A Model of Rebalancing Cascades

1. Motivating Examples

Trading strategies can interact with one another to amplify small initial shocks to fundamentals:

Quant Crisis, Aug 2007: “During the week of August 6, 2007, a number of [quantitative hedge funds] experienced unprecedented losses… Initial losses [were] due to the forced liquidation of one or more large equity market-neutral portfolios… and the subsequent price impact… caused other similarly constructed portfolios to experience losses. These losses, in turn, caused other funds to deleverage their portfolios [and] led to further losses [and] more deleveraging and so on.”
Flash Crash, May 2010: “The Dow Jones industrial average plunged more than 600 points in a matter of minutes that day and then recovered in a blink… [it] began with the sale by Waddell & Reed of $75,000$ E-Mini S&P 500 futures contracts… late in the trading day… [with] many of the contracts bought by… computerized traders who… [then] traded contracts back and forth [like a] ‘hot potato’.”
Drop in Oil Prices, May 2011: “Never before had crude oil plummeted so deeply during the course of a day… prices were off by nearly $\mathdollar 13$ a barrel… [and] market players were unable to identify any single bank or fund orchestrating a massive sale to liquidate positions… [rather] computerized trading just kicked in when key price levels were reached.”
End-of-Day Volume, Oct 2011: “In the last $18$ minutes of trading, the S&P 500-stock index jumped more than $10$ points with no news to account for the rally. If you were left scratching your head, you were not alone… [and, the] culprit behind the late-day market swings: exchange-traded funds or ETFs.”
Sterling after Brexit, Oct 2016: “If a country’s exchange rate represents international investors’ confidence in its government’s policies, the markets have given Britain the thumbs-down… The most likely explanation for the plunge lies in the action of algorithmic trades… These sales can be contagious, with one program’s trades setting off the sell signals of other algorithms.”

I refer to these sorts of events as rebalancing cascades. Stock $1$ ‘s fundamentals change, so a trading strategy sells stock $1$ and replaces it with stock $2$ . This purchase of stock $2$ forces another trading strategy to buy stock $2$ and sell stock $3$ . And, this sale forces…

The examples above seem to suggest that you don’t need a very big initial change in stock $1$ ‘s fundamentals to trigger a cascade. For instance, when oil prices suddenly dropped in May 2011, traders were “unable to identify any single bank or fund orchestrating a massive sale”. They were left “scratching [their] heads”, to use the language of the next example. So, in this post, I write down a random-networks model à la Watts (2002) to understand when we should expect small changes in fundamentals to trigger these sorts of long rebalancing cascades.

2. Market Structure

Consider a market with $S$ stocks, $s = 1, \, 2, \, \ldots, \, S$ , where $S$ is a really big number. If a change in the fundamentals of stock $s'$ will force some trading strategy to rebalance and buy stock $s$ instead, then let’s say that stock $s'$ and stock $s$ are neighbors. Suppose that two randomly selected stocks are neighbors with probability $\sfrac{\lambda}{S}$ . This uniform-random-matching assumption means that the number of neighbors that each stock has, $N_s$ , is Poisson distributed with mean $\lambda \overset{\scriptscriptstyle \mathrm{def}}{=} \mathrm{E}[N_s]$ ,

(1) $\begin{equation*} N_s \sim \mathrm{Pois}(\lambda). \end{equation*}$

The fact that each stock is only neighbors with a fraction of the market captures the idea that different trading strategies rebalance in different ways. For technical reasons, let’s assume that $\lambda = \mathrm{O}[\log(S)]$ . The figure below gives examples of this sort of random network when $S=100$ .

