Pairs trading by transforming two cointegrated series into a mean-reverting process?

Question

I am slightly confused about the following.

Let us assume I have two cointegrated time-series. I would like to model their 'cointegration' by a mean-reverting Ornstein-Uhlenbeck process since if they cointegrated it would imply that their linear combination is equal to a stationary process, hence with a stationary mean.

• If $x_t$ and $y_t$ are the two cointegrated time-series, then there exists a stationary process $u_t = y_t - \beta x_t$.

In practice I would have an equity and its corresponding swap derivative which I assume are cointegrated, and I would like to observe the equity movements (not trade it) and hope it gives me a signal whether the corresponding swap derivative is over-or-under priced relative to the equity based on their historical cointegrating factor.

This is how I imagined to do it:

• Do a regression between the two time-series to find the 'beta parameter coefficient'
• Use historical data to generate the points for $u_t$ based on the equation above
• $u_t$ should be a mean-reverting process now, hence find the parameters of the Ornstein-Uhlenbeck process by calibrating them by its discrete counterpart; the AR process.

Does my logic make any sense?

Answer 1

1. Yes, but there are better ways to find beta. If you’re checking cointegration anyway, why not take the eigenvector you get from conintegration tests as the beta?

2. Yes

3. AR process is not the counterpart to OU, since it’s not inherently mean reverting.

Many people get this wrong, the spread itself would never follow something as structured as OU, it’ll be driven by a latent variable that follows OU, and spread would be latent variable + some multiple of noise(white noise) to add stochasticity. This now would become a state space model and you can solve it using a Kalman Filter