Comparing Kyle and Grossman-Stiglitz

1. Motivation

New information-based asset-pricing models are often extensions of either Kyle (1985) or Grossman-Stiglitz (1980). At first glance, these two canonical models look quite similar. Both price an asset with an unknown payout, like a stock or bond, and both analyze the strategic behavior of informed traders in the presence of demand noise. Yet, in spite of these similarities, the internal logic in each model is quite different.

On one hand, Kyle (1985) studies the behavior a single, large, risk-neutral, informed trader who recognizes that his own trading impacts the price. That is, he knows that, if he tries to buy more shares of the asset, then the market maker will think to herself: “Why is there more demand for this asset? Well, the informed trader probably found out some good news, so I better raise the stock price.” As a result, the informed trader buys until his expected profit from holding another share is exactly offset by the price impact of this purchase. The informed trader’s strategic behavior in Kyle (1985) is all about managing price impact.

On the other hand, Grossman-Stiglitz (1980) studies many, small, risk-averse, informed traders who take the equilibrium price as given. Rather than submitting a single request for a certain number of shares, informed traders in Grossman-Stiglitz (1980) submit a menu of price-quantity pairs that they’d be happy to buy—for example, an informed trader might be willing to buy $500$ shares at $\mathdollar 3.00$ per share, $400$ shares at $\mathdollar 3.50$ per share, $300$ shares at $\mathdollar 3.75$ per share, and so on. Informed traders choose this menu of price-quantity pairs such that, at each price level, their expected utility gain from holding another share of the stock is exactly offset by the disutility they realize from the extra variance that holding this share adds to their portfolio. Informed traders’ strategic behavior in Grossman-Stiglitz (1980) is all about managing risk.

Even though both these models end up conveying the same key idea—namely, that informed traders are more aggressive when their is more demand noise—each model arrives at this conclusion for very different reasons. So, when making modeling decisions, it’s useful to have a sense of each model’s predictions and assumptions. In this post, I work through a static version of each model with this goal in mind.

2. Kyle (1985)

Let’s start the analysis by looking at a static version of Kyle (1985). In this model, there is a single asset with an unknown payout, $\hat{v} \sim \mathrm{N}(0,\sigma_v^2)$ , where the payout is composed of two independent parts,

(1) $\begin{align*} \hat{v} &= \hat{f} + \hat{e}, \end{align*}$

with $\hat{f} \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0,\sigma_f^2)$ , $\hat{e} \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0,\sigma_e^2)$ , and $\sigma_f^2 = \lambda \cdot \sigma_v^2$ for some $\lambda \in (0,1)$ . Here, $\hat{f}$ is the knowable component of the asset’s payout, and $\hat{e}$ is the idiosyncratic component of the asset’s payout. So, for example, if $\lambda = 0.50$ , then the half of the variation in the asset’s payout can be learned and half is due to the accidents of life.

This is how trading works in Kyle (1985). An informed trader (think: arbitrageur) and a noise trader both submit market orders to a market maker—that is, the informed trader might tell the market maker, “I want to buy $500$ shares.”, and the noise trader might tell the market maker, “I want to sell $200$ shares.” While the informed trader has private information about the value of the asset and uses this information when trading, the noise trader just submits random orders, $\hat{z} \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(\mu_z,\sigma_z^2)$ . The challenge facing the market maker is that she only gets to see aggregate order flow,

(2) $\begin{align*} a = x + \hat{z}. \end{align*}$

So, while she has beliefs about how likely it is that an order comes from the informed trader, she doesn’t know whether any particular order comes from the informed trader or the noise trader. To the market maker, the order flow described above just looks like, “People want to buy $300$ shares.”

