The Continuous Limit of Kyle (1985) Ends With A Bang

Suppose you uncover good news about a stock’s fundamental value. When you start trading on this information, market makers will notice a spike in aggregate demand, infer that someone must have discovered good news, and adjust the stock’s price upward accordingly. How much will this price impact cut into your profits? Well, you could measure the damage by estimating the change in the stock’s price per $100$ shares you buy. If the number is small, the asset’s liquid; if it’s big, it ain’t. Price impact = 1/liquidity.

Of course, if the stock you’re interested in is very illiquid, then maybe you should just make yourself less conspicuous by spreading out your order flow. You can always sacrifice immediacy to gain liquidity. It’s a simple enough idea. But, figuring out precisely how you should spread out your order flow is a hard problem because it introduces a feedback loop. When you spread out your order flow, market makers are less likely to notice a spike in aggregate demand, so your equilibrium price impact will be smaller. But, if your price impact is smaller, then you have less of an incentive to spread out your order flow in the first place.

Where does this feedback loop end? How does strategic trading affect market liquidity in equilibrium? Kyle (1985) provides a nice model that answers these questions. In the model, there’s a single informed agent who places market orders for a risky asset over the course of $N \geq 1$ auctions. And, a risk-neutral market maker sets the market-clearing price in each auction after observing a noisy version of the informed agent’s demand. Solving the model reveals precisely how the informed agent should optimally adjust his order flow from auction to auction, thereby maximizing his expected profits given the equilibrium price impact.

One of the things I really like about the original paper is its analysis of the continuous-time limit. It turns out that, as the number of auctions during a fixed unit of time tends to infinity, $N \to \infty$ , the informed agent strategically adjusts his demand from auction to auction in a way that keeps the equilibrium price impact constant. This is surprising. In general, you’d expect your price impact at 2:00pm to depend on what’s happened during the previous hour because market makers learn from aggregate order flow. But, if you can trade infinitely often, then the model indicates that you will strategically trade in a way that offsets this learning. Obviously, we don’t think that high-frequency trading actually makes liquidity constant. This is just a really clever way of showing how important equilibrium feedback loops are.

While I really like this analysis, I also think it’s created a misconception about how the informed agent behaves in the model. A constant price impact does not imply that the market is stable… quite the opposite actually. A Kyle (1985) with a large number of auctions describes a market that ends with a lightning fast flurry of intense trading activity. When there are many auctions, $N \gg 1$ , the informed agent trades smoothly and steadily until the final few auctions. Then, all hell breaks loose. This post illustrates why.

Structure

In the model, there’s a single informed agent who places market orders for a risky asset in $N \geq 1$ auctions that take place during a fixed time interval. Let $\Delta t = \sfrac{1}{N}$ denote the time between auctions in minutes. For concreteness, I’m going to think about this time interval as 1:00-2:00pm. So, if $N = 2$ , then $\Delta t = 30$ minutes and there are auctions at 1:30 and 2:00pm; whereas, if $N = 4$ , then $\Delta t = 15$ minutes and auctions occur at 1:15, 1:30, 1:45, and 2:00pm. This risky asset has fundamental value $v \sim \mathrm{Normal}(0, \, 1)$ , and the informed agent knows $v$ prior to the start of the first auction. Based on this information, he submits market orders of size $\Delta x_n$ in each auction. So, $x_n = \sum_{\check{n}=1}^n \Delta x_{\check{n}}$ represents his cumulative demand during the first $n$ auctions.

The equilibrium price in each auction, $p_n$ , is determined by a market maker after observing the aggregate order flow, which is the sum of both the informed agent’s demand and a random demand shock:

$\begin{equation*} \Delta a_n = \Delta x_n + \Delta u_n \qquad \text{where} \qquad \Delta u_n \overset{\scriptscriptstyle \text{iid}}{\sim} \mathrm{Normal}(0,\! \sqrt{\Delta t}) \end{equation*}$

What’s important here is that the market maker only observes $\Delta a_n$ . He can’t separately observe $\Delta x_n$ or $\Delta u_n$ on its own. As a result, when there’s a lot of demand, he might suspect that the informed agent has uncovered good news, but he can’t be certain. The order flow could always just be due to random noise. Given the market maker’s pricing rule, let $\pi_n = \sum_{\hat{n}=n}^N (v - p_{\hat{n}}) \cdot \Delta x_{\hat{n}}$ denote the informed agent’s future profit on all positions acquired at auctions $\{n, \, \ldots, \, N\}$ .

