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

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?