Two commodities which are normal distributed and perfectly correlated

Question

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 1

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:

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