This post shows how to solve for the equilibrium price impact and demand coefficients in a 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 period model in closed form, and then give the general period form of the dynamic multi-period model. Any intuition that I can get with a dynamic model usually comes in the first periods, so I find myself frequently working out the period model explicitly. Here is that model.
2. Market description
I begin by outlining the market setting. Consider a world with trading periods and a single asset whose fundamental value is given by:
in units of dollars per share. There are 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, , each period:
where denotes the order flow from the informed traders and 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:
Prior to the start of the first trading period, informed traders see an unbiased signal about the asset’s fundamental value:
so that . In period , these traders choose the number of shares to demand from the market maker, , to solve:
where denotes their value function entering period . Similarly, in period these traders optimize:
The extra term shows up in informed traders’ time optimization problem but not their time 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:
and a linear market maker pricing rule in each period:
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 . Using DeGroot (1969)-style updating, it’s possible to compute their posterior beliefs:
After observing aggregate order flow in period , 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:
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 as follows:
It’s also possible to view the aggregate order flow in time as a signal about the informed traders’ signal rather than the fundamental value of the asset:
yielding posterior beliefs:
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 problem. Since the market maker faces perfect competition, the time price has to satisfy the condition:
As a result, and
However, this is about all we can say without knowing more about how the informed traders behave.
Moving to the informed traders’ time problem, we see that they optimize over the size of their time market order with knowledge of their private signal, , and the time price, , as follows:
Taking the first order condition yields an expression for their optimal time demand:
Informed traders place market orders in period that are linearly increasing in the size of their private signal; what’s more, if we hold the equilibrium value of constant, they will trade more aggressively when they have a more accurate private signal (i.e., ).
If we now return to the market makers’ problem, we can partially solve for the price impact coefficient in period :
However, to go any further and solve for or , we need to know how aggressively traders will act on their private information in period … we need to know .
5. First Period Solution
To solve the informed traders’ time problem, I first make an educated guess about the functional form of their value function:
We can now solve for the time equilibrium parameter values by plugging in the linear price impact and demand coefficients to the informed traders’ optimization problem:
Taking the first order condition with respect to the informed traders’ time demand gives:
Evaluating their expectation operator yields:
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:
Finally, using the same projection formula as above, we can solve for the market makers’ price impact rule:
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 problem, and substituting in the equilibrium coefficients we get:
Using the fact that and then leads to:
Adding and subtracting in the first term simplifies things even further:
Thus, informed traders’ continuation value is quadratic in the distance between their expectation of the fundamental value and the period price:
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 . To compute the equilibrium coefficient values, make an initial guess, , and use this guess to compute the values of the other equilibrium coefficients:
Then, just iterate on the initial guess numerically until you find that:
since we know that 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).