Correlation Bounds for Indirect Relationships Between Three Assets

Summary

The question considers three stocks where X is cointegrated with Y and Y with Z, but X and Z are not reported as cointegrated and no three-variable cointegrating relation is available. It asks whether the two pair relationships can support a dynamic portfolio generalization of pairs trading.

The answer does not provide a trading strategy or portfolio construction method. Instead, assuming Pearson correlations, it gives a triangle-inequality bound: the square root of one minus the X–Z correlation is no greater than the sum of the corresponding quantities for X–Y and Y–Z. This offers a mathematical constraint on pairwise correlation relationships, but correlation is distinct from cointegration, and the short response does not establish a cointegrating relation or explain how to trade the three assets.

Key ideas

• The setup has two stated pairwise cointegrating relationships but no direct X–Z or joint relation.
• Under the answer’s Pearson-correlation assumption, a triangle inequality bounds the X–Z correlation distance using the other two pairs.
• The bound concerns correlation and does not itself establish cointegration or define a trading strategy.

Tags

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


# 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.

## Answer by Joseph Zambrano (score 3)

https://quant.stackexchange.com/a/46781

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,

$(1-\rho(X,Z))^{1/2}\le (1-\rho(X,Y))^{1/2} + (1-\rho(Y,Z))^{1/2}$

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