Mean reversion and adjusted beta for pairs trading

Question

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 1

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