# Why is the holder of a basket call long correlation?

I'm told that the holder of a basket call is long correlation.

I understand that an increase in correlation leads to an variance of a portfolio.

But with one "degree of freedom" (high positive correlation), if one asset falls, so do the others. And with multiple degrees of freedom (correlation near zero), if one falls, the others may rise and bring the average into the money.

Is this reasoning wrong?

• Yes, it is wrong. "the others may rise": they may, but it is unlikely right? Since they are moving independently of each other most likely about half will move up and half will move down, offsetting each other to a large extent. The average will not move much. – Alex C Apr 26 '18 at 13:29
• Alternatively, consider a particular case where the basket is made up of only $2$ components with correlation $-1$ such that their moves cancel each other. In that case the value of the basket will remain constant and therefore the value of the optionality is null. – Daneel Olivaw Apr 26 '18 at 14:59

• To add a little bit of math. $Cov(\alpha X, (1-\alpha) Y) = \alpha^2 Var(X) + (1-\alpha)^2 Var(Y) + 2 \alpha (1-\alpha) Cov(X, Y)$ Clearly $\alpha (1-\alpha)$ is positive. – Will Gu Apr 27 '18 at 6:19
• oops typo alert: left side should be $Cov(αX + (1−α)Y)$. – Will Gu Apr 27 '18 at 6:30