Notes on Kyle (1989)

1. Motivating Example

In several earlier posts (e.g., here and here) I’ve talked about the two well-known information-based asset-pricing models, Grossman and Stiglitz (1980) and Kyle (1985). But, there are lots of situations that don’t really fit with either of these two models. For one thing, uninformed speculators often recognize that they’re going to have a price impact, so it’s at odds with Grossman and Stiglitz (1980). For another thing, uninformed speculators typically use limit orders, so it’s at odds with Kyle (1985).

This post outlines the Kyle (1989) model which studies speculators that place limit orders and recognize their own price impact.

2. Market Structure

Assets. There is a single trading period and a single risky asset with a price of $p$ . This risky asset’s liquidation value is $v \sim \mathrm{N}(0, \, \tau_v^{-1})$ . For example, you might think about the asset as a stock that’ll have a value of $v$ after some important news announcement tomorrow. It’s just that, right now, you don’t know which direction the news will go.

Traders. There are $3$ kinds of traders: noise traders, informed speculators, and uninformed speculators. Noise traders demand $-z \sim \mathrm{N}(0, \, \tau_z^{-1})$ shares of the risky asset. There are $N$ informed speculators and $M$ uninformed speculators. Both informed and uninformed traders have an initial endowment of $\mathdollar 0$ (this is just a normalization) and exponential utility with risk-aversion parameter $\rho > 0$ ,

(1) $\begin{align*} - \, \exp\left\{ \, - \, \rho \cdot (v - p) \cdot x \, \right\}, \end{align*}$

where $x$ denotes the number of shares demanded by a speculator.

Information. Prior to trading, each informed speculator gets a private signal $s_n$ and has a demand schedule $\mathrm{X}_{I,n}(p,\,s_n)$ . That is, he has in mind a function which tells him how many shares to demand at each possible price, $p$ , given his private signal, $s_n$ . Assume that the informed speculators’ signals can be written as

(2) $\begin{align*} s_n = v + \epsilon_n \end{align*}$

where $\epsilon_n \sim \mathrm{N}(0, \, \tau_{\epsilon}^{-1})$ . Each uninformed speculator has a demand schedule $\mathrm{X}_{U,m}(p)$ .

3. Equilibrium Concept

Definition. An equilibrium is a set of demand schedules, $X_{I,n}(p,\,s_n)$ for the $n=1,\,2,\,\ldots,\,N$ informed speculators and $X_{U,m}(p)$ for the $m=1,\,2,\,\ldots,\,M$ uninformed speculators, and a price function $P(v,\,z)$ such that (a) markets clear,

(3) $\begin{align*} z &= {\textstyle \sum_{n=1}^N} X_{I,n}(p,\,s_n) + {\textstyle \sum_{m=1}^M} \cdot X_{U,m}(p), \end{align*}$

and (b) both informed and uninformed speculators optimize,

(4) $\begin{align*} X_{I,n}(p,\,s_n) &\in \arg \max_x \left\{ \, \mathrm{E}[ \, - \, \exp\left\{ \, \rho \cdot (v - p) \cdot x \right\} \, | \, p, \, s_n \, ] \, \right\} \text{ for all } n = 1, \, 2, \, \ldots, \, N \\ \text{and} \qquad X_{U,m}(p) &\in \arg \max_x \left\{ \, \mathrm{E}[ \, - \, \exp\left\{ \, \rho \cdot (v - p) \cdot x \right\} \, | \, p \, ] \, \right\} \text{ for all } m = 1, \, 2, \, \ldots, \, M. \end{align*}$

Both informed and uninformed speculators understand the relationship between prices and the random variables $v$ and $z$ . Prices will not be fully revealing due to the presence of noise-trader demand, $z$ .

Refinements. I make a pair of additional restrictions on the set of equilibria. Namely, I look only for linear, symmetric equilibria where the informed speculators’ demand schedules can be written as

(5) $\begin{align*} \mathrm{X}_I(p,\, s_n) &= \alpha_I - \beta_I \cdot p + \gamma_I \cdot s_n, \end{align*}$

the uninformed-speculators’ demand schedules can be written as

(6) $\begin{align*} \mathrm{X}_U(p) &= \alpha_U - \beta_U \cdot p, \end{align*}$

and the price can be written as

(7) $\begin{align*} \mathrm{P}(v,\,z) &= \theta_0 + \theta_v \cdot v - \theta_z \cdot z. \end{align*}$

4. Information Updating

Price Impact. If we substitute the linear demand schedules for the informed and uninformed speculators into the market-clearing condition, then we get a formula for the price,

