A Tell-Tale Sign of Short-Run Trading

Motivation

Trading has gotten a lot faster over the last two decades. The term “short-run trader” used to refer to people who traded multiple times a day. Now, it refers to algorithms that trade multiple times a second.

Some people are worried about this new breed of short-run trader making stock prices less informative about firm fundamentals by trading so often on information that’s unrelated to companies’ long-term prospects. But, this is a red herring. By the same logic, market-neutral strategies should be making market indexes like the Russell 3000 less informative about macro fundamentals. And, no one believes this.

Short-run traders aren’t ignoring fundamentals; they’re learning about fundamentals before everyone else by studying order flow. And, this post shows why they also make trading volume look asymmetric and lumpy as a result. The logic is simple. If short-run traders get additional information from order flow, then they’ll use this information to cluster their trading at times when everyone else is moving in the opposite direction.

Benchmark Model

Consider a market with a single company that’s going to pay a dividend, $d_t$ , in future periods $t = 1, \, 2, \, \ldots$ And, suppose that there’s a unit mass of small agents, $i \in (0, \, 1]$ , who have noisy priors about these dividends,

$\begin{equation*} d_t \overset{\scriptscriptstyle \text{iid}}{\sim} \mathrm{N}( \mu_t^{(i)}, \, \sfrac{\sigma^2\!}{2} ) \quad \text{for some} \quad \sigma^2 > 0, \end{equation*}$

which are correct on average, $d_t = \int_0^1 \mu_t^{(i)} \cdot \mathrm{d}i$ . This assumption means that, by aggregating agents’ demand, equilibrium prices can contain information about dividends that isn’t known by any individual agent.

This unit mass of agents is split into two different groups: long-term investors and short-run traders. At time $t=0$ , each group of agents trades shares in its own separate fund, $f \in \{L, \, H\}$ , that offers frequency-specific exposure to the company’s dividends at times $t=1,\,2$ . The long-term investors, $i \in (0, \, \sfrac{1}{2}]$ , trade the low-frequency fund which has a payout:

$\begin{equation*} d_L \overset{\scriptscriptstyle \text{def}}{=} d_1 + d_2 \end{equation*}$

And, the short-run traders, $i \in (\sfrac{1}{2}, \, 1]$ , trade the high-frequency fund which has a payout:

$\begin{equation*} d_H \overset{\scriptscriptstyle \text{def}}{=} d_1 - d_2 \end{equation*}$

At time $t=0$ , each agent observes the equilibrium price of a frequency-specific fund, $p_f$ , and then chooses the number of shares to buy, $x_f^{(i)}$ , in order to maximize his expected utility at the end of time $t=2$ :

(1) $\begin{equation*} \max_{x_f^{(i)}} \mathrm{E}^{(i)}\left[ \, - e^{ \, - \alpha \cdot \{d_f - p_f\} \cdot x_f^{(i)} \, } \, \middle| \, p_f \, \right] \quad \text{for some} \quad \alpha > 0 \end{equation*}$

Above, $\mathrm{E}^{(i)}[\cdot|p_f]$ denotes agent $i$ ‘s conditional expectation, and $\alpha$ denotes his risk-aversion parameter.

Let $z_t$ denote the number of shares of the company’s stock that are available for purchase at time $t$ . We say that “markets clear” at time $t=0$ if the dividend payout from each available share at times $t=1,\,2$ has been unambiguously assigned to exactly one trader via their fund holdings:

(2) $\begin{align*} {\textstyle \int_0^{\sfrac{1}{2}}} x_L^{(i)} \cdot \mathrm{d}i + {\textstyle \int_{\sfrac{1}{2}}^1} x_H^{(i)} \cdot \mathrm{d}i &= z_1 \\ \text{and} \quad {\textstyle \int_0^{\sfrac{1}{2}}} x_L^{(i)} \cdot \mathrm{d}i - {\textstyle \int_{\sfrac{1}{2}}^1} x_H^{(i)} \cdot \mathrm{d}i &= z_2 \end{align*}$

Let $z_L \overset{\scriptscriptstyle \text{def}}{=} z_1 + z_2$ and $z_H \overset{\scriptscriptstyle \text{def}}{=} z_1 - z_2$ denote the number of available shares at each frequency.

An equilibrium then consists of a demand rule, $\mathrm{X}(\mathrm{E}^{(i)}[d_f|p_f], \, p_f) = x_f^{(i)}$ , and a price function, $\mathrm{P}(d_f, \, z_f) = p_f$ , such that 1) demand maximizes the expected utility of each agent given the price and 2) markets clear.

