The change in the price of Apples each month is normally distributed with a standard deviation of 10 cents. The change in the price of Bananas each month is normally distributed with a standard deviation of 20 cents. You can assume that there is only a single Apple price that all apples follow, and there is a single Banana price that all bananas follow. These monthly price changes of Apples and Bananas have correlation of 1.0. Assuming these facts hold true going forward, if the price of Apples goes up by 10 cents, what would the change in the difference in price between Bananas and Apples in cents?
My first reaction to this is how can you tell exactly? Because the price difference will be distributed like
$$B_t - A_t \sim N(\mu_X+\mu_Y, 0.1^2+0.2^2)$$
So it must have something to do with the correlation being equal to 1, which gives us this
$$0.02 = Cov(A_t,B_t)$$
I cannot see where to go from here. I dont see how we can that we are certain that the price of Bananas would go up 20 cents just because the correlation is 1. Even if we know that the correlation is 1 we dont know by how much the variable will go up.
With a perfect correlation of $1$, with probability $1$ you can say there is a linear relationship where $B_t=c+dA_t$ for some $c$ and $d$.
From the ratio of the standard deviations, you can say $d=\frac{20}{10}=2$.
You can then say this implies $B_t-A_t=c+(d-1)A_t$ and that $d-1=1$
This suggests that if the price of Apples goes up by $10$ cents, then the difference in price between Bananas and Apples also goes up by $10$ cents.