I struggle with the interpretation of a process I derive from Ito's Lemma. Let's say I have function f(S,t) which is twice differentiable wrt S.
I thus can apply Ito's Lemma to get $df(S,t)$. So far so good. If we start from
$dS_t = \mu(S,t) dt + \sigma(S,t) dW_t$
then the volatility term of $df(S,t)$ is just
$df(S,t) \propto \sigma(S,t) f'(S,t) dW_t$
Now I struggle a bit. Because this means that the volatility of df(S,t) will be negative, whenever $f'(S,t)<0$, as $\sigma(S,t)>0$.
So Ito's lemma can result in negative volatilities whenever $f'(S,t)<0$?
Is this true, or where do I make a mistake?