Be Careful When Modeling Spreads As Univariate Processes

In my previous notes on the multivariate Ornstein-Uhlenbeck process, I focused on the fact that correlations in this framework generally depend on the sampling frequency – an important point for traders and risk managers.

But perhaps an even more important point for traders is that a spread consisting of two instruments that follow an OU process generally can’t be represented as a univariate OU process. If we mistakenly model one of these spreads with a univariate OU process, we’ll be using a misspecified model – with consequences for our P&L.

We can motivate this fact in two ways – with some math and with some pictures. Let’s have a quick look at the math first.

Consider two variables, x and y, which follow univariate Ornstein-Uhlenbeck processes given by the stochastic differential equations

dx = -a x dt + v dw

dy = -b y dt + u dq

where v and u are constants that scale the Brownian motions, w and q.

Let’s define z as the spread between x and y – ie, z=x-y.

In this case, after applying a multivariate version of Ito’s lemma, we get the stochastic differential equation for z:

dz = (-ax + by) dt + some diffusion terms

For z to follow a univariate OU process, we need to be able to write its stochastic differential equation in the form

dz = -c z dt + some diffusion term

for some constant, c.

But we can only do this if a and b are equal. In other words, the spread, x-y, can be represented by a univariate OU process only if x and y revert around their means at the same speed (and assuming there are no interaction effects between x and y).

The situation is illustrated in the graph below. Imagine x currently equals 5 and y currently equals 2. Assume x reverts toward a mean of zero with a half-life of three months, and assume y reverts toward a mean of zero with a half-life of six months. The two lines in this graph represent the path of expected values for x and y.

In this figure, x and y follow univariate Ornstein-Uhlenbeck processes, with x reverting around a mean of zero with a half-life of three months, and with y reverting around a mean of zero with a half-life of six months.

Note that at some point x goes from being greater than y to being less than y. As a result, the spread goes from being positive to being negative, after which it approaches its long-run mean of zero from below, as illustrated in this next graph.

The expected spread, x-y, declines toward its long-run mean of zero but then passes beyond its mean before approaching zero from below. These dynamics can't be represented by a univariate Ornstein-Uhlenbeck process.

In this example, not only is the path of expected values for the spread not following a simple exponential decay; it’s not even monotonic. Clearly, we’re not going to be able to represent the spread, x-y, using a univariate OU process.

What happens if we mistakenly model the spread, z, as a univariate Ornstein-Uhlenbeck process? We’ll misspecify both the mean and the variance at each point in time (though eventually the mean will converge to the correct value). As a result, we’ll misspecify both the risk and the expected return in the trade, misallocating capital in the process.

The preferred approach in this case is to calculate the mean and variance of the spread directly from the multivariate Ornstein-Uhlenbeck process used to model x and y. In this way, we’ll correctly specify our risk and return over time, allowing us to apply our capital optimally, in line with our objectives.

Applying multivariate Ornstein-Uhlenbeck process to the individual legs of a spread trade does require a bit more effort than applying the univariate Ornstein-Uhlenbeck process to the spread. But the better trading results are well worth the effort.