Modeling a Cointegrated Pair as a Mean-Reverting Trading Spread
Summary
The document considers whether two cointegrated price series can be combined into a stationary spread and modeled with an Ornstein-Uhlenbeck process. The proposed workflow estimates a hedge coefficient through regression, constructs the residual spread, and calibrates mean-reversion parameters using a discrete-time model. The example is an equity and a related swap, where deviations in the spread might signal relative mispricing in the swap.
The response agrees that a stationary combination is the intended starting point, but suggests deriving the coefficient from a cointegration test’s eigenvector. It cautions that an ordinary autoregressive process is not automatically mean reverting. It further argues that the observed spread may include white-noise variation around a latent mean-reverting component, motivating a state-space model and Kalman filtering. The exchange gives conceptual guidance but no data, calibration, trading results, or guarantee that the assumed pair is cointegrated or that the resulting signal is profitable.
Key ideas
- Cointegration implies a stationary linear combination of the two series, which can be studied as a spread.
- A cointegration test may provide a hedge coefficient through its estimated eigenvector.
- An autoregressive representation is not necessarily mean reverting without suitable parameter restrictions.
- The observed spread may combine a latent mean-reverting process with additional noise.
- A state-space model with Kalman filtering is suggested for separating latent spread dynamics from noise.
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# Pairs trading by transforming two cointegrated series into a mean-reverting process? # 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? ## Answer by Dhruv Mahajan (score 1, accepted) https://quant.stackexchange.com/a/59892 - 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? - Yes - 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
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