Now, suppose that $\Delta_s \in \{ 0, \, 1\}$ is an indicator variable for whether or not stock $s$ ‘s fundamentals have changed. If a bunch of strategies start trading stock $s$ because of changes in its neighboring stocks’ fundamentals, then this additional trading activity can affect stock $s$ ‘s fundamentals. For example, if lots of funds buy a stock and push it into the S&P 500, then the stock will have a higher market beta. Let’s model neighboring stocks’ effect on stock $s$ ‘s fundamentals as follows,

(2) $\begin{equation*} \Delta_s = \begin{cases} 1 &\text{if } (\sfrac{1}{N_{s}}) \cdot {\textstyle \sum_{s' \in \mathcal{N}_s}} \Delta_s' \geq \phi \\ 0 &\text{else} \end{cases}, \end{equation*}$

where $\phi \in (0, \, 1]$ captures the vulnerability of a stock’s fundamentals to rebalancing. If there are lots of different strategies trading stock $s$ in lots of different ways, $N_s > N^\star$ , then no single rebalancing decision will be important enough to change stock $s$ ‘s fundamentals. But, if stock $s$ has at most $N^\star \overset{\scriptscriptstyle \mathrm{def}}{=} \lfloor \sfrac{1}{\phi} \rfloor$ neighbors, then a change in the fundamentals of a single neighbor will generate enough rebalancing to cause stock $s$ ‘s fundamentals to change too. Let’s say that stock $s$ has $V_s \overset{\scriptscriptstyle \mathrm{def}}{=} \sum_{s' \in \mathcal{N}_s} 1_{\{ \, N_{s'} \leq N^\star \}}$ such vulnerable neighbors.

Here’s the exercise I have in mind. Imagine that we select a stock at random, $s$ , and exogenously change its fundamentals, $\Delta_s = 1$ . If $s$ happens to have a vulnerable neighbor, $s'$ , then the rebalancing caused by our initial shock will change the fundamentals of a second stock, $\Delta_{s'} = 1$ . And, if $s'$ happens to have an additional vulnerable neighbor of its own, $s''$ , then the second wave of rebalancing caused by our initial shock to stock $s$ will change the fundamentals of a third stock as well, $\Delta_{s''} = 1$ . If stock $s''$ doesn’t have any additional vulnerable neighbors, then we will have triggered a rebalancing cascade of length $3$ with our initial shock to a single stock’s fundamentals. I want to characterize the distribution of cascade lengths for a randomly selected initial stock $s$ ,

(3) $\begin{equation*} C_s = \Delta_s + 1_{\{ \mathcal{V}_s \neq \emptyset \}} \cdot \left\{ \, {\textstyle \sum_{s' \in \mathcal{V}_s}} \left( \, \Delta_{s'} + 1_{\{ \mathcal{V}_{s'} \neq \emptyset \}} \cdot \left\{ \, {\textstyle \sum_{s'' \in \mathcal{V}_{s'}}} \left( \, \Delta_{s''} + \cdots \, \right) \, \right\} \, \right) \, \right\}. \end{equation*}$

as a function of the market’s average connectivity, $\lambda$ , and vulnerability threshold, $\phi$ .

3. Generating Functions

Generating functions make it possible to compute the distribution of cascade lengths. Here’s the basic idea. Take a look at Graham, Knuth, and Patashnik (1994, Ch 7) for more details. Suppose we’re flipping coins and counting the number of heads. The distribution of the number of heads, $h$ , after one flip is given by:

(4) $\begin{equation*} \text{\# of heads}|\text{1 flip} = \begin{cases} 1 &\text{w/ prob $q$, and} \\ 0 &\text{w/ prob $(1-q)$.} \end{cases} \end{equation*}$

If $q = \sfrac{1}{2}$ , then the coin is fair. The generating function for this same distribution is:

(5) $\begin{align*} \mathrm{G}(x|\text{1 flip}) &= {\textstyle \sum_{h=0}^1} \, p_h \cdot x^h. \end{align*}$

Each term in the series is associated with one possible outcome for the total number of heads. $p_0 = (1 - q)$ is the probability of realizing $h=0$ heads. $p_1 = q$ is the probability of realizing $h=1$ heads. We say that $\mathrm{G}(x|\text{1 flip})$ generates the distribution because we can compute all its moments by evaluating the derivatives of the generating function at $x=1$ . For example, the $0$ th-order derivative, $\left. \mathrm{G}(x|\text{1 flip})\right|_{x=1} = 1$ , says that the coin never lands on its edge or winks out of existence. And, if we want to compute the expected number of heads, then we can use the $1$ st-order derivative:

(6) $\begin{align*} \mathrm{E}[h|\text{1 flip}] &= \left. x \cdot \mathrm{G}'(x|\text{1 flip}) \right|_{x=1} \\ &= \left. x \cdot {\textstyle \sum_{h=0}^1} \, p_h \cdot h \cdot x^{h-1} \right|_{x=1} \\ &= \left. {\textstyle \sum_{h=0}^1} \, p_h \cdot h \cdot x^h \right|_{x=1} \\ &= p_0 \cdot 0 \cdot 1^0 + p_1 \cdot 1 \cdot 1^1 \\ &= q. \end{align*}$

The fact that derivatives of the generating function give the moments of the associated distribution will be useful below. Let’s call this Property 1: derivatives are moments.

Here are two additional properties to keep in mind. Property 2: multiple samples. If we raise the generating function for the number of heads in one flip to the $n$ th power, then we get the generating function for the total number of heads in $n$ flips. To illustrate, look at what happens if we square the generating function for the number of heads in one flip:

(7) $\begin{align*} \mathrm{G}(x|\text{1 flip})^2 &= \left\{ \, p_0 \cdot x^0 + p_1 \cdot x^1 \, \right\} \times \left\{ \, p_0 \cdot x^0 + p_1 \cdot x^1 \, \right\} \\ &= p_0^2 \cdot x^0 + 2 \cdot p_0 \cdot p_1 \cdot x^1 + p_1^2 \cdot x^2 \\ &= (1 - q)^2 \cdot x^0 + 2 \cdot (1 - q) \cdot q \cdot x^1 + q^2 \cdot x^2. \end{align*}$

The result is just the generating function for the number of heads in two flips, $\mathrm{G}(x|\text{2 flips})$ .

Property 3: partial information. If we multiply the generating function for the total number of heads in $(n-1)$ flips by $x^1$ , then we have the generating function for the total number of heads in all $n$ flips conditional on having already seen heads on the first flip. To illustrate, notice what happens when we multiply the generating function for the number of heads in one flip by $x^1$ :

(8) $\begin{align*} \mathrm{G}(x|\text{2 flips}, \, \text{heads on 1st flip}) &= x^1 \cdot \mathrm{G}(x|\text{1 flip}) \\ &= p_0 \cdot x^1 + p_1 \cdot x^2 \\ &= (1 - q) \cdot x^1 + q \cdot x^2. \end{align*}$

If we’ve already seen heads on the first flip, then there’s no way to realize $h=0$ heads. The lowest we can go is $h=1$ heads now. So, the first term is now $h=1$ . And, this outcome occurs if we see tails on the second flip, which happens with probability $(1-q)$ .

4. Cascade Lengths

Now, let’s return to our original problem. Let $\mathrm{G}_c(x)$ be the generating function distribution of cascade lengths that we would start with an exogenous shock to stock $s$ ‘s fundamentals:

(9) $\begin{equation*} \mathrm{G}_c(x) \overset{\scriptscriptstyle \mathrm{def}}{=} {\textstyle \sum_{c=1}^S} \, q_c \cdot x^c. \end{equation*}$

The coefficient $q_c$ gives the probability that a shock to stock $s$ ‘s fundamentals would set off a cascade of length $C_s = c$ . If stock $s$ doesn’t have any vulnerable neighbors, then a shock to stock $s$ ‘s fundamentals can only affect stock $s$ , $c=1$ . Whereas, if a shock to stock $s$ ‘s fundamentals would set off a cascade affecting every other stock in the market, then $c=S$ . Next, let $\mathrm{G}_v(x)$ be the generating function for the number of vulnerable neighbors that stock $s$ has:

(10) $\begin{equation*} \mathrm{G}_v(x) \overset{\scriptscriptstyle \mathrm{def}}{=} {\textstyle \sum_{v=0}^{S-1}} \, p_v \cdot x^v. \end{equation*}$

So, the coefficient $p_v$ is the probability that stock $s$ has $v$ vulnerable neighbors.