(8) $\begin{align*} p &= \lambda \times \left( \, \left\{ \, N \cdot \alpha_I + M \cdot \alpha_U \, \right\} + \gamma_I \cdot {\textstyle \sum_{n=1}^N} s_n - z \, \right) \quad \text{where} \quad \lambda = {\textstyle \frac{1}{N \cdot \beta_I + M \cdot \beta_U}}. \end{align*}$

Thus, if noise traders supply one additional share, then the price drops by $\mathdollar \lambda$ . Next, I define the same object for informed and uninformed speculators. That is, taking the demand schedules of the other speculators as given, how much will the price change if the $n$ th informed speculator or $m$ th uninformed speculator increases his demand by $1$ share? This question defines the residual supply curves,

(9) $\begin{align*} p &= \hat{p}_{I,n} + \lambda_I \times X_I(p,\,s_n) \\ \text{and} \qquad p &= \hat{p}_{U} + \lambda_U \times X_U(p). \end{align*}$

Imperfect competition is present because each trader recognizes that if he submits a different schedule, the resulting equilibrium price may change.

Forecast Precision. Informed speculator’s forecast precision is given via Bayesian updating as:

(10) $\begin{align*} \tau_I = \left( \, \mathrm{Var}[v|p,\, s_n] \, \right)^{-1} = \tau_v + \tau_{\epsilon} + \varphi_I \times (N-1) \cdot \tau_{\epsilon} \quad \text{where} \quad \varphi_I = {\textstyle \frac{(N - 1) \cdot \gamma_I^2 \cdot \tau_z}{(N - 1) \cdot \gamma_I^2 \cdot \tau_z + \tau_{\epsilon}}}. \end{align*}$

$\varphi_I$ represents the fraction of the precision from the other $(N-1)$ informed speculators revealed to the $n$ th informed speculator by the price. The corresponding forecast precision for uninformed speculator is:

(11) $\begin{align*} \tau_U = \left( \, \mathrm{Var}[v|p] \, \right)^{-1} = \tau_v + \varphi_U \times N \cdot \tau_{\epsilon} \quad \text{where} \quad \varphi_U = {\textstyle \frac{N \cdot \gamma_I^2 \cdot \tau_z}{N \cdot \gamma_I^2 \cdot \tau_z + \tau_{\epsilon}}}. \end{align*}$

$\varphi_U$ represents the fraction of the precision of the $N$ informed speculators revealed to the $m$ th uninformed speculator by the price. Clearly, prices become perfectly revealing as $\varphi_I, \, \varphi_U \to 1$ .

Posterior Beliefs. The $n$ th informed speculator’s posterior beliefs about the risky asset’s liquidation value is a weighted average of the public price signal and his private signal,

(12) $\begin{align*} \mathrm{E}[v|p, \, s_n] &= \left( {\textstyle \frac{\varphi_I }{\tau_I} \cdot \frac{\tau_{\epsilon}}{\gamma_I}} \right) \times \left( \, \lambda^{-1} \cdot p - \{N \cdot \alpha_I + M \cdot \alpha_U\} \, \right) + \left( {\textstyle \frac{(1 - \varphi_I) \cdot \gamma_I}{\tau_I} \cdot \frac{\tau_{\epsilon}}{\gamma_I}} \right) \times s_n \end{align*}$

Uninformed speculators form their posterior beliefs solely on the public price signal,

(13) $\begin{align*} \mathrm{E}[v|p] &= \left( {\textstyle \frac{\varphi_U}{\tau_U} \cdot \frac{\tau_{\epsilon}}{\gamma_I}} \right) \times \left( \, \lambda^{-1} \cdot p - \{N \cdot \alpha_I + M \cdot \alpha_U\} \, \right). \end{align*}$

5. Optimal Demand

Informed Demand. The $n$ th informed speculator observes his private signal, $s_n$ , and the price implied by the demand of the other traders, $\hat{p}_{I,n}$ . He then solves the optimization problem below,

(14) $\begin{align*} \max_x \left\{ \, (\mathrm{E}[v|\hat{p}_{I,n}, \, s_n] - \hat{p}_{I,n}) \cdot x - (\lambda_I + {\textstyle \frac{\rho}{2}} \cdot \mathrm{Var}[v|\hat{p}_{I,n}, \, s_n]) \cdot x^2 \, \right\} \quad \Rightarrow \quad x = {\textstyle \frac{\mathrm{E}[v|\hat{p}_{I,n}, \, s_n] - \hat{p}_{I,n}}{2 \cdot \lambda_I + \rho \cdot \mathrm{Var}[v|\hat{p}_{I,n}, \, s_n]}}. \end{align*}$

