Multiscale Noisy-Rational-Expectations Equilibrium

1. Motivation

Evolutionarily Slow. In modern financial markets, people simultaneously trade the exact same assets on vastly different timescales. For example, a Jegadeesh and Titman (1993)-style momentum portfolio turns over half its holdings once every 6 months. By contrast, Kirilenko, Kyle, Samadi, and Tuzun (2014) estimate that “high-frequency traders (HFTs) reduce half of their net holdings in 137 seconds.” These two horizons differ by 5 orders of magnitude:

(1)   \begin{align*}    \frac{\sfrac{1}{2} \, {\scriptstyle \mathrm{holdings}}}{137 \, {\scriptstyle \mathrm{seconds}}}    \times    \frac{60 \, {\scriptstyle \mathrm{seconds}}}{1 \, {\scriptstyle \mathrm{minute}}}    \times    \frac{60 \, {\scriptstyle \mathrm{minutes}}}{1 \, {\scriptstyle \mathrm{hour}}}    \times    \frac{24 \, {\scriptstyle \mathrm{hours}}}{1 \, {\scriptstyle \mathrm{day}}}    \times    \frac{30 \, {\scriptstyle \mathrm{days}}}{1 \, {\scriptstyle \mathrm{month}}}    \times    \frac{6 \, {\scriptstyle \mathrm{months}}}{\sfrac{1}{2} \, {\scriptstyle \mathrm{holdings}}}    =   1.1 \times 10^5. \end{align*}

This is a big number. To put it in perspective, there have only been around 2 \times 10^5 generations since the human/chimpanzee divergence. See MacKay (2003, C4). So, in a quite literal sense, momentum traders are evolutionarily slow by HFT standards. This difference in timescales has a couple of important implications.

Different Strokes. First, because their respective timescales are so different, short- and long-run traders value assets differently. One reason is purely mechanical: short-run traders have to close out their positions at the end of each day. Why should they be interested in a company’s quarterly cash flows? Another reason is more subtle: a trader’s timescale affects the kind of information he can process. Short-run traders operate at speeds faster than any human can handle, so these traders have to focus on machine-readable information; whereas, long-run traders operate on more human timescales, so these traders tend to focus on soft information. Traders at different timescales also have very different educational backgrounds. As the pace of trading gets faster, traders become more likely to have a background in mathematics, computer science, or engineering than a background in economics, finance, or accounting.

Predictable Demand. Second, because long-run traders’ timescale is so slow, their order flow will look slightly predictable to short-run traders. If long-run traders space their orders out over the course of a month, then demand in the previous minute is going to give short-run traders a little bit of information about what demand will be like in the next minute. The SEC points out in a 2010 report that short-run traders often use “sophisticated pattern recognition software to ascertain from publicly available information the existence of a large buyer (seller)” or “ping different market centers in an attempt to locate and trade in front of large buyers (sellers).” Even if you didn’t know that the last common ancestor we shared with chimpanzees would eventually evolve into humans in the long-run, at each step along the way, you could look at two parents and still have a pretty good idea of what their offspring were going to be like.

What It All Means. These two observations imply that short-run traders will want to adjust their own trading behavior due to the presence of long-run traders and vice versa, affecting the returns we observe at each horizon. Put another way, a pair of assets with the same fundamentals might realize different returns because their traders operate at different timescales. This post outlines an asset-pricing model studying precisely this effect. It’s an overview of the model in my paper, When Fast Trading Looks Like Priced Noise.

2. Market Structure

Long-Run Value. There are T trading periods and a single asset that has a long-run fundamental value of

(2)   \begin{align*} v \sim \mathrm{N}(\mu_v, \, \sigma_v^2). \end{align*}

For instance, you might think about v as a liquidating dividend. There is the same amount of variance in the long-run fundamental value regardless of the number of trading periods. After all, a factory’s production doesn’t get more erratic if people start to trade shares in the company that owns it more quickly.

Short-Run Value. This asset also realizes short-run, transient, value shocks

(3)   \begin{align*} \epsilon_t \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0, \, \sigma_{\epsilon}^2) \end{align*}

that are irrelevant in the long-run. For instance, you might think about each \epsilon_t as a payment from an exchange for providing liquidity. Importantly, short-run traders care about both the permanent and the transient parts of an asset’s short-run value when choosing their portfolio holdings,

(4)   \begin{align*} \tilde{v}_t = v + \epsilon_t. \end{align*}

Algorithmic traders have to close out their positions overnight. So, they can’t justify an awful P&L statement on Monday by telling their MD that holding the position for a month will return it to the black.