Because the equilibrium price of each fund only depends on its promised payout and the number of available shares, if agents knew the number of available shares, then they could reverse engineer a fund’s future payout at times $t=1, \, 2$ by studying its equilibrium price at time $t=0$ . And, an equilibrium in such a market wouldn’t be well-defined. So, to make sure that equilibrium prices at time $t=0$ aren’t fully revealing, let’s assume that the number of available shares in each period is a random variable:

$\begin{equation*} z_t \overset{\scriptscriptstyle \text{iid}}{\sim} \mathrm{N}( 0, \, \sfrac{1}{2}) \end{equation*}$

This means thinking about “available” shares as shares that haven’t already been purchased by noise traders.

The key fact about this benchmark model is that agents don’t use order-flow information at time $t=1$ to update their time $t=0$ beliefs. As a result, the model really isn’t about short-run traders in spite of how the variables are named. Crouzet/Dew-Becker/Nathanson shows that with some clever relabeling you could just as easily think about the low- and high-frequency funds as index and market-neutral funds, respectively.

Trading Volume

Although agents are only active at time $t=0$ in the benchmark model, each fund has to trade at times $t=1,\,2$ in order to deliver frequency-specific payouts. Let’s use $x_L$ and $x_H$ to denote aggregate demand:

$\begin{align*} x_L &\overset{\scriptscriptstyle \text{def}}{=} {\textstyle \int_0^{\sfrac{1}{2}}} x_L^{(i)} \cdot \mathrm{d}i \\ \text{and} \quad x_H &\overset{\scriptscriptstyle \text{def}}{=} {\textstyle \int_{\sfrac{1}{2}}^1} \, x_H^{(i)} \cdot \mathrm{d}i \end{align*}$

To deliver $d_L$ to every one of its shareholders, the low-frequency fund has to buy $x_L$ shares of the company’s stock between times $t=0$ and $t=1$ and then liquidate this position between $t=2$ and $t=3$ . And, to deliver $d_H$ to every one of its shareholders, the high-frequency fund has to buy $x_H$ shares between times $t=0$ and $t=1$ , sell $2 \cdot x_H$ shares between $t=1$ and $t=2$ , and then buy back $x_H$ shares between times $t=2$ and $t=3$ .

So, trading volume in the benchmark model is:

$\begin{align*} \mathit{vlm}_{0|1} &\overset{\scriptscriptstyle \text{def}}{=} |x_L| + |x_H| \\ \mathit{vlm}_{1|2} &\overset{\scriptscriptstyle \text{def}}{=} 2 \cdot |x_H| \\ \text{and} \quad \mathit{vlm}_{2|3} &\overset{\scriptscriptstyle \text{def}}{=} |x_L| + |x_H| \end{align*}$

The key thing to notice is that trading volume is symmetric, $\mathit{vlm}_{0|1} = \mathit{vlm}_{2|3}$ , because short-run traders don’t get any new information after time $t=0$ .

Model Solution

So, how many shares of each frequency-specific fund are agents going to demand in the benchmark model? To solve the model and answer this question, let’s first guess that the price function is linear:

$\begin{equation*} \mathrm{P}(d_f, \, z_f) = d_f - \sqrt{\sfrac{\sigma^2\!}{\mathit{SNR}}} \cdot z_f \quad \text{for some} \quad \mathit{SNR} > 0 \end{equation*}$

This guess introduces a new parameter, $\mathit{SNR}$ , which represents the signal-to-noise ratio of fund prices at time $t=0$ . If this parameter is large, then the time $t=0$ prices of the low- and high-frequency funds will reveal a lot of the information about the company’s time $t=1, \, 2$ dividend payouts.

Here’s the upshot of this guess. It implies that each fund’s price is a normally-distributed signal about its future payout, $d_f \sim \mathrm{N}(p_f, \, \sfrac{\sigma^2\!}{\mathit{SNR}})$ . And, with normally-distributed signals, we know how to compute agents’ posterior beliefs about $d_f$ after seeing $p_f$ :

$\begin{equation*} \mathrm{E}^{(i)}[d_f|p_f] = {\textstyle \left\{ \frac{1}{1 + \mathit{SNR}}\right\}} \cdot \mu_f^{(i)} + {\textstyle \left\{ \frac{\mathit{SNR}}{1 + \mathit{SNR}}\right\}} \cdot p_f \end{equation*}$

Above, $\mu_f^{(i)}$ denotes agent $i$ ‘s priors about a particular fund, either $\mu_L^{(i)} \overset{\scriptscriptstyle \text{def}}{=} \mu_1^{(i)} + \mu_2^{(i)}$ or $\mu_H^{(i)} \overset{\scriptscriptstyle \text{def}}{=} \mu_1^{(i)} - \mu_2^{(i)}$ . And, we can then use these posterior beliefs to compute agents’ equilibrium demand rule by solving the first-order condition of Equation (1) with respect to $x_f^{(i)}$ :