But, the residual demand curve, $\hat{p}_{I,n}$ , is related to the actual price, $\hat{p}_{I,n} = p - \lambda_I \cdot X_I(p,\,s_n)$ . So, after a little bit of rearranging, we can write the informed speculators’ optimal demand schedules as

(15) $\begin{align*} X_I(p,\,s_n) &= {\textstyle \frac{\mathrm{E}[v|p, \, s_n] - p}{\lambda_I + \sfrac{\rho}{\tau_I}}}. \end{align*}$

Uninformed Demand. The $m$ th uninformed speculator observes only the price implied by the demand of the other traders, $\hat{p}_U$ . He then solves the optimization problem below,

(16) $\begin{align*} \max_x \left\{ \, (\mathrm{E}[v|\hat{p}_U] - \hat{p}_U) \cdot x - (\lambda_U + {\textstyle \frac{\rho}{2}} \cdot \mathrm{Var}[v|\hat{p}_U]) \cdot x^2 \, \right\} \quad \Rightarrow \quad x = {\textstyle \frac{\mathrm{E}[v|\hat{p}_U] - \hat{p}_U}{2 \cdot \lambda_U + \rho \cdot \mathrm{Var}[v|\hat{p}_U]}}. \end{align*}$

Using the exact same tricks, we can write the uninformed speculators’ optimal demand schedule as

(17) $\begin{align*} X_U(p) &= {\textstyle \frac{\mathrm{E}[v|p] - p}{\lambda_U + \sfrac{\rho}{\tau_U}}}. \end{align*}$

6. Endogenous Parameters

Next, it’s useful to define a couple of additional parameters.

Information Incidence. First, define the information incidence as

(18) $\begin{align*} \zeta &= \left( {\textstyle \frac{\tau_\epsilon}{\tau_I}} \right)^{-1} \times (\lambda \cdot \gamma_I). \end{align*}$

This new parameter represents the increase in the equilibrium price when the $n$ th informed speculator’s valuation of the risky asset goes up by $\mathdollar 1$ as a result of a higher signal realization, $s_n$ . For trader $n$ ‘s valuation to rise by $\mathdollar 1$ , his private signal must rise by a factor of $(\frac{\tau_\epsilon}{\tau_I})^{-1}$ , and prices move by a factor of $(\lambda \cdot \gamma_I)$ for every $\mathdollar 1$ increase in the $n$ th informed speculator’s private signal, $s_n$ . In equilibrium, it turns out that $\zeta \leq \sfrac{1}{2}$ .

Marginal Market Share. Next, define two parameters capturing the marginal market share of the informed and uninformed speculators,

(19) $\begin{align*} \xi_I = \beta_I \cdot \lambda \quad \text{and} \quad \xi_U = \beta_U \cdot \lambda. \end{align*}$

Here’s how you interpret $\xi_I$ : if noise traders demand $1$ additional share, then the quantity traded by each informed speculator increases by $\xi_I$ shares. Likewise, $\xi_U$ captures the amount of additional trading that each uninformed speculator does in response to a $1$ share increase in noise-trader demand.

7. Model Solution

Kyle (1989) shows that there exists a unique symmetric linear equilibrium if $N \geq 2$ , $M \geq 1$ , $\tau_z^{-1} > 0$ , and $\tau_{\epsilon} > 0$ . This equilibrium is characterized by a system of $4$ equations and $4$ unknowns, $\{ \, \gamma_I, \, \zeta, \, \xi_I, \, \xi_U \, \}$ , subject to the constraints that $\gamma_I > 0$ , $0 < \zeta \leq \sfrac{1}{2}$ , $\sfrac{\varphi_U}{N} < \beta_I \cdot \lambda < \sfrac{1}{N}$ , and $0 < \beta_U \cdot \lambda < \sfrac{(1 - \varphi_U)}{M}$ .