Given this setup, the informed trader chooses how many shares, $x$ , to demand from the market maker,

(3) $\begin{align*} \max_{x} \, \mathrm{E}\left( \, \left( \hat{v} - p \right) \cdot x \, \middle| \, \hat{f} \, \right), \end{align*}$

knowing that, if he demands more shares, then the market maker will see this additional demand and use this information to adjust the price. The market maker tries to set the price as accurately as possible,

(4) $\begin{align*} \min_{p} \, \mathrm{E}\left( \, \left(\hat{v} - p\right)^2 \, \middle| \, a \, \right), \end{align*}$

after seeing the aggregate demand for the asset. To keep things tractable, let’s assume that both the market maker’s pricing rule, $p = \alpha + \beta \cdot a$ , and the informed trader’s demand rule, $x = \gamma + \delta \cdot \hat{f}$ , are linear. We’ll see shortly that these guesses are correct.

From here, solving the model is straightforward. First, let’s plug the functional form of the market maker’s pricing rule into the informed trader’s optimization problem:

(5) $\begin{align*} 0 = \mathrm{E}\left( \, (\hat{v} - \alpha) - 2 \cdot \beta \cdot x + \beta \cdot (\hat{z} - \mu_z) \, \middle| \, \hat{f} \, \right) &= \underbrace{\hat{f} - (\alpha + \beta \cdot x)}_{\substack{\text{Expected profit from} \\ \text{buying the $x$th share.}}} - \underbrace{\beta \cdot x}_{\substack{\text{Cost due} \\ \text{to price} \\ \text{impact of} \\ \text{$x$th share.}}}. \end{align*}$

This equation says that the informed trader will keep on trading until the expected profit that he’ll earn from buying the last share is exactly offset by the price impact of this purchase. Rearranging the terms gives a linear demand rule, $x = - \, \sfrac{\alpha}{(2 \cdot \beta)} + \sfrac{1}{(2 \cdot \beta)} \cdot \hat{f}$ , just as we guessed.

Then, let’s study to the market maker’s problem to solve for the equilibrium coefficient values. The market maker sets the price equal to her conditional expectation of the asset’s value given the aggregate demand, $\mathrm{E}( \hat{v} | a ) = \alpha + \beta \cdot a$ where $\alpha = \mathrm{E}(\hat{v}) - \beta \cdot (\mathrm{E}(x) - \mu_z)$ . But, this means that equilibrium $\beta$ is just a regression coefficient with the following functional form:

(6) $\begin{align*} \beta &= \frac{\mathrm{Cov}(\hat{v},a)}{\mathrm{Var}(a)} = \frac{1}{2} \cdot \frac{\sigma_f}{\sigma_z}. \end{align*}$

Noticing that $\delta = \sfrac{1}{(2 \cdot \beta)}$ gives the remaining coefficients:

(7) $\begin{align*} \begin{matrix} \alpha = \sqrt{\lambda} \cdot \left\{ \sfrac{\mu_z}{\sigma_z} \right\} \cdot \sigma_v, & \ & \beta = \sfrac{\sqrt{\lambda}}{2} \cdot (\sfrac{1}{\sigma_z}) \cdot \sigma_v, & \ & \gamma = - \mu_z, & \ & \text{and} & \ & \delta = \sfrac{1}{\sqrt{\lambda}} \cdot (\sfrac{1}{\sigma_v}) \cdot \sigma_z. \end{matrix} \end{align*}$

Although Kyle (1985) is no longer an accurate description of how real-world traders interact with market makers, people still use it because it’s an incredibly simple way of capturing a key fact: informed traders trade more aggressively (i.e., $\delta \nearrow$ ) when there is more noise trading (i.e., when $\sigma_z \nearrow$ ). For more detailed analysis, see my earlier posts on the 2-period Kyle (1985) model and its geometric interpretation

3. Grossman-Stiglitz (1980)

Next, let’s analyze a simplified version of Grossman-Stiglitz (1980) where there is a single asset with the same payout structure as above, $\hat{v} = \hat{f} + \hat{e}$ . In this model, there are now many informed traders take the equilibrium price as given and solve the optimization problem below,

