How to determine ratios for mean-reverting basket

Suppose I have a basket of 3 securities A, B, and C. I believe that the basket is cointegrated and I want to create a mean-reverting trade. I fit the model: $\log(A)=\beta_b*\log(B)+\beta_c*\log(C)+\alpha$ where A, B, and C are the prices of the securities.

This gives me estimates of $\alpha$, $\beta_b$ and $\beta_c$.

Now suppose that I believe that the spread is out of line. I want to sell \$1 of A and buy \$1 of the B and C basket. How should I allocate that dollar to B and C? Is it simply $\beta_b*\$1$units of B and$\beta_c*\$1$ units of C or is it more complex?

Related, is it more correct to regress log prices or raw prices when fitting the model?

(I know that this is related to How to build a mean reverting basket? but the answers there weren't very detailed and this is a more specific question).

• You generally regress returns, not prices... May 2, 2013 at 21:39
• The classic pairs trading paper Gatev & Goetzmann 2006 regresses prices May 2, 2013 at 22:46
• Can get it out with a mass simulation if you can't figure it out. Nov 18, 2013 at 12:42
• quant.stackexchange.com/questions/21994/… Dec 25, 2016 at 3:16

In a recent paper - Cointegration and Relative Value Arbitrage by Binh Do and Robert Faff, the issue of relative value arbitrage with three stocks is addressed. On page 27 they formulate the cointegrating relation similarly to how you did: $$\ p_{1t} = \alpha + \gamma p_{2t} + \beta p_{3t} + \epsilon_t$$
$$w_1/w_2 = p_{1t}/-\gamma p_{2t}$$ $$w_1/w_3 = p_{1t}/-\beta p_{3t}$$ $$|w_1| + |w_2| + |w_3| = 2$$