Equation 1. The first equation is the easiest. If noise traders demand an additional share, then someone has to sell it to them. Informed speculators tend to adjust their demand by $\xi_I$ shares, and uninformed speculators tend to adjust their demand by $\xi_U$ shares. Thus, because there are $N$ informed speculators and $M$ uninformed speculators, we have the market-clearing condition below:

(20) $\begin{align*} 1 &= N \cdot \xi_I + M \cdot \xi_U. \end{align*}$

Equation 2. Next, we turn to the second equation, which characterizes the informed speculator’s demand response to price changes, $\beta_I$ , via the endogenous parameter $\xi_I$ ,

(21) $\begin{align*} (1 - \zeta) &= (1 - \varphi_I) \times (1 - \xi_I). \end{align*}$

This equation links how unresponsive prices are to a $\mathdollar 1$ increase in an informed speculator’s private signal, $(1 - \zeta)$ , to the product of how uninformative prices are about other informed speculators’ signals, $(1 - \varphi_I)$ , and how little each informed speculator has to trade in response to a $1$ share increase in noise-trader demand, $(1 - \xi_I)$ . After all, if informed speculators don’t have to trade that often—i.e., $(1 - \xi_I) \approx 1$ —and prices don’t really reveal much of their private signal to other informed speculators when they do—i.e., $(1 - \varphi_I) \approx 1$ , then prices shouldn’t be moving that much in response to private shocks—i.e., $(1 - \zeta) \approx 1$ .

Equation 3. The third equation is much more directly an equilibrium characterization of $\gamma_I$ ,

(22) $\begin{align*} \gamma_I &= \tau_{\epsilon} \times \left( {\textstyle \frac{1}{\rho}} \right) \times (1 - \varphi_I) \times \left( {\textstyle \frac{1 - 2 \cdot \zeta}{1 - \zeta}} \right). \end{align*}$

Informed speculators are going to trade more aggressively in response to a $\mathdollar 1$ increase in their private signal when their private signal is more precise (i.e., $\tau_{\epsilon}$ is big), when they are closer to risk neutral (i.e., $\rho$ is small), when prices don’t reveal much about their private signal to other informed speculators (i.e., $(1 - \varphi_I) \approx 1$ because $\varphi_I \approx 0$ ), or when prices don’t move much when informed speculators trade on their private information (i.e., $\frac{1 - 2 \cdot \zeta}{1 - \zeta} \approx 1$ because $\zeta \approx 0$ ). Notice that this last effect is second order when $\zeta$ is small.

Equation 4. Finally, let’s have a look at equation, which characterizes the uninformed speculator’s demand response to price changes, $\beta_U$ , via the endogenous parameter $\xi_U$ ,

(23) $\begin{align*} \zeta \cdot \tau_U - \varphi_U \cdot \tau_I &= \xi_U \times \underset{> 0}{\left( {\textstyle \frac{\zeta \cdot \tau_U}{1 - \xi_U}} + {\textstyle \frac{\rho \cdot \gamma_I \cdot \tau_I}{\tau_{\epsilon}}} \right)}. \end{align*}$

I don’t have any clean way to analyze the right-hand side of this equation, but it is possible to show that the right-hand side will only be $0$ if $\xi_U = 0$ —that is, if there are lots of small uninformed speculators. What’s more, we know from Equation (13) that prices will only be an unbiased estimate of the uninformed speculators’ beliefs if:

(24) $\begin{align*} 1 &= \left( {\textstyle \frac{\varphi_U \cdot \tau_{\epsilon}}{\gamma_I \cdot \tau_U}} \right) \times \lambda^{-1}. \end{align*}$

If we rearrange the left-hand side of the equation a bit,

(25) $\begin{align*} 0 &= \zeta \cdot \tau_U - \varphi_U \cdot \tau_I \\ \varphi_U \cdot \tau_I &= \zeta \cdot \tau_U \\ \varphi_U \cdot \tau_I &= \left( {\textstyle \frac{\tau_\epsilon}{\tau_I}} \right)^{-1} \cdot (\lambda \cdot \gamma_I) \cdot \tau_U \\ \left( {\textstyle \frac{\varphi_U \cdot \tau_{\epsilon}}{\gamma_I \cdot \tau_U}} \right) \times \lambda^{-1} &= 1, \end{align*}$

we see that prices can only be unbiased if there are lots of small uninformed speculators, just like in Grossman and Stiglitz (1980). Otherwise, prices overreact—that is, $\theta = \left( {\textstyle \frac{\varphi_U \cdot \tau_{\epsilon}}{\gamma_I \cdot \tau_U}} \right) \times \lambda^{-1} < 1$ .

8. Numerical Analysis

To make sure that I’ve understood how to solve the model correctly, I solved for the equilibrium parameters numerically when $\rho=1$ , $N=2$ , $M=1$ , $\tau_{\epsilon}=2$ , and $\tau_v=1$ as the precision of noise-trader demand volatility ranges from $\tau_z = 0$ to $\tau_z = 2$ . You can find the code here. I’ve plotted some of the results below.