(8) $\begin{align*} \max_x \, \mathrm{E}\left( \, - e^{- \phi \cdot w} \, \middle| \, \hat{f}, \, p \, \right) \quad \text{subject to} \quad w = \bar{w} + (\hat{v} - p) \cdot x, \end{align*}$

where $\phi$ is the informed traders’ risk-aversion parameter in units of $\sfrac{1}{\mathdollar}$ . Note that this means the traders are now trying to maximize utility rather than profits. Solving this problem, we see that

(9) $\begin{align*} 0 &= \underbrace{\hat{f} - p}_{\substack{\text{Expected utils} \\ \text{gained from} \\ \text{buying $x$th} \\ \text{share at} \\ \text{price $p$.}}} - \underbrace{\phi \cdot \sigma_e^2 \cdot x}_{\substack{\text{Utils lost due} \\ \text{to increased risk} \\ \text{from buying $x$th} \\ \text{share at price $p$.}}}. \end{align*}$

That is, at each price level, the informed traders trade until their expected utility gain from holding another share of the risky asset is exactly offset by the disutility they realize from the extra variance that holding this share adds to their portfolio. Rearranging this expression then gives their optimal portfolio position at each price level, $x = (\phi \cdot \sigma_e)^{-1} \times \{\sfrac{(\hat{f} - p)}{\sigma_e}\}$ . So, informed traders buy shares whenever the equilibrium price is below their signal about the asset’s expected value, and they buy relatively more shares when they are less risk averse and when their information is more precise.

There is no explicit market maker learning from aggregate order flow in Grossman-Stiglitz (1980) like there is in Kyle (1980); instead, there is a collection of uninformed traders who observe the equilibrium price of the asset, $p$ , and use this information to guide their trading. Each of these uninformed traders solves the same optimization problem as the informed traders,

(10) $\begin{align*} \max_y \, \mathrm{E}\left( \, - e^{- \varphi \cdot w} \, \middle| \, p \, \right) \quad \text{subject to} \quad w = \bar{w} + (\hat{v} - p) \cdot y, \end{align*}$

where $\varphi$ is the informed traders’ risk-aversion parameter in units of $\sfrac{1}{\mathdollar}$ . Solving this problem, we see that

(11) $\begin{align*} 0 &= \underbrace{\mathrm{E}(\hat{f}|p) - p}_{\substack{\text{Expected utils} \\ \text{gained from} \\ \text{buying $y$th} \\ \text{share at price $p$.}}} - \underbrace{\varphi \cdot (\mathrm{Var}(\hat{f}|p) + \sigma_e^2) \cdot y}_{\substack{\text{Utils lost due} \\ \text{to increased risk} \\ \text{from buying $y$th} \\ \text{share at price $p$.}}}. \end{align*}$

That is, at each price level, the uninformed traders trade until their expected utility gain from holding another share of the risky asset is exactly offset by the disutility they realize from the extra variance that holding this share adds to their portfolio. This is exactly the same demand rule that the informed traders use. The only difference is that they don’t have a private signal, $\hat{f}$ , about the asset’s fundamental value. Just like before, rearranging this expression then gives the uninformed traders’ optimal portfolio position at each price level, $y = \varphi^{-1} \cdot \mathrm{StD}(\hat{v}|p)^{-1} \times \{ \sfrac{(\mathrm{E}(\hat{f}|p) - p)}{\mathrm{StD}(\hat{v}|p)}\}$ .

Yet, there is something slightly puzzling about the setup of the Grossman-Stiglitz (1980) model so far. What does it mean for traders to condition on the price? How can they already know this? The short answer is: they don’t. Rather than submitting market orders like in Kyle (1980), traders in Grossman-Stiglitz (1980) submit a menu of price-quantity pairs that they’d be happy trading and then the equilibrium price is pinned down by a menu auction. If there was some uncertainty as to how many shares of the asset were available due to noise trader demand and all traders (both informed and uninformed) submitted a menu of price-quantity pairs,

(12) $\begin{align*} \hat{z} &= \int x_i \cdot di + \int y_u \cdot du \end{align*}$

then the market-clearing price and demand that would be chosen by an auctioneer is the same as the Grossman-Stiglitz (1980) price and demand.