Pricing Rule. Each period, uninformed market makers set the asset’s price equal to their conditional expectation of the asset’s long-run fundamental value after observing the aggregate demand for the asset,

(5)   \begin{align*} p_t = \mathrm{E}[v|a_t] = \lambda_0 + \lambda_a \times a_t \qquad \text{with} \qquad a_t = x_t + y + z_t. \end{align*}

Here, x_t denotes the number of shares bought by the short-run trader, y denotes the number of shares bought by the long-run trader, and z_t \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0, \, \sigma_z^2) denotes the number of shares bought by noise traders.

To offset their expected losses from trading with more informed agents, market makers charge a fee, \kappa, to each group of traders where

(6)   \begin{align*} \kappa &= \mathrm{E}[ \, {\textstyle \frac{1}{\sigma_a} \cdot \sum_{t=1}^T} (v - p_t) \cdot a_t \, ]. \end{align*}

So, the monthly price we observe in the data is \hat{p} = (\frac{1}{T} \cdot \sum_{t=1}^T p_t) - \kappa. As Brennan and Subrahmanyam (1996) write, “privately informed investors create significant illiquidily costs for uninformed investors, implying that the required rates of return should be higher for securities that are relatively illiquid.” A fee is a simple way of building this price effect into the model. Alternatively, you could add in this price effect by getting rid of the market maker and modeling risk-averse informed traders facing imperfect competition a la Kyle (1989), but this approach leads to a much more complicated analysis without any additional insight.

3. Short-Run Traders

Optimization Problem. There are T groups of short-run traders that each trade in a single period. Because long-run traders trade so gradually compared to short-run traders, their demand will be slightly predictable and, thus, give away some of their private information about long-run fundamentals. So, prior to trading, short-run traders observe not only the combined short-run value of the asset, \tilde{v}_t, but also a signal about the asset’s long-run fundamental value, s_t. These traders then solve the static optimization problem below,

(7)   \begin{align*} \max_{x_t} \, \mathrm{E}\left[ \, (\tilde{v}_t - p_t) \cdot x_t \, \middle| \, \tilde{v}_t, \, s_t \, \right]. \end{align*}

The short-run traders’ signal each period is centered around the asset’s long-run fundamental value and gets more precise when long-run traders trade more aggressively,

(8)   \begin{align*}   s_t &\overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}( v, \, \sfrac{\sigma_z^2}{\gamma_v^2} ), \end{align*}

where \gamma_v is the long-run traders’ demand response which I define in the next section. So, the more long-run traders act on their private information, the more this information leaks out to short-run traders.

Demand Rule. I look for a linear demand rule:

(9)   \begin{align*}   x_t &= \beta_0 + \beta_{\tilde{v}} \times \tilde{v}_t + \beta_s \times s_t. \end{align*}

The coefficient \beta_0 has units of \mathrm{shares} and captures how many shares a short-run trader will demand on average. The coefficient \beta_{\tilde{v}} has units of \sfrac{(\mathrm{shares})^2\!}{\mathdollar} and captures how many additional shares a short-run trader will demand in the current period if the total short-run value increases by \mathdollar 1 per share. The coefficient \beta_s also has units of \sfrac{(\mathrm{shares})^2\!}{\mathdollar} but captures how many additional shares a short-run trader will demand if his signal about the long-run fundamental value of the asset gets \mathdollar 1 per share more optimistic.

4. Long-Run Traders

Optimization Problem. There is a single group of long-run traders who observe the long-run fundamental value of the asset, v. These traders solve the static optimization problem below,

(10)   \begin{align*} \max_y \, \mathrm{E}\left[ \, {\textstyle \sum_{t=1}^T} (v - p_t) \cdot y \, \middle| \, v \, \right]. \end{align*}

Long-run traders have to demand the exact same number of shares every single time. This is what makes them “long-run” and why there is no time subscript on their demand, y. For instance, you can think about long-run traders as traders who slowly adjust their positions using instruments like time-weighted average pricing rules, which automatically execute trades over a predetermined timeframe.

Demand Rule. Again, I look for a linear demand rule:

(11)   \begin{align*}  y = \gamma_0 + \gamma_v \times v \end{align*}

The coefficient \gamma_0 has units of \mathrm{shares} and captures how many shares the long-run trader will demand on average. The coefficient \gamma_v has units of \sfrac{(\mathrm{shares})^2\!}{\mathdollar} and captures how many additional shares the long-run trader will demand in each trading period if the long-run fundamental value increases by \mathdollar 1 per share.

