Two Period Kyle (1985) Model

1. Motivation

This post shows how to solve for the equilibrium price impact and demand coefficients in a 2 period Kyle (1985)-type model where informed traders see a noisy signal about the fundamental value of a single asset. There are various other places where you can see how to solve this sort of model. e.g., take a look at Markus Brunnermeier’s class notes or Laura Veldkamp’s excellent textbook. Both these sources solve the static 1 period model in closed form, and then give the general T \geq 1 period form of the dynamic multi-period model. Any intuition that I can get with a dynamic model usually comes in the first 2 periods, so I find myself frequently working out the 2 period model explicitly. Here is that model.

2. Market description

I begin by outlining the market setting. Consider a world with 2 trading periods t = 1, 2 and a single asset whose fundamental value is given by:

(1)   \begin{align*} v \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0, \sigma_{v}^2) \end{align*}

in units of dollars per share. There are 2 kinds of agents: informed traders and noise traders. Both kinds of traders submit market orders to a group of market makers who see only the aggregate order flow, \Delta x_t, each period:

(2)   \begin{align*} \Delta x_t &= \Delta y_t + \Delta z_t \end{align*}

where \Delta y_t denotes the order flow from the informed traders and \Delta z_t \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0, \sigma_{\Delta z}^2) denotes the order flow from the noise traders. The market makers face perfect competition, so they have to set the price each period equal to their expectation of the fundamental value of the asset given aggregate demand:

(3)   \begin{align*} p_1 &= \mathrm{E}[v|\Delta x_1] \qquad \text{and} \qquad p_2 = \mathrm{E}[v|\Delta x_1, \Delta x_2] \end{align*}

Prior to the start of the first trading period, informed traders see an unbiased signal s about the asset’s fundamental value:

(4)   \begin{align*} s = v + \epsilon \qquad \text{where} \qquad \epsilon \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0,\sigma_{\epsilon}^2) \end{align*}

so that s \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(v,\sigma_{\epsilon}^2). In period 1, these traders choose the number of shares to demand from the market maker, \Delta y_1, to solve:

(5)   \begin{align*} \mathrm{H}_0 = \max_{\Delta y_1} \, \mathrm{E}\left[ \, (v - p_1) \cdot \Delta y_1 + \mathrm{H}_1 \, \middle| \, s \, \right] \end{align*}

where \mathrm{H}_{t-1} denotes their value function entering period t. Similarly, in period 2 these traders optimize:

(6)   \begin{align*} \mathrm{H}_1 = \max_{\Delta y_2} \, \mathrm{E} \left[ \, (v - p_2) \cdot \Delta y_2 \, \middle| \, s, \, p_1  \ \right] \end{align*}

The extra H_1 term shows up in informed traders’ time t=1 optimization problem but not their time t=2 optimization problem because the model ends after the second trading period.

An equilibrium is a linear demand rule for the informed traders in each period:

(7)   \begin{align*}  \Delta y_t = \alpha_{t-1} + \beta_{t-1} \cdot s \end{align*}

and a linear market maker pricing rule in each period:

(8)   \begin{align*}  p_t = \kappa_{t-1} + \lambda_{t-1} \cdot \Delta x_t \end{align*}

such that given the demand rule in each period the pricing rule solves the market maker’s problem, and given the market maker pricing rule in each period the demand rule solves the trader’s problem.

3. Information and Updating

The informed traders need to update their beliefs about the fundamental value of the asset after observing their signal s. Using DeGroot (1969)-style updating, it’s possible to compute their posterior beliefs:

(9)   \begin{align*} \sigma_{v|s}^2 &= \left( \frac{\sigma_{\epsilon}^2}{\sigma_v^2 + \sigma_{\epsilon}^2} \right) \times \sigma_v^2 \qquad \text{and} \qquad \mu_{v|s} = \underbrace{\left( \frac{\sigma_v^2}{\sigma_v^2 + \sigma_{\epsilon}^2} \right)}_{\theta} \times s \end{align*}

After observing aggregate order flow in period t=1, market makers need to update their beliefs about the true value of the asset. Using the linearity of informed traders’ demand rule, we can rewrite the aggregate demand as a signal about the fundamental value as follows:

