# Reference for "an increase in volatility increases value/price of american options"

I'm looking for a textbook/journal article reference for the well-known result that an increase in volatility increases the value/price of a standard American (call and put) option. In the case of continuous time with geometric brownian motion dynamics, I know how to prove the result directly from standard comparative-statics on the value function; however, I'm looking for a formal reference as every time I read something related on an article it is taken as given with no formal reference to the fact or only numerical examples are provided.

• Thanks, that is in fact enlightening. I was curious because I see it everywhere as a hard-known fact but I've never seen an explicit argument about it, and I was wondering if someone established it as a general fact. As I did it was by computing the value function for the infinite-time horizon in explicit form from the dynamic programming problem and just taking a partial derivative with respect to $\sigma$, so I guess it is as good as it gets. Apr 15, 2020 at 20:46