5. Solving the Model

Short-Run Demand. When you account for long-run trader’s demand rule, short-run traders’ problem morphs into:

(12)   \begin{align*} \max_{x_t} \, \mathrm{E}\left[ \, (\tilde{v}_t - \lambda_0 - \lambda_a \cdot \{ x_t + \gamma_0 + \gamma_v \cdot v + z_t\}) \cdot x_t \, \middle| \, \tilde{v}_t, \, s_t \, \right]. \end{align*}

After observing the combined short-run asset value and their signal about long-run fundamental value, short-run traders have posterior beliefs of

(13)   \begin{align*} \mathrm{E}[v|\tilde{v}_t,s_t] &= \psi \cdot \omega \cdot \mu_v + (1 - \psi) \cdot \omega \cdot \tilde{v}_t +  (1 - \omega) \cdot s_t \\ \text{and} \quad \mathrm{Var}[v|\tilde{v}_t,s_t] &=  \psi^2 \cdot \omega^2 \cdot \sigma_v^2 + (1 - \psi)^2 \cdot \omega^2 \cdot \sigma_{\epsilon}^2 + (1 - \omega)^2 \cdot {\textstyle \frac{\sigma_z^2}{\gamma_v^2}} \end{align*}

where \psi = \sfrac{\sigma_{\epsilon}^2}{(\sigma_v^2 + \sigma_{\epsilon}^2)} and \omega = \sfrac{(\sfrac{\sigma_z^2}{\gamma_v^2})}{(\psi \cdot \sigma_v^2 + \sfrac{\sigma_z^2}{\gamma_v^2})}. Thus, the short-run traders’ optimal demand is given by:

(14)   \begin{align*} x_t  &=  \left( \, - \, {\textstyle \frac{1}{2 \cdot \lambda_a}} \cdot \{ \lambda_0 + \lambda_a \cdot ( \gamma_0 + \gamma_v \cdot \psi \cdot \omega \cdot \mu_v ) \} \, \right) \\ &\qquad \qquad + \, \left( \, {\textstyle \frac{1}{2 \cdot \lambda_a}} \cdot \{ 1 - \lambda_a \cdot \gamma_v \cdot (1 - \psi) \cdot \omega \} \, \right) \times \tilde{v}_t \\ &\qquad \qquad \qquad \qquad + \, \left( \, - \, {\textstyle \frac{\gamma_v}{2}} \cdot (1 - \omega) \, \right) \times s_t. \end{align*}

Long-Run Demand. When you account for short-run traders’ demand rule, the long-run trader’s problem turns into:

(15)   \begin{align*} \max_y \, \mathrm{E}\Big[ \, T \cdot \Big( \, v -  \{ \lambda_0 + \lambda_a \cdot (\beta_0 + \beta_{\tilde{v}} \cdot \underset{=v}{{\textstyle \frac{1}{T} \cdot \sum_{t=1}^T} \tilde{v}_t} + \beta_s \cdot \underset{=v}{{\textstyle \frac{1}{T} \cdot \sum_{t=1}^T} s_t} + y + \underset{=0}{{\textstyle \frac{1}{T} \cdot \sum_{t=1}^T} z_t}) \} \, \Big) \cdot y \, \Big| \, v \, \Big] \end{align*}

Notice that, as the number of short-run trading periods grows large, T \nearrow \infty, the long-run trader’s problem becomes completely deterministic because they only care about the average price. From the point of view of the long-run trader, transient asset-value fluctuations blur together and average out in the same way that you can’t tell with the naked eye that incandescent lights actually flicker at 60Hz. Thus, the long-run trader’s optimal demand is given by:

(16)   \begin{align*} y &=  \left( \, - \, {\textstyle \frac{1}{2 \cdot \lambda_a}} \cdot \{ \lambda_0 + \lambda_a \cdot \beta_0 \} \, \right) + \left( \, {\textstyle \frac{1}{2 \cdot \lambda_a}} \cdot \{ 1 - \lambda_a \cdot (\beta_{\tilde{v}} + \beta_s) \} \, \right) \times v. \end{align*}

The short-run traders’ demand coefficients still show up in the long-run trader’s demand rule. So, even though all of the short-run, transient, value shocks average out, there are still echoes of the short-run traders’ demand rule in the long-run.