An equilibrium consists of two sequences of functions, $\mathsf{X} = \langle \mathrm{X}_1(\cdot), \, \ldots, \, \mathrm{X}_N(\cdot) \rangle$ and $\mathsf{P} = \langle \mathrm{P}_1(\cdot), \, \ldots, \, \mathrm{P}_N(\cdot) \rangle$ . The first sequence defines the informed agent’s trading strategy, $x_n = \mathrm{X}_n(p_1, \, \ldots, \, p_{n-1}, \, v)$ . The second sequence defines the market maker’s pricing rule, $p_n = \mathrm{P}_n(\Delta a_1, \, \ldots, \, \Delta a_n)$ . And, we say that $(\mathsf{X}^\star\!, \, \mathsf{P}^\star)$ is an equilibrium if the following two conditions hold. First, the informed agent’s trading strategy has to be profit maximizing. i.e., if you were to pick any other alternative trading strategy, $\mathsf{X}^{\text{alt}}$ , that’s identical to $\mathsf{X}^\star$ in the first $(n-1)$ auctions, then it must be the case that:

$\begin{equation*} \mathrm{E}[ \, \pi_n(\mathsf{X}^\star\!,\,\mathsf{P}^\star) \mid p_1^\star, \, \ldots, \, p_{n-1}^\star, \, v \, ] \geq \mathrm{E}[ \, \pi_n(\mathsf{X}^\text{alt},\,\mathsf{P}^\star) \mid p_1^\star, \, \ldots, \, p_{n-1}^\star, \, v \, ] \end{equation*}$

Second, the market maker’s pricing rule has to be efficient:

$\begin{equation*} p_n^\star = \mathrm{E}[ \, v \mid \Delta a_1^\star, \, \ldots, \, \Delta a_n^\star \, ] \qquad \text{where} \qquad \Delta a_n^\star = \Delta x_n^\star+ \Delta u_n^{\phantom{\star}} \end{equation*}$

i.e., price must be equal to the conditional expectation of fundamental value given aggregate order flow.

Solution

Theorem 2 of Kyle (1985) gives the solution to this model. If you restrict yourself to only considering pricing rules of the form

$\begin{equation*} p_n = p_{n-1} + \lambda_n \cdot \Delta a_n \qquad \text{and} \qquad p_0 = \mathrm{E}[v] = 0 \end{equation*}$

then there’s a unique equilibrium $(\mathsf{X}^\star\!,\,\mathsf{P}^\star)$ given by

$\begin{align*} \Delta x_n^\star &= \beta_n^{\phantom{\star}}\! \cdot (v - p_{n-1}^\star) \cdot \Delta t \\ \Delta p_n^\star &= \lambda_n^{\phantom{\star}}\! \cdot (\Delta x_n^\star + \Delta u_n^{\phantom{\star}}) \\ \sigma_n^2 &= \mathrm{Var}[ \, v \mid \Delta x_1^\star + \Delta u_1^{\phantom{\star}}, \, \ldots, \, \Delta x_n^\star + \Delta u_n^{\phantom{\star}} \,\! ] \\ \mathrm{E}[ \, \pi_{n+1}^{\phantom{\star}} \mid p_1^\star, \, \ldots, \, p_n^\star, \, v \, ] &= \alpha_n^{\phantom{\star}}\! \cdot (v - p_n^\star)^2 + \delta_n^{\phantom{\star}} \end{align*}$

where $\beta_n$ , $\sigma_n^2$ , $\lambda_n$ , $\alpha_n$ , and $\delta_n$ are the solution to the following system of difference equations