(3) $\begin{equation*} \mathrm{X}(\mathrm{E}^{(i)}[d_f|p_f], \, p_f) = \{1 + \mathit{SNR}\} \cdot {\textstyle \big\{ \frac{\mathrm{E}^{(i)}[d_f|p_f] - p_f}{\sigma} \big\}} \cdot {\textstyle \big\{ \frac{1}{\alpha \cdot \sigma} \big\}} \end{equation*}$

Finally, to verify that our original guess about a linear price function was correct, we can plug this equilibrium demand rule into the market-clearing conditions in Equation (2) and solve for $p_f$ :

$\begin{equation*} \mathrm{P}(d_f, \, z_f) = d_f - \alpha \cdot \sigma^2 \cdot z_f \end{equation*}$

The resulting price function is indeed linear, so our solution is internally consistent (…though not unique). And, by matching coefficients, we can solve for the equilibrium signal-to-noise ratio, $\mathit{SNR} = \{ \alpha \cdot \sigma \}^{-2}$ . In other words, fund prices at time $t=0$ reveal more information about a company’s dividend at times $t=1, \, 2$ when agents are less risk averse ( $\alpha$ small) or when they have more precise priors ( $\sigma$ small).

Order-Flow Info

How would this solution have to change if short-run traders could learn from time $t=1$ order flow?

For markets to clear at time $t=1$ , the aggregate demand for the low-frequency fund plus the aggregate demand for the high-frequency fund has to equal the total number of available shares, $x_L + x_H = z_1$ . And, from Equation (3), we know that the aggregate demand for the low-frequency fund is related to the company’s total dividend payout:

$\begin{equation*} x_L = \sqrt{\sfrac{\mathit{SNR}}{\sigma^2}} \cdot \{ d_L - p_L \} \end{equation*}$

So, by looking at the time $t=1$ order flow, short-run traders can get a signal about the company’s time $t=2$ dividend since $d_2 = d_L - d_1$ :

$\begin{equation*} d_2 \sim \mathrm{N}\left( \{p_L - d_1\} - \sqrt{\sfrac{\sigma^2\!}{\mathit{SNR}}} \cdot x_H, \, \sfrac{\sigma^2\!}{\mathit{SNR}} \right), \end{equation*}$

And, this information about the time $t=2$ dividend is helpful since $\mathrm{Cov}[d_2, \, d_H] = - \,\sigma^2$ .

With this additional signal, the short-run traders who previously invested in the high-frequency fund would now rather trade the company’s stock directly at times $t=1, \, 2$ . Let $\tilde{x}_H$ denote their demand at time $t=2$ . When they observe a high price for the low-frequency asset at time $t=0$ , $p_L > 0$ , and a large dividend payout at time $t=1$ , $d_1 > 0$ , they know that $d_2$ is likely small. And, as a result, they’ll short more shares at time $t=2$ , $|\tilde{x}_H| > |x_H|$ . By contrast, when they observe a high price for the low-frequency asset at time $t=0$ , $p_L > 0$ , and a small dividend payout at time $t=1$ , $d_1 < 0$ , they know that $d_2$ is likely large. So, they’ll short fewer shares at time $t=2$ , $|\tilde{x}_H| < |x_{H}|$ .

Either way, trading volume is going to look asymmetric and lumpy as a result, with relatively more of the trading volume clustered at one of the end points. If $p_L > 0$ and $d_1 > 0$ , then relatively more of the trading volume will occur at between time $t=2$ and $t=3$ because $\mathit{vlm}_{0|1}$ is unchanged and:

$\begin{equation*} \mathit{vlm}_{2|3} = |x_L| + |x_H| < |x_L| + |\tilde{x}_H| \overset{\scriptscriptstyle \text{def}}{=} \widetilde{\mathit{vlm}}_{2|3} \end{equation*}$

Whereas, if $p_L > 0$ and $d_1 < 0$ , then relatively more of the trading volume will occur between time $t=0$ and $t=1$ because $\mathit{vlm}_{0|1}$ is unchanged and now $\mathit{vlm}_{2|3} > \widetilde{\mathit{vlm}}_{2|3}$ .

What’s more, to long-term investors who can’t see short-run order flow, short-run traders are going to add execution risk. The price at which the low-frequency fund executes its time $t=2$ orders will now vary. And, this variation will be related to the magnitude (but not the sign) of their time $t=0$ demand.

Finally, note that this analysis shows why it’s easier to model indexers and stock pickers than long-term investors and short-run traders. An equilibrium in either model has to contain a demand rule and a price function (e.g., see the setup in the benchmark model). But, an equilibrium in a model with multi-frequency trade also has to contain a rule for how long-term investors think short-run traders will affect their order execution. And, this rule is the crux of any model with long-term investors and short-run traders.