(10)   \begin{align*} \frac{\Delta x_1}{\beta_0} &= v + \left( \epsilon + \frac{\Delta z_1}{\beta_0} \right) \end{align*}

Note that both the signal error and noise trader demand cloud the market makers’ inference. Using the same DeGroot updating strategy, it’s possible to compute the market makers’ posterior beliefs about v as follows:

(11)   \begin{align*} \sigma_{v|\Delta x_1}^2 = \left( \frac{\beta_0^2 \cdot \sigma_{\epsilon}^2 + \sigma_{\Delta z}^2}{\beta_0^2 \cdot \sigma_s^2 + \sigma_{\Delta z}^2} \right) \times \sigma_v^2 \quad \text{and} \quad \mu_{v|\Delta x_1} = \left( \frac{\beta_0^2 \cdot \sigma_v^2}{\beta_0^2 \cdot \sigma_s^2 + \sigma_{\Delta z}^2} \right) \times \Delta x_1 \end{align*}

It’s also possible to view the aggregate order flow in time t=1 as a signal about the informed traders’ signal rather than the fundamental value of the asset:

(12)   \begin{align*} \frac{\Delta x_1}{\beta_0} &= s + \frac{\Delta z_1}{\beta_0} \end{align*}

yielding posterior beliefs:

(13)   \begin{align*} \sigma_{s|\Delta x_1}^2 = \left( \frac{\sigma_{\Delta z}^2}{\sigma_{\Delta z}^2 + \beta_0^2 \cdot \sigma_s^2} \right) \times \sigma_s^2 \quad \text{and} \quad \mu_{s|\Delta x_1} = \left( \frac{\beta_0^2 \cdot \sigma_s^2}{\sigma_{\Delta z}^2 + \beta_0^2 \cdot \sigma_s^2} \right) \times \Delta x_1 \end{align*}

4. Second Period Solution

With the market description and information sets in place, I can now solve the model by working backwards. Let’s start with the market makers’ time t=2 problem. Since the market maker faces perfect competition, the time t=1 price has to satisfy the condition:

(14)   \begin{align*} \mathrm{E}[v|\Delta x_1] &= p_1 \end{align*}

As a result, \kappa_0 = 0 and

(15)   \begin{align*} \kappa_1  &= \mathrm{E}[v|\Delta x_1] - \lambda_1 \cdot \mathrm{E}[\Delta x_2|\Delta x_1] = p_1 - \underbrace{(\theta \cdot \mu_{s | \Delta x_1} - p_1)}_{=0} = p_1 \end{align*}

However, this is about all we can say without knowing more about how the informed traders behave.

Moving to the informed traders’ time t=2 problem, we see that they optimize over the size of their time t=2 market order with knowledge of their private signal, s, and the time t=1 price, p_1, as follows:

(16)   \begin{align*} \mathrm{H}_1 &= \max_{\Delta y_2} \ \mathrm{E} \left[ \, \left(v - \kappa_1 - \lambda_1 \cdot \Delta x_2 \right) \cdot \Delta y_2 \, \middle| \, s, p_1  \, \right] \end{align*}

Taking the first order condition yields an expression for their optimal time t=2 demand:

(17)   \begin{align*} \Delta y_2 &= \underbrace{- \, \frac{p_1}{2 \cdot \lambda_1}}_{\alpha_1} + \underbrace{\frac{\theta}{2 \cdot \lambda_1}}_{\beta_1} \cdot s \end{align*}

Informed traders place market orders in period t=2 that are linearly increasing in the size of their private signal; what’s more, if we hold the equilibrium value of \lambda_1 constant, they will trade more aggressively when they have a more accurate private signal (i.e., \sigma_{\epsilon}^2 \searrow 0).