Notice how these two generating functions are linked. If stock $s$ has $v=1$ vulnerable neighbor, $s'$ , then an initial shock to stock $s$ ‘s fundamentals will set off a cascade of length $C_s = c$ if a shock to its one vulnerable neighbor will set off a cascade of length $C_{s'} = c - 1$ excluding stock $s$ . If stock $s$ has $v=2$ vulnerable neighbors, $s'$ and $s''$ , then an initial shock to stock $s$ will set off a cascade of length $C_s = c$ if shocks to its two vulnerable neighbors will set off cascades of combined length $C_{s'} + C_{s''} = c - 1$ excluding stock $s$ . And, if stock $s$ has $v=3$ vulnerable neighbors, then an initial shock to stock $s$ will set off a cascade of length $C_s = c$ if shocks to its three vulnerable neighbors will set off cascades of combined length $C_{s'} + C_{s''} + C_{s'''} = c - 1$ .

We know from the previous section (property 2: multiple samples) that $\mathrm{G}_c(x)^v$ is the generating function for the probability that $v$ different shocks set of cascades of combined length $c$ . And, we also know from the previous section (property 3: partial information) that we have to multiply through by $x^1$ if we want the generating function for the probability that $v$ different shocks set of cascades of combined length $(c-1)$ . So, the generating function for the distribution of cascade lengths has to satisfy the following internal-consistency condition as the number of stocks gets large, $S \to \infty$ :

(11) $\begin{align*} \mathrm{G}_c(x) &= p_0 \cdot x + p_1 \cdot x \cdot \mathrm{G}_c(x) + p_2 \cdot x \cdot \mathrm{G}_c(x)^2 + p_3 \cdot x \cdot \mathrm{G}_c(x)^3 + \cdots \\ &= x \cdot \left\{ \, p_0 \cdot \mathrm{G}_c(x)^0 + p_1 \cdot \mathrm{G}_c(x)^1 + p_2 \cdot \mathrm{G}_c(x)^2 + p_3 \cdot \mathrm{G}_c(x)^3 + \cdots \, \right\} \\ &= x \cdot \mathrm{G}_v\left(\mathrm{G}_c(x)\right). \end{align*}$

The outer function, $\mathrm{G}_v(\cdot)$ , gives the probability that the initial stock $s$ has $v$ vulnerable neighbors. The inner function, $\mathrm{G}_c(x)$ , gives the probability that shocks to these vulnerable neighbors would set of cascades of combined length $c$ . And, the multiplication by $x$ accounts for the fact that we want to compute the probability that shocks to these vulnerable neighbors would set of cascades of combined length $(c-1)$ not $c$ .

With this equation in hand, we can now compute the expected length of the rebalancing cascade that would follow from an initial shock to randomly selected stock $s$ :

(12) $\begin{align*} \mathrm{E}[C_s] = \left. x \cdot \mathrm{G}_c'(x) \right|_{x=1} &= 1 + \mathrm{G}_v'(1) \cdot \mathrm{G}_c'(1) \\ &= 1 + \mathrm{G}_v'(1) \cdot \mathrm{E}[C_s]. \end{align*}$

Rearranging yields an expression for the expected cascade length:

(13) $\begin{align*} \mathrm{E}[C_s] = \frac{1}{1 - \mathrm{G}_v'(1)}. \end{align*}$

And, in the exact same way that $\mathrm{G}_c'(1) = \mathrm{E}[C_s]$ (property 1: derivatives are moments), the expected number of vulnerable neighbors that each stock has is given by $\mathrm{G}_v'(1) = \mathrm{E}[V_s]$ . When stocks typically have less than $1$ vulnerable neighbor, $\mathrm{E}[V_s] < 1$ , we have an expression for the average rebalancing-cascade length as a function of the market’s connectivity, $\lambda$ , and vulnerability threshold, $\phi$ .

