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?