This auction-like equilibrium concept means that informed traders are solving a fundamentally different problem in Grossman-Stiglitz (1980). In Grossman-Stiglitz (1980), the informed traders don’t know how many shares of the risky asset they will end up buying and have to optimally choose an entire menu of price-quantity pairs. In Kyle (1980), by contrast, the informed trader knows exactly how many shares of the risky asset he will buy, he just doesn’t know what the resulting price will be. He solving a much simpler problem in Kyle (1980) since he only has to optimally pick a single number rather than an entire menu.

Once we’ve setup the model, solving it is again relatively straightforward. Just like in Kyle (1980), let’s guess that the equilibrium pricing rule is linear, $p = \eta + \theta \cdot \hat{f} - \beta \cdot \hat{z}$ , where $\eta$ has units of $\sfrac{\mathdollar}{\text{share}}$ , $\theta$ is dimensionless, and $\beta$ has units of $\sfrac{\mathdollar}{\text{shares}^2}$ . Thus, if the uninformed traders only condition on the price of the risky asset when making their portfolio choice, they will have the following signal about $\hat{f}$ :

(13) $\begin{align*} \frac{p - (\eta - \beta \cdot \mu_z)}{\theta} &= \hat{f} - \frac{\beta}{\theta} \cdot (\hat{z} - \mu_z). \end{align*}$

This means that they will have beliefs,

(14) $\begin{align*} \mathrm{Var}(\hat{f}|p) &= \left\{ \frac{\beta^2 \cdot \sigma_z^2}{\beta^2 \cdot \sigma_z^2 + \theta^2 \cdot \sigma_f^2} \right\} \times \sigma_f^2 \\ \mathrm{E}(\hat{f}|p) &= \left\{ \frac{\theta^2 \cdot \sigma_f^2}{\beta^2 \cdot \sigma_z^2 + \theta^2 \cdot \sigma_f^2} \right\} \times \frac{p - (\eta - \beta \cdot \mu_z)}{\theta}. \end{align*}$

To simplify the notation below, let’s define a new dimensionless constant, $\kappa = \sfrac{(\theta^2 \cdot \sigma_f^2)}{(\beta^2 \cdot \sigma_z^2 + \theta^2 \cdot \sigma_f^2)}$ , which represents how much weight the uninformed traders put on the price signal. When $\kappa \approx 1$ , uninformed traders learn a lot from the price; whereas, when $\kappa \approx 0$ , uninformed traders learn very little from the price.

To solve for the coefficients $\eta$ , $\theta$ , and $\beta$ we then need to enforce the market-clearing condition,

(15) $\begin{align*} \hat{z} &= \int_0^{\sfrac{1}{2}} x_i \cdot di + \int_{\sfrac{1}{2}}^1 y_u \cdot du, \end{align*}$

where I’m assuming for simplicity that half of the traders are informed and half are uninformed. Allowing traders to optimally choose their information like in the original Grossman-Stiglitz (1980) model doesn’t change the basic comparison to the Kyle (1980) setup and complicates the analysis. If we substitute the functional forms for traders’ demand into the market-clearing condition, then we get:

(16) $\begin{align*} 2 \cdot \varphi \cdot \hat{z} &= \frac{\varphi}{\phi} \cdot \left( \frac{\hat{f} - p}{\sigma_e^2} \right) + \left( \frac{\mathrm{E}(\hat{f}|p) - p}{\mathrm{Var}(\hat{f}|p) + \sigma_e^2} \right). \end{align*}$

Rearranging this expression then gives us an expression for the equilibrium price that is linear in the price signal, $\hat{f}$ , and then noise trader demand, $\hat{z}$ ,

