# Trading 3 stocks X Y Z where X cointegrated to Y, Y to Z, but no other cointegration is available

Suppose you have 3 stocks, say X Y Z. You also know that

X is cointegrated to Y using some test (say ADF)

and

Y is cointegrated to Z.

However, no transitivity, and no threesome cointegration whatsoever (in other words, neither X is directly cointegration to Z, nor is there a 3 symbols cointegration using Johansen).

Is there a way to generalize pair trading to make a dynamic portfolio of X Y Z?

Intuitively I would say yes, by thinking of two pairs, XY and YZ. But I don't see yet a good strategy managing both efficiently.

• Are you okay with upper and lower bounds on the other correlations?
– will
Jul 23 '19 at 22:39

Assuming we are talking about Pearson correlation, then we may apply the triangle inequality. Let $$\rho(X,Y)$$ denote the correlation between $$X$$ and $$Y$$. Then,
$$(1-\rho(X,Z))^{1/2}\le (1-\rho(X,Y))^{1/2} + (1-\rho(Y,Z))^{1/2}$$