How much of the insider's private information is incorporated into prices in Kyle's single auction equilibrium model?

Question

While desribing properties of the single auction equilibrium defined by theorem 1 in "Continuous Auctions and Insider Trading" A. Kyle conjectured that

one-half of the insider's private information is incorporated into prices and the volatility of prices is unaffected by the level of noise trading $\sigma^2_u$.

Mathematically, the above conjecture is expressed in the text as the following formula: $$\Sigma_1 = \mathrm{var}\{{\tilde{v}\mid \tilde{p}}\} = \frac{1}{2}\Sigma_0 = \frac{1}{2}\mathrm{var}\{{\tilde{v}}\} \tag{1} \label{one}$$ There $\tilde{v}$ denotes the ex post liquidation value ( a normal variable with mean $p_0$ and variance $\Sigma_0$) and $\tilde{p}$ denotes the price which is set by the market maker.

The model assumes that the insider and noise traders submit their market orders to the market maker who then sets the price at which they trade the quantity necessary to clear the market.

It is assumed that the market maker set the price deterministically, as a function of joint volume of orders submitted by the inisder $\tilde{x}$ and by the noise traders $\tilde{u}$: $$\label{two} \tilde{p} = p_0 + \lambda(\tilde{x} + \tilde{u}) \tag{2}$$

where $\lambda=2\Big(\frac{\Sigma_0}{\sigma_u^2}\Big)^\frac{1}{2}$

The insider define the size of its order $\tilde{x}$ deterministically as a function of the ex post liquidation value which he "observes" as an insider:

$$\label{three} \tilde{x} = \beta(\tilde{v} - p_0) \tag{3}$$

where $\beta=(\frac{\sigma_u^2}{\Sigma_0})^\frac{1}{2}$.

The volume of noise traders $\tilde{u}$ is a normally distributed random variable with mean zero and variance $\sigma^2_u$. Note that $\eqref{three}$ makes $\tilde{x}$ distributed as $\tilde{u}$ i.e. $\tilde{x}$ has zero mean and $\sigma_u$ variance too!

Substituting $\eqref{three}$ into $\eqref{two}$ and rearranging we have:

$$ \tilde{v} = p_0 + \frac{\tilde{p} - p_0}{\lambda \beta} - \frac{\tilde{u}}{\beta} \tag{4}$$

Thus

$$ \Sigma_1 = \mathrm{var}\{{\tilde{v}\mid \tilde{p}=p}\} = \mathrm{var}\{\frac{\tilde{u}}{\beta}\} = \frac{\sigma_0^2}{\frac{\sigma_0^2}{\Sigma_0}} = \Sigma_0 \tag{5}$$

so none of the insider's private information is incorporated into prices.

Where am I wrong?

Answer 1

Finally I concluded that the confusion is due to (one of many small) typos in the article.

$\Sigma_1$ shoud refer to $\mathbf{var} \{\tilde{v} \mid \tilde{x} + \tilde{u}\}$, not to $\mathbf{var} \{\tilde{v} \mid \tilde{p}\}$. It is the volume that gives information about the ex post liquidation value $\tilde{v}$ to the market maker, not the price.

This conclusion is consistent with the calculation and interpretation of $\Sigma_n$ in the Theorem 2 later in the article.

Of course I agree with the author that:

a simple calculations shows that $\Sigma_1 = \frac{1}{2}\Sigma_0$

It might be interesting to note that while $\tilde{x} + \tilde{u}$ is a normally distributed with mean zero and variance $\sigma_u$, i.e. looks similar to the volume generated by nose traders $\tilde{u}$, it has non-zero correlation with $\tilde{v}$, which does not depend on $\sigma_u$ : $$\mathbf{cor}\{\tilde{v}, \tilde{x} + \tilde{u}\} = \sqrt{\frac{\beta^2\Sigma_0}{\beta^2\Sigma_0+\sigma_u^2}} = \frac{1}{\sqrt{2}}$$

The Kyle's model is amazing!

Answer 2

I read this as the short term volatility of noise trading is subsumed by the longer term volatility of measured price changes, which might be a reasonable approximation if $\sigma^2_u \ll \sigma^2_v$. In this case, the noise trading also has no effect on the mean, and therefore 100% of the price reflects only insider information. However, it doesn’t follow necessarily that 100% of insider information is reflected into price.

Intuitively, even if price only reflects insider information, not all insider is necessarily reflected into price... insiders might hold some things back. There could a number of possible reasons for this including fear of litigation or other limits to arbitrage.

So I think we agree that the model provides no specificity about the conjecture that “one-half of the insider's private information is incorporated into prices”. However, I think we disagree on the implications regarding how much insider information is factored into price, since it only follows that some or all insider information is incorporated into prices.