The figure above plots the average length of the rebalancing-cascade that would emerge if we selected an initial stock $s$ at random and shocked its fundamentals. It’s got a really interesting shape. A little math shows exactly why. We should expect short rebalancing cascades whenever stocks don’t have that many vulnerable neighbors. Here’s the expression for the average number of vulnerable neighbors that each stock has:

(14) $\begin{align*} \mathrm{E}[V_s] = \mathrm{G}_v'(1) &= \lambda \times \mathrm{Pr}[N_s \leq N^\star] \\ &= \lambda \times \mathrm{Pr}[N_s < (\lfloor \sfrac{1}{\phi} \rfloor +1)] \\ &=\lambda \times \left\{ \, \sum_{n < (\lfloor \sfrac{1}{\phi} \rfloor+1)} e^{-\lambda} \cdot \frac{\lambda^n}{n!} \, \right\}. \end{align*}$

Notice that stocks can have less than $1$ vulnerable neighbor on average for either of two reasons. First, they could have very few neighbors—that is, $\lambda$ could be less than $1$ . Think about this as a fragmented market where very few people trade. This is the region on the bottom of the figure. Second, even if there are lots of people trading, stocks could have fundamentals that aren’t very vulnerable to the effects of rebalancing—that is, $\phi$ is large. This is the region in the upper right of the figure. But, if the market isn’t too fragmented and stocks’ fundamentals are a little vulnerable to the effects of rebalancing, then long rebalancing cascades can emerge. In fact, they can be infinitely long…

5. Infinite Cascades

…but what does that even mean? It’s actually much more reasonable than it first sounds. In a large market, $S \to \infty$ , an infinitely long rebalancing cascade is just a cascade that affects a non-infinitesimal fraction of all stocks. If we now specify that $\mathrm{G}_c(x)$ is the generating function for the distribution of finite-length rebalancing cascades, then we can define $\theta$ as the fraction of all stocks affected by an infinitely long rebalancing cascade,

(15) $\begin{align*} \mathrm{G}_c(1) \overset{\scriptscriptstyle \mathrm{def}}{=} 1 - \theta. \end{align*}$

Think back to the coin-flipping example where we said that $\left. \mathrm{G}(x|\text{1 flip})\right|_{x=1} = 1$ because the coin never landed on its edge or magically winked out of existence. If the coin didn’t obey the laws of physics and disappeared $20{\scriptstyle \%}$ of the time, then we would have said that $\left. \mathrm{G}(x|\text{1 flip})\right|_{x=1} = 0.80$ . So, if $\mathrm{G}_c(x)$ is the generating function for the distribution of finite-length rebalancing cascades, then realizing an infinitely long cascade is like realizing a magical event that’s not characterized by $\mathrm{G}_c(x)$ . And, this way of framing the problem, $\mathrm{G}_c(1) = 1 - \theta = \mathrm{G}_v(\mathrm{G}_c(1))$ , gives us a way to solve for the fraction of the market that’s typically affected by an infinitely long rebalancing cascade, $\theta = 1 - e^{\lambda \cdot \theta}$ .

Finally, notice how sharp the phase transition is. Tiny changes in the market’s connectivity, $\lambda$ , and vulnerability, $\phi$ , can make all the difference between expecting infinitely long rebalancing cascades and expecting $16$ -stock long rebalancing cascades. Take a look at the figure above. Each panel has $S =1000$ stocks (the dots) and represents a single realization of trading-strategy rebalancing rules (the lines) in markets where each stock has $\lambda = 6.40$ neighbors (left) and $\lambda = 6.41$ neighbors (right) respectively. Both these markets are observationally equivalent. But, as shown in the figure below, when $\phi=0.30$ an initial shock to stock $s$ ‘s fundamentals will yield a huge rebalancing cascade when $\lambda = 6.40$ (left) but not when $\lambda = 6.41$ (right). Small changes that push a market over the transition point where $\mathrm{E}[V_s] = 1$ can have huge effects on the cascade-length distribution. What’s more, right at this transition point where $\mathrm{E}[V_s] = 1$ , the sizes of the rebalancing cascades follow a power-law distribution,

(16) $\begin{align*} \mathrm{Pr}[C_s = c] \sim c^{-\sfrac{3}{2}}, \end{align*}$

as shown in Newman et al. (2002). Slight differences in how the market happens to be wired up today can affect whether or not a stock on the other side of the market will be affected by an initial shock to stock $s$ ‘s fundamentals.