$\begin{align*} \beta_n \cdot \Delta t &= {\textstyle \frac{1}{2 \cdot \lambda_n} \times \left(\frac{1 - 2 \cdot \alpha_n \cdot \lambda_n}{1 - \alpha_n \cdot \lambda_n} \right)} \\ \sigma_n^2 &= \sigma_{n-1}^2 \cdot (1 - \lambda_n \cdot [\beta_n \cdot \Delta t]) \\ \lambda_n &= \sigma_n^2 \cdot \beta_n \\ \alpha_{n-1} &= {\textstyle \frac{1}{4 \cdot \lambda_n} \times \left( \frac{1}{1 - \alpha_n \cdot \lambda_n}\right)} \\ \delta_{n-1} &= \delta_n + \alpha_n \cdot \lambda_n^2 \cdot \Delta t \end{align*}$

under the conditions $p_0 = \alpha_N = \delta_N = 0$ , $\sigma_0^2 = 1$ , and $\lambda_n \cdot (1 - \alpha_n \cdot \lambda_n) > 0$ .

On the right, I’ve plotted these constants for different $N$ s. You can see how the equilibrium values change as the number of auctions increases by moving the slider at the top. To verify consistency, note that when there are $N=4$ auctions, the values of $\lambda_n$ and $\sigma_n^2$ to the right are the same as those depicted by the four $\square \, \square \, \square \, \square$ s in Figure 1 of Kyle (1985).

N→∞ Limit

What’s more, by pushing the slider all the way to the right so that $N=60$ and auctions occur every $\Delta t = 1$ minute, you get something close to the continuous-limit result I described in the introduction. As $N \to \infty$ , the price-impact coefficient $\lambda_n \to 1$ due to the way that the informed agent spreads out his order flow across auctions. But, does this mean that informed demand is smooth?

No.

You can see as much in the animation to the right. The top panel shows realizations of the informed agent’s trading volume, $|\Delta x_n|$ . As you push the slider towards $N = 60$ in this figure, you can watch the informed agent’s demand fluctuate more and more as he approaches the last 2:00pm auction. He’s spreading out his demand, but his demand’s clearly not smooth. When $N$ is large, the market ends with a short intense burst of trading activity. In fact, you might even think about this short intense burst of activity as a tell-tale sign of short-horizon trading.

Here’s another way of looking at what’s happening. The noise shocks are the same regardless of which auction they arrive in, $\Delta u_n \overset{\scriptscriptstyle \text{iid}}{\sim} \mathrm{Normal}(0,\! \sqrt{\Delta t})$ . Noise doesn’t care whether it’s auction $n=1$ or auction $n=60$ . So, the typical noise shock will always be on the order of $\sqrt{\Delta t}$ . As a result, the typical change in the equilibrium price in auction $(n-1)$ due to demand noise will be $\lambda_{n-1} \cdot \! \sqrt{\Delta t}$ . And, this random price change will, in turn, change the informed agent’s demand in auction $n$ since $\Delta x_n = \beta_n \cdot (v - p_{n-1}) \cdot \Delta t$ :

$\begin{equation*} \mu_n = \beta_n \cdot (\lambda_{n-1} \cdot \! \sqrt{\Delta t}) \cdot \Delta t \end{equation*}$

By plugging in the equilibrium conditions from the previous section, we can rewrite the typical change in the informed agent’s demand due to random fluctuations in the previous auction as follows:

$\begin{align*} \mu_n &= {\textstyle \frac{\lambda_{n-1}}{2 \cdot \lambda_n} \cdot \left(\frac{1 - 2 \cdot \alpha_n \cdot \lambda_n}{1 - \alpha_n \cdot \lambda_n} \right)} \times \! \sqrt{\Delta t} \\ &= 2 \cdot \alpha_{n-1} \cdot \lambda_{n-1} \cdot (1 - 2 \cdot \alpha_n \cdot \lambda_n) \times \! \sqrt{\Delta t} \end{align*}$

The bottom panel in the figure to the right plots the quantity. When $N = 60$ , the informed agent responds more and more aggressively to noise shocks in the previous auction as $n \to 60$ . This behavior corresponds to the hyperbolic growth in $\beta_n$ as $n \to 60$ plotted above. Constant price impact does not imply smooth trading.