# Ito's lemma results in negative volatility processes

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?

Best, Florian

• I think you are identifying the parameter in front of $dW_t$ with the term volatility. The variance rate of your new process is $\sigma(S_t,t)^2f'(S_t,t)^2dt$, which is indeed positive as required. Jun 25, 2021 at 9:57
• To further illustrate @Kermittfrog's point (which is the right answer): If you consider the Heston (1993) model, variance is modelled by an SDE whose solution follows a $\chi^2$ distribution. Hence, the variance is positive in the model. If you use Itô's Lemma, you find that the volatility in the model is normally distributed and could thus be negative. However, only variance (not volatility) matters for option pricing. Jun 25, 2021 at 10:45
• @Kermittfrog: Thanks! Yes exactly, I just refer to the volatility of $f(S,t)$ as the term in front of $dW_t$, so actually the diffusion coefficient of the process. This diffusion coefficient is negative if $f'(S,t)<0$. And I am struggling a bit with the interpretation of this negative diffusion coefficient for a process in finance. Jun 25, 2021 at 11:45
• It should be interpreted in absolute value, then. Jun 25, 2021 at 14:50