Estimating Mean-Reverting Spreads and Adjusted Beta in Pairs Trading

Summary

The document examines beta estimation in a pairs-trading model where the return of one stock depends on the return of another as well as a mean-reverting component. It argues that a straightforward regression of the two stock returns can produce a distorted beta because the unobserved spread movement remains in the residual. The question is how to estimate beta while accounting for that component.

The response models the spread as an Ornstein–Uhlenbeck process and expresses it through cumulative returns of the two assets, a drift term, and beta. Substituting this relationship into the return equation yields a regression involving the assets’ cumulative returns and time, with the spread’s reversion speed entering as a coefficient. The proposed route is to estimate the resulting system using multidimensional linear regression. The exchange gives a modeling suggestion, but no data example, diagnostic checks, or evidence that the assumptions fit a particular pair; the stated equations should also be checked carefully when implementing them.

Key ideas

• A return regression may confound pairs beta with movements in an unobserved mean-reverting spread.
• The response assumes the spread follows an Ornstein–Uhlenbeck process.
• Expressing the spread through cumulative asset returns creates regression terms that account for mean reversion.
• The suggested estimation approach is multidimensional linear regression.
• The exchange provides no empirical validation or guidance on testing the model assumptions.

Tags

# Mean reversion and adjusted beta for pairs trading


# Mean reversion and adjusted beta for pairs trading












Trying to evaluate model for pairs trading. Consider classic formula:

$\frac{dP}{P} = adt+b\frac{dQ}{Q}+dX$,

where $P$ and $Q$ are stock prices, and $X$ is a mean reverting process (MRP) and $a$ is close to zero.

Using real world example I would like to evaluate parameters of MRP. Practically we cannot observe MRP, rather we can derive it from $P$ and $Q$. If we go straightforward and calculate $\hat{b}$ as least squares estimator of $\frac{dP}P$ against $\frac{dQ}Q$, then we have residual estimation including MRP., i.e.

$$ \frac{dP/P}{dQ/Q} = \hat{b}, $$

so that our beta $\hat{b}$ captures change in stock prices together with MRP

$$ \frac{dP/P - dX}{dQ/Q} = \hat{b}. $$

This gives skewed estimation of $\hat{b}$.

My question is: how to get estimation of $\hat{b}$ adjusted for MRP?

## Answer by M. Jeunesse (score 1)

https://quant.stackexchange.com/a/25259

let define $$ \text{RP}_t = \sum_{u< t} \frac{dP_u}{P_u}$$ $$ \text{RQ}_t =\sum_{u<t} \frac{dP_u}{P_u}$$ $X$ is a mean reverting process so : $$ dX = \alpha (\mu - X)dt + \sigma dB $$ where $B$ is a brownian motion

meanwhile using your relationship you get : $$ X_t = \text{RP}_t - b \text{RQ}_t - a t $$

you use $X$ dynamics with this and you get: $$\begin{split}\frac{dP}{P} &= a dt + b \frac{dQ}{Q} + dX \\ &= (a+\alpha\mu) dt + b \frac{dQ}{Q} - \alpha \text{RP} dt - \alpha b \text{RQ}_t dt - a\alpha t dt + \sigma dB \end{split}$$ you are now in the case of a classical multi dimensionnal linear regression

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