System of Equations. Given the short- and long-run traders’ optimal demand rules, the market maker in each trading period sets the price equal to his conditional expectation of the asset’s long-run fundamental value after observing the aggregate demand:

(17)   \begin{align*} p_t  &=  \left( \left\{ 1 - \lambda_a \cdot ( \beta_{\tilde{v}} + \beta_s + \gamma_v ) \right\} \cdot \mu_v - \lambda_a \cdot \{\beta_0 + \gamma_0\} \right) + \left( {\textstyle \frac{(\beta_{\tilde{v}} + \beta_s + \gamma_v) \cdot \sigma_v^2}{(\beta_{\tilde{v}} + \beta_s + \gamma_v)^2 \cdot \sigma_v^2 + \beta_{\tilde{v}}^2 \cdot \sigma_{\epsilon}^2 +  \{1 + \sfrac{\beta_s^2}{\gamma_v^2}\} \cdot \sigma_z^2}}\right) \times a_t. \end{align*}

Thus, the equilibrium slope coefficients are defined by the following system of 4 equations and 4 unknowns:

(18)   \begin{align*} \beta_{\tilde{v}} &= \phantom{-} {\textstyle \frac{1}{2 \cdot \lambda_a}} \cdot ( 1 - \lambda_a \cdot \gamma_v \cdot (1 - \psi) \cdot \omega ), \\ \beta_s &= - \, \,  {\textstyle \frac{\gamma_v}{2}} \, \, \cdot (1 - \omega), \\ \gamma_v &= \phantom{-} {\textstyle \frac{1}{2 \cdot \lambda_a}} \cdot ( 1 - \lambda_a \cdot \{\beta_{\tilde{v}} + \beta_s\} ), \\ \text{and} \qquad \lambda_a &= \phantom{-} {\textstyle \frac{(\beta_{\tilde{v}} + \beta_s + \gamma_v) \cdot \sigma_v^2}{(\beta_{\tilde{v}} + \beta_s + \gamma_v)^2 \cdot \sigma_v^2 + \beta_{\tilde{v}}^2 \cdot \sigma_{\epsilon}^2 +  \{1 + \sfrac{\beta_s^2}{\gamma_v^2}\} \cdot \sigma_z^2}}. \end{align*}

After solving the slop coefficients, the equilibrium level coefficients are then pinned down by a further system of 3 equations and 3 unknowns:

(19)   \begin{align*} \beta_0 &= - \, {\textstyle \frac{1}{2 \cdot \lambda_a}} \cdot \{ \lambda_0 + \lambda_a \cdot ( \gamma_0 + \gamma_v \cdot \psi \cdot \omega \cdot \mu_v ) \}, \\ \gamma_0 &= - \, {\textstyle \frac{1}{2 \cdot \lambda_a}} \cdot \{ \lambda_0 + \lambda_a \cdot \beta_0 \}, \\ \text{and} \qquad \lambda_0 &= \phantom{-} \,\! \left( \left\{ 1 - \lambda_a \cdot ( \beta_{\tilde{v}} + \beta_s + \gamma_v ) \right\} \cdot \mu_v - \lambda_a \cdot \{\beta_0 + \gamma_0\} \right) = \mu_v. \end{align*}

6. Comparative Statics

Equilibrium Parameters. I solve for the equilibrium parameters numerically (code). The figure below shows how the slope parameters vary as I increase the ratio of short- to long-run asset-value volatility from \sfrac{\sigma_{\epsilon}}{\sigma_v} = 0 to \sfrac{\sigma_{\epsilon}}{\sigma_v} = 1.5. When \sfrac{\sigma_{\epsilon}}{\sigma_v} = 0, there are no short-run, transient, value shocks and the model collapses to the standard Kyle (1985) model. As \sfrac{\sigma_{\epsilon}}{\sigma_v} increases, there are more short-run value shocks for short-run traders to trade on (upward-sloping curve in left-most panel labeled \beta_{\tilde{v}}), meaning that there is effectively more noise trading. As a result, long-run traders trade more aggressively on their private information (upward-sloping curve in right-center panel label \gamma_v) and market makers respond less aggressively to aggregate demand shocks (downward-sloping curve in right-most panel labeled \lambda_a).


Expected Profits. The figure below plots each trader type’s expected profit as I again increase the ratio of short- to long-run asset-value volatility from \sfrac{\sigma_{\epsilon}}{\sigma_v} = 0 to \sfrac{\sigma_{\epsilon}}{\sigma_v} = 1.5. Because the long-run trader’s demand looks slightly predictable to short-run traders, the long-run trader can’t fully exploit the noise that short-run traders provide. So, the long-run trader can’t fully exploit the camouflage provided by short-run traders’ extra demand. Nevertheless, the long-run trader’s expected profit each trading period still goes up when there is more short-run trading activity (upward-sloping curve in left-center panel). This is exactly the point Cliff Asness and Michael Mendelson make in their Wall Street Journal op-ed when they write that, “How do we feel about high-frequency trading? We think it helps us. It seems to have reduced our costs and may enable us to manage more investment dollars.”


Notes on Information Aversion


1. Motivation In spite of how they are modeled in Merton (1971), traders don't pay attention to their portfolio every second of every day. What's more, this lumpy rebalancing behavior has important asset-pricing implications. If traders aren't … [Continue reading]