(17) $\begin{align*} p &= - \, \{ \xi^{-1} \cdot (1 - \lambda) \cdot (\sfrac{\kappa}{\theta}) \} \cdot (\eta - \beta \cdot \mu_z) \\ &\qquad + \, \{ \xi^{-1} \cdot (\sfrac{\varphi}{\phi}) \cdot (1 - \kappa \cdot \lambda) \} \cdot \hat{f} \\ &\qquad \qquad - \, 2 \cdot \{\xi^{-1} \cdot (1 - \lambda) \cdot (1 - \kappa \cdot \lambda)\} \cdot \varphi \cdot \sigma_v^2 \cdot \hat{z}, \end{align*}$

with $\xi = (\sfrac{\varphi}{\phi}) \cdot (1 - \kappa \cdot \lambda) - \sfrac{(\kappa - \theta)}{\theta} \cdot (1 - \lambda)$ . Matching the coefficients and solving then yields the equilibrium values

(18) $\begin{align*} \begin{matrix} \eta = \left\{ \frac{\kappa \cdot (1 - \lambda)}{\xi \cdot \theta + \kappa \cdot (1 - \lambda)} \right\} \cdot \beta \cdot \mu_z, & & \theta = \xi^{-1} \cdot (\sfrac{\varphi}{\phi}) \cdot (1 - \kappa \cdot \lambda), & \ \text{and} \ & \beta = 2 \cdot \{\theta \cdot (1 - \lambda)\} \cdot \phi \cdot \sigma_v^2. \end{matrix} \end{align*}$

4. Comparative Statics

Now that we’ve seen how each model works, let’s look at some comparative statics to better understand how each model’s predictions differ. Unless stated otherwise, I use the following parameter values in the plots below: $\sigma_v = \sfrac{\mathdollar 1}{\text{sh}}$ , $\mu_z = 1{\scriptstyle \mathrm{sh}}$ , $\sigma_z = 1{\scriptstyle \mathrm{sh}}$ , $\lambda = \sfrac{1}{2}$ , $\phi = \sfrac{1}{\mathdollar}$ , and $\varphi = \sfrac{1}{\mathdollar}$ . For each outcome variable, I look at how the predictions of each model as I vary the quality of the informed traders’ information, $\lambda$ , from $0 \to 1$ , the level of the informed traders’ risk aversion, $\phi$ , from $0 \to 2$ , and the volatility of noise traders’ demand, $\sigma_z$ , from $0 \to 2$ . If you’d like to look at other parameterizations, you can find all of the code here.

Let’s begin by looking at informed traders’ demand response in each model. That is, if the informed traders’ private signal about the value of risky asset increases by $\mathdollar 1$ per share, then how many additional shares will the informed trader typically demand? In the Kyle (1980), this is just the equilibrium $\beta$ :

(19) $\begin{align*} \mathrm{E}\left( \frac{\partial x}{\partial \hat{f}} \right) &= \beta = \frac{1}{\sqrt{\lambda}} \cdot \frac{1}{\sigma_v} \cdot \sigma_z. \end{align*}$

By contrast, this expression takes on a more complicated functional form in Grossman-Stiglitz (1980):

(20) $\begin{align*} \mathrm{E}\left( \frac{\partial x}{\partial \hat{f}} \right) &= \frac{1}{\phi} \cdot \frac{1}{\sigma_v^2} \cdot \left\{ \frac{1 - \theta}{1 - \lambda} \right\}. \end{align*}$

The left-most panel of the figure below shows that, as the quality of the informed traders’ private signal increases, they they trade less aggressively in both models. That is, informed traders demand fewer additional shares of the risky asset per $\mathdollar 1$ increase in the private signal, $\hat{f}$ .

Similarly, the right-most panel of the figure shows that, as the noise-trader-demand volatility increases, informed traders trade more aggressively in both models. It’s important to note, however, that this effect is driven by very different forces in each model. In Kyle (1980), the informed trader trades more aggressively when there is more demand noise because the additional noise lowers his price impact. By comtrast, in Grossman-Stiglitz (1980) informed traders trade more aggressively when there is more demand noise because this additional noise makes the equilibrium price signal less informative for the uninformed traders.