If we now return to the market makers’ problem, we can partially solve for the price impact coefficient in period t=2:

(18)   \begin{align*} \lambda_1  &= \frac{\mathrm{Cov}[ \Delta x_2, v | \Delta x_1]}{\mathrm{Var}[ \Delta x_2| \Delta x_1]} = \frac{\mathrm{Cov}\left[ \, \alpha_1 + \beta_1 \cdot s + \Delta z_2, v \, \middle| \, \Delta x_1 \, \right]}{\mathrm{Var}\left[ \, \alpha_1 + \beta_1 \cdot s + \Delta z_2 \, \middle| \, \Delta x_1 \, \right]} = \frac{\beta_1 \cdot \sigma_{v|\Delta x_1}^2}{\beta_1^2 \cdot \sigma_{s|\Delta x_1}^2 + \sigma_{\Delta z}^2} \end{align*}

However, to go any further and solve for \sigma_{v|\Delta x_1}^2 or \sigma_{s|\Delta x_1}^2, we need to know how aggressively traders will act on their private information in period t=1… we need to know \beta_0.

5. First Period Solution

To solve the informed traders’ time t=1 problem, I first make an educated guess about the functional form of their value function:

(19)   \begin{align*} \mathrm{E}[\mathrm{H}_1|s] &= \psi_1 + \omega_1 \cdot \left( \mu_{v|s} - p_1 \right)^2 \end{align*}

We can now solve for the time t=1 equilibrium parameter values by plugging in the linear price impact and demand coefficients to the informed traders’ optimization problem:

(20)   \begin{align*} \mathrm{H}_0 &= \max_{\Delta y_1} \, \mathrm{E}\left[ \, (v - p_1) \cdot \Delta y_1 + \psi_1 + \omega_1 \cdot \left( \theta \cdot s - p_1 \right)^2 \, \middle| \, s \, \right] \end{align*}

Taking the first order condition with respect to the informed traders’ time t=1 demand gives:

(21)   \begin{align*} 0 &= \mathrm{E}\left[ \, \left(v - 2 \cdot \lambda_0 \cdot \Delta y_1 - \lambda_0 \cdot \Delta z_1 \right)   - 2 \cdot \omega_1 \cdot \lambda_0 \cdot \left( \theta \cdot s - \lambda_0 \cdot \{ \Delta y_1 + \Delta z_1  \} \right) \, \middle| \, s \, \right] \end{align*}

Evaluating their expectation operator yields:

(22)   \begin{align*} 0 &= \theta \cdot s - 2 \cdot \lambda_0 \cdot \Delta y_1 - 2 \cdot \omega_1 \cdot \lambda_0 \cdot \left\{   \theta \cdot s - \lambda_0 \cdot \Delta y_1 \right\}  \end{align*}

Rearranging terms then gives the informed traders’ demand rule which is a linear function of the signal they got about the asset’s fundamental value:

(23)   \begin{align*} \Delta y_1 &= \frac{\theta}{2 \cdot \lambda_0} \cdot \left( \frac{1 - 2 \cdot \omega_1 \cdot \lambda_0}{1 - \omega_1 \cdot \lambda_0} \right) \cdot s \end{align*}

Finally, using the same projection formula as above, we can solve for the market makers’ price impact rule:

(24)   \begin{align*} \lambda_0 &= \frac{\mathrm{Cov}[ \Delta x_1, v]}{\mathrm{Var}[ \Delta x_1]} = \frac{\mathrm{Cov}[\alpha_0 + \beta_0 \cdot (v + \epsilon) + \Delta z_1, v]}{\mathrm{Var}[ \alpha_0 + \beta_0 \cdot s + \Delta z_1]} = \frac{\beta_0 \cdot \sigma_v^2}{\beta_0^2 \cdot \sigma_s^2 + \sigma_{\Delta z}^2} \end{align*}

6. Guess Verification

To wrap things up, let’s now check that my guess about the value function is consistent. Looking at the informed traders’ time t=2 problem, and substituting in the equilibrium coefficients we get:

(25)   \begin{align*} \mathrm{H}_1 &= \mathrm{E} \left[ \, \left(v - p_2 \right) \cdot \Delta y_2 \, \middle| \, s  \, \right] \\ &= \mathrm{E} \left[ \, \left(v - \left\{p_1 + \lambda_1 \cdot \left( \alpha_1 + \beta_1 \cdot s + \Delta z_2 \right) \right\}  \right) \times \left( \alpha_1 + \beta_1 \cdot s \right) \, \middle| \, s  \, \right] \end{align*}

