Chapter 2

Mean Reversion

What Is Mean Reversion and How Does It Help Us?

Mean reversion is one of the most fundamental concepts underpinning relative value analysis. But while mean reversion is widely understood at an intuitive level, surprisingly few analysts are familiar with the specific tools available for characterizing mean-reverting processes.

In this chapter, we discuss some of the key characteristics of mean-reverting processes and the mean reversion tools that can be used to identify attractive trading opportunities. In particular, we address:

• model selection;
• model estimation;
• calculating conditional expectations and probabilities;
• calculating ex ante risk-adjusted returns, particularly Sharpe ratios;
• calculating first passage times, also known as stopping times.

For each concept, we start with a verbal and intuitive description of the concept, followed by a mathematical definition of the concept, and finish with an example application of the concept to market data.

A variable is said to exhibit mean reversion if it shows a tendency to return to its long-term average over time. Mathematicians will object that this definition is simply an exercise in replacing the words "exhibit", "mean", and "reversion" with the synonyms "shows", "long-term average", and "return". To address such objections, we'll provide a more mathematical definition shortly. But first we'll attempt to establish some further intuition about mean-reverting processes. To some extent, Justice Stewart's famous maxim on pornography, "I know it when I see it", applies to mean reversion. With that in mind, let's take a look at some processes that exhibit mean reversion and a few that don't.

Figure 2.1 and Figure 2.2 show two simulated time series. Both have an initial value of zero, and both have identical volatilities. But one is constructed to be a simple random walk, with zero drift, while the other is constructed to have a tendency to return toward its long-run mean, constructed to be zero in this example. In fact, the two series were constructed with identical normal random variates. In the case of the random walk, the mean of each observation was the value of the previous observation, so that the process was a martingale. In the case of the mean-reverting process, the mean of each observation was set to reflect the tendency for the process to return to the mean. At this point, we'd hope most readers would identify Figure 2.2 as the one with the mean-reverting variable. If we observe both figures closely, we can see that the mean-reverting process is in some sense a transformation of the random walk in Figure 2.1.

FIGURE 2.1 Simulated random walk.

Source: Authors.

FIGURE 2.2 Simulated mean-reverting process.

Source: Authors.

The speed with which a variable tends to revert toward its mean can vary. For example, Figure 2.3 and Figure 2.4 show time series that were simulated using the same random normal variates that generated the mean-reverting variable in Figure 2.2 but with an important difference. The variable in Figure 2.3 was constructed to have a faster speed of mean reversion than the variable in Figure 2.2, while the variable in Figure 2.4 was constructed to have a still faster speed of mean reversion.

FIGURE 2.3 Simulated mean-reverting process: Faster mean reversion.

Source: Authors.

FIGURE 2.4 Simulated mean-reverting process: Even faster mean reversion.

Source: Authors.

While it's well and good to consider variables simulated via known equations by a computer, traders and analysts have to make judgments about real-world data, which are almost always messier in some respects than simulated data. So it's also useful to consider a few real-world examples.

Figure 2.5 shows the spot price of gold in US dollars since January 1975. In our view, the strong upward drift exhibited in this series makes it a poor candidate to be modeled by a mean-reverting process.

FIGURE 2.5 Spot price of gold in US dollars since January 1975.

Source: Bloomberg.

Figure 2.6 shows the realized volatility of the ten-year (10Y) US Treasury yield since January 1962. Given that this series has repeatedly returned to a long-run mean in the past, it appears to be a relatively good candidate for modeling with a mean-reverting process.

FIGURE 2.6 Realized volatility of 10Y US Treasury bond yield (bp/year).

Source: Bloomberg.

As another example, Figure 2.7 shows the 2Y/5Y/10Y butterfly spread along the USD swap curve since 1998. Given the number of times during the sample that this spread returns to its long-run mean, we consider it another good candidate for modeling with a mean-reverting process.

FIGURE 2.7 2/5/10 butterfly spread along USD swap curve since 1988.

Source: Bloomberg.

Mathematical Definitions

Having provided verbal and graphical intuition regarding mean reversion, it's time to attempt to provide a few useful mathematical definitions.

Stochastic Differential Equation

First, we'll provide a brief definition of a stochastic differential equation (SDE). In practice, this term is fairly simple to define, as most of the definition is contained within the name. In other words, it's an equation that characterizes the random behavior of a variable over an infinitesimal period of time. As a result, it gives us the data-generating mechanism for the variable. For example, the equation would allow us to simulate the variable over time using a computer.

For example, dxt = k(µ-xt)dt + sdWt is the SDE for an Ornstein-Uhlenbeck (OU) process, the continuous-time limit of a first-order autoregressive process. The OU process is a popular SDE for modeling mean-reverting variables, as it has moments and densities that can be expressed analytically. In this equation, dxt is the change in the value of the random variable x at time t, over the infinitesimal interval dt. The speed of mean reversion is given by the parameter k and the long-run mean of the variable is given by µ. The instantaneous volatility of the variable is given by s, and the term dWt is the change in the value of Wt over the instantaneous time interval dt. In fact, Wt is ultimately the source of randomness that drives the process in this equation. In particular, Wt is a pure random walk, often referred to as Gaussian white noise. Wt is also referred to as a Wiener process, after the American mathematician Norbert Wiener.

In general, SDEs take the form

The term f(xt) is the drift coefficient of the equation, and it defines the mean of the process. The term g(xt) is the diffusion coefficient of the equation, and it defines the volatility of the process.

Conditional Density

Next, we'll define the conditional density of a process, also referred to as a transition density. In particular, the conditional density gives us the probability density for the future value of a random variable conditional on knowing some other information about the variable. In the case of a time series process, the conditioning information is usually some earlier value of the variable. For example, in the case of the OU process, the transition density of xt+t for t > 0 is a normal density with mean given by µ + (xt - µ)e-kt and with a variance given by .

Unconditional Density

The unconditional density of a process is the probability density for the future value of a random variable without being able to condition the density on any additional information. You could think of the unconditional density as the histogram that would result from simulating the process over an infinitely long period. More precisely, it's the limit of the conditional density p(xt+t) as t goes to infinity. So in the case of the OU process, the unconditional density is a normal density with mean given by µ and with variance given by .

Stationary Densities and Mean-Reverting Processes

In some cases, a variable will have a conditional density, but it won't have an unconditional density. In other words, the limit of the conditional density p(xt) won't converge to a limiting density.

A simple example of this would be a random walk with drift, given by the SDE dxt = ?dt + sdWt. The transition density or unconditional density for this process is normal...