In addition to informed traders’ demand responses, there are lots of other statistics of interest. For instance, how does the average price vary with the amount of noise trading? In Kyle (1980), the answer is simple: it doesn’t. In fact, the expected price of the asset is always equal to its expected value:

(21) $\begin{align*} \mathrm{E}(p) &= \alpha - \beta \cdot (\gamma - \mu_z) = 0. \end{align*}$

In Grossman-Stiglitz (1980), however, the average price of the asset,

(22) $\begin{align*} \mathrm{E}(p) &= \eta - \beta \cdot \mu_z, \end{align*}$

can vary quite a bit with the level of noise-trader-demand volatility as shown in the right-most panel of the figure below. This risk premium comes from the fact that, if there is more noise-trader-demand volatility, then the risk-averse traders have to bear more risk, so they will be willing to pay less for the asset.

Finally, let’s take a look at the equilibrium price response, $\mathrm{E}(\sfrac{\partial p}{\partial\! \hat{f}})$ , in each model. That is, on average, how much higher is the price when the informed traders’ private signal is $\mathdollar 1$ per share higher? For the Kyle (1980) model, the answer is always:

(23) $\begin{align*} \sfrac{1}{2} = \beta \cdot \delta. \end{align*}$

In the Grossman-Stiglitz (1980) model, the answer corresponds to the equilibrium $\theta$ parameter. We see in the right-most panel of the figure below that, if there is more noise-trader-demand volatility, then the equilibrium price in the Grossman-Stiglitz (1980) model becomes less informative. Note that there is absolutely no effect in the Kyle (1980) model, where the informed trader strategically dials up the aggressiveness of his demand rule whenever noise-trader-demand volatility increases to ensure that $\sfrac{1}{2} = \beta \cdot \delta$ . So, if you want to make predictions linking the information content of prices to noise-trader-demand volatility, you’d better use Grossman-Stiglitz (1980).

5. Fine Tuning

Let me conclude by making one last observation about trying to reverse engineer the models to line up more precisely. The middle panel of each of the 3 figures above shows how the predictions of the Kyle (1980) and Grossman-Stiglitz (1980) models change as I increase informed traders’ risk aversion parameter. In every one of these panels, the predictions of the Kyle (1980) model are always flat. They have to be. The informed trader in Kyle (1980) is risk-neutral, so varying this parameter can’t have any affect of the model’s predictions. Is there any way to tune this parameter to get the models to line up more precisely?

Yes and no. No in the sense that, if you look at the middle panels in the average price and price response plots, you can see that increasing the informed traders’ risk aversion has different effects on these two outcomes, so you can’t get the two models to agree on both of these predictions by monkeying around with one tuning parameter. You’d need to fine tune the risk-aversion parameters of both the informed and the uninformed traders to do this. So it is possible if you really want to.

Even when the models give qualitatively similar predictions, like for informed traders’ demand responses, fine tuning informed traders’ risk-aversion parameter to get the models to agree quantitatively means making really strong assumptions about other parameters of the model. Specifically, notice that the demand response in Grossman-Stiglitz (1980) is identical to the demand response in Kyle (1980) if the informed traders’ risk-aversion parameter is precisely:

(24) $\begin{align*} \phi &= \left\{ \frac{\sqrt{\lambda}}{1 - \lambda} \right\} \cdot \frac{1 - \theta}{\sigma_v \cdot \sigma_z}. \end{align*}$

But, this means that deep parameters like the asset’s payout volatility, $\sigma_v$ , and the volatility of noise-trader demand, $\sigma_z$ , can only occur in certain pairs. For instance, the plot below shows the permissible values when $\lambda = \sfrac{1}{2}$ and $\varphi = \sfrac{1}{\mathdollar}$ . We see that, in order to make Kyle (1980) and Grossman-Stiglitz (1980) give quantitatively similar predictions about informed traders’ demand responses, payout volatility and noise-trader-demand volatility have to be inversely related and often orders of magnitude apart.