Using the fact that \alpha_1 = -\sfrac{p_1}{(2 \cdot \lambda_1)} and \beta_1 = \sfrac{\theta}{(2 \cdot \lambda_1)} then leads to:

(26)   \begin{align*} \mathrm{H}_1 &= \mathrm{E} \left[ \, \frac{1}{2 \cdot \lambda_1} \times \left( \left\{ v - p_1 \right\} - \frac{1}{2} \cdot \left\{ \theta \cdot s - p_1 \right\} - \lambda_1 \cdot \Delta z_2 \right) \times \left( \theta \cdot s - p_1 \right) \, \middle| \, s  \, \right] \end{align*}

Adding and subtracting \mu_{s | \Delta x_1} = \theta \cdot s in the first term simplifies things even further:

(27)   \begin{align*} \mathrm{H}_1 &= \mathrm{E} \left[ \, \frac{1}{2 \cdot \lambda_1} \times \left( \left\{ v - \theta \cdot s \right\} + \frac{1}{2} \cdot \left\{ \theta \cdot s - p_1 \right\} - \lambda_1 \cdot \Delta z_2 \right) \times \left( \theta \cdot s - p_1 \right) \, \middle| \, s  \, \right] \end{align*}

Thus, informed traders’ continuation value is quadratic in the distance between their expectation of the fundamental value and the period t=1 price:

(28)   \begin{align*} \mathrm{H}_1 &= \text{Const.} + \underbrace{\frac{1}{4 \cdot \lambda_1}}_{\omega_1} \cdot \left( \mu_{v|s} - p_1 \right)^2 \end{align*}

which is consistent with the original linear quadratic guess. Boom.

7. Numerical Analysis

Given the analysis above, we could derive the correct values of all the other equilibrium coefficients if we knew the optimal \beta_0. To compute the equilibrium coefficient values, make an initial guess, \widehat{\beta}_0, and use this guess to compute the values of the other equilibrium coefficients:

(29)   \begin{align*} \widehat{\lambda}_0 &\leftarrow \frac{\widehat{\beta}_0 \cdot \sigma_v^2}{\widehat{\beta}_0^2 \cdot \sigma_s^2 + \sigma_{\Delta z}^2} \\ \widehat{\sigma}_{v|\Delta x_1}^2 &\leftarrow \left( \frac{\widehat{\beta}_0^2 \cdot \sigma_{\epsilon}^2 + \sigma_{\Delta z}^2}{\widehat{\beta}_0^2 \cdot \sigma_s^2 + \sigma_{\Delta z}^2} \right) \cdot \sigma_v^2 \\ \widehat{\sigma}_{s|\Delta x_1}^2 &\leftarrow \left( \frac{\sigma_{\Delta z}^2}{\widehat{\beta}_0^2 \cdot \sigma_s^2 + \sigma_{\Delta z}^2} \right) \cdot \sigma_s^2 \\ \widehat{\lambda}_1 &\leftarrow \frac{1}{\sigma_{\Delta z}} \cdot \sqrt{ \frac{\theta}{2} \cdot \left( \widehat{\sigma}_{v|\Delta x_1}^2 - \frac{\theta}{2} \cdot \widehat{\sigma}_{s|\Delta x_1}^2 \right) } \end{align*}

Then, just iterate on the initial guess numerically until you find that:

(30)   \begin{align*} \widehat{\beta}_0 &= \frac{\theta}{2 \cdot \widehat{\lambda}_0} \cdot \left( \frac{1 - 2 \cdot \widehat{\omega}_1 \cdot \widehat{\lambda}_0}{1 - \widehat{\omega}_1 \cdot \widehat{\lambda}_0} \right) \end{align*}

since we know that \beta_0 must satisfy this condition in equilibrium.

The figure below plots the coefficient values at various levels of noise trader demand and signal error for inspection. Here is the code. The informed traders are more aggressive with there is more noise trader demand (i.e., moving across panels from left to right) and in the second trading period (i.e., blue vs red). The trade less aggressively as their signal quality degrades (i.e., moving within panel from left to right).

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