$ against the $N\to \infty$ limit $3.4142$. The insider’s marginal auction is worth exponentially less.
3. The naive continuum diverges linearly in $\sqrt{N}$ (${\lambda }_1=100$ at $N=10^4$), while Back’s continuous-time identity ${\lambda }^2{\sigma }_u^{2}T={\Sigma }_0$ holds exactly - and the integral of information release equals ${\Sigma }_0$ to the last digit. The two “limits” are not the same limit, and only one of them exists.

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4. Failure Modes & First-Principles Breakdowns

1. Confusing the two limits. “Kyle’s continuous-time model” is Back’s (1992) linear-release equilibrium, not the $N\to \infty$ limit of the discrete auction game. Anyone who calibrates a high-frequency impact parameter by “taking $\tau \to 0$” from the discrete recursion will get an impact coefficient that diverges.
2. $\lambda$ is not a physical constant. It is $\sqrt{{\Sigma }_0/({\sigma }_u^{2}T)}$: it depends on the prior uncertainty, the noise volume, and the horizon. Double the noise volume and depth ($1/\lambda$) rises by $\sqrt{2}$; double the horizon and depth rises by $\sqrt{2}$. Impact estimates that ignore ${\Sigma }_0$ are not comparable across names.
3. The half-information result is a knife-edge of normality. $\mathrm{Var}[v\mid P]={\Sigma }_0/2$ follows from joint normality. Under fat tails or a discrete value the “exactly half” property fails, and with it every argument that uses it as a calibration anchor.
4. One strategic insider is not a market. With $K$ informed traders the revelation rate scales with $K\beta$ and $\lambda$ falls toward zero (page 06). A single-insider $\lambda$ fit on a market with many informed participants systematically over-states impact.
5. Noise is not a fixed budget. The model treats ${\sigma }_u^2$ as exogenous and constant. In reality noise volume responds to realised volatility and to the presence of informed flow - the parameter you calibrated is a function of the thing you are modelling.
6. The insider’s optimality is never tested. The model says the insider should trade $\beta (v-p_0)$; it does not say anyone does. Empirically, informed flow is autocorrelated and clustered in time, which the one-shot Gaussian insider does not generate.

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5. References

• Kyle, Albert S. - “Continuous auctions and insider trading,” Econometrica 53(6), 1315–1335 (1985). Sections 1–3: the batch auction, $\lambda$, ${\Sigma }_0/2$, the $N$-auction recursion, and the insider’s profit.
• Back, Kerry - “Insider trading in continuous time,” Review of Financial Studies 5(3), 387–409 (1992). *The rigorous continuous-time limit; the founding application of filtering to microstructure.
• Holden, Craig W.; Subrahmanyam, Avanidhar - “Long-lived private information and imperfect competition,” Journal of Finance 47(1), 247–270 (1992). What happens with many informed traders; see page 06.
• Kühn, Christoph; Lorenz, Christopher - “Insider trading in discrete time Kyle games,” Mathematics and Financial Economics 19, 39–66 (2025). Exactly the question this page turns on: which discrete games converge to the continuous-time Kyle–Back equilibrium (and which do not).
• Hasbrouck, Joel - Empirical Market Microstructure (2007)
• Grossman, Sanford J.; Stiglitz, Joseph E. - “On the impossibility of informationally efficient markets,” American Economic Review 70(3), 393–408 (1980). *Why some noise must exist for information to be worth acquiring
• Kyle, Albert S.; Obizhaeva, Anna A. - “Market microstructure invariance,” Econometrica 84(3), 975–1024 (2016). What happens when you insist that $\lambda$ obey dimensional invariance across assets.

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6. Connected Graph Bridges

• Back: 01 - From Zero · Index Hub
• Forward: 03 - Glosten–Milgrom (the sequential-trade, two-point-value counterpart of the same equilibrium)
• Sibling at the same λ: Market Impact & Depth (measuring Kyle’s $\lambda$, the square-root law, transient impact)
• Scheduling on top of it: Optimal Execution & Almgren–Chriss (uses $\eta ,\gamma$, not $\lambda$ - the link is the calibration)
• Mathematical machinery: Probability & Measure Theory · Stochastic Calculus (the filtering identity in §2.4)