Perfect Correlation with Unequal Volatility in Bivariate Normal Variables

Summary

The document examines two normally distributed commodity price changes with zero means, unequal standard deviations, and perfect positive correlation. It asks whether their relative value changes and whether one commodity’s move is simply a fixed multiple of the other’s. The key tool is the conditional distribution of one variable given the observed value of the other.

For a bivariate normal pair, the conditional mean depends on the correlation and the ratio of their standard deviations, while the conditional variance is proportional to one minus squared correlation. At perfect correlation, that conditional variance is zero: once one price change is known, the other is determined by a linear relationship, scaled by relative volatility. Thus, perfect correlation does not mean equal changes when volatilities differ. The discussion supplies the conditional formula but does not work through price levels, accumulated returns, or the effects of changing correlation over time; its result applies to the stated normal model.

Key ideas

• Perfect correlation means one variable’s move is a deterministic linear function of the other in the bivariate normal model.
• When volatilities differ, perfectly correlated price changes need not be equal in size.
• The conditional mean scales the observed move by the ratio of the variables’ standard deviations.
• The conditional variance becomes zero when correlation is exactly positive or negative one.
• Correlation alone does not imply that two commodity price levels maintain a fixed ratio over time.

Tags

# Two commodities which are normal distributed and perfectly correlated


# Two commodities which are normal distributed and perfectly correlated












The daily price change in commodity 1 is distributed $N(0,0.15^2)$ and the daily price change in commodity 2 is distributed $N(0,0.3^2)$. The two commodities are 100% correlated.

1) Does the relative value of commodity 1 vs commodity 2 change over the next year?

I would have thought no as the relative value is distributed $N(0-0,0.15^2+0.3^2)$ but a quick sketch of the problem suggests otherwise.

2) Is the change of value of commodity 1 the same as the change of value of 2x commodity 2? or the change of value of 2x commodity 1 the same as the change of value of commodity 2?

My first thought here is that is cant be as we are sampling from a curved distribution, but then they are perfectly correlated. Anyone answer this better?

## Answer by Magic is in the chain (score 1, accepted)

https://quant.stackexchange.com/a/50611

This is a case of bivariate normal as there are two normal variables (as opposed to two variables driven by the same common random factor). The answer to your question is the conditional distribution of one of the variables given the other variable:

$Y|X \sim N\left(\mu_y+\rho \sigma_y \frac{x-\mu_x}{\sigma_x}, \sigma_y^2 \left(1-\rho^2\right) \right)$

For zero means and perfectly correlated case, this becomes:

$Y|X \sim N\left(\sigma_y \frac{x}{\sigma_x}, 0\right)$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.