Ito’s Lemma and the Sign of a Diffusion Coefficient

Summary

The document asks whether applying Ito’s lemma to a function of a stochastic process can produce a negative volatility. Starting from a process with positive diffusion coefficient, it identifies the transformed process’s Brownian-motion term as the original coefficient multiplied by the function’s derivative. The question focuses on what happens when that derivative is negative.

It does not include an answer or supporting examples, so it leaves the interpretation unresolved. In particular, it does not distinguish the signed coefficient on the Brownian increment from volatility understood as a nonnegative magnitude. Readers should treat it as a conceptual question about interpreting Ito’s formula, rather than a worked explanation.

Key ideas

• The transformed process’s diffusion term depends on the derivative of the function.
• A negative derivative can make the coefficient of the Brownian increment negative.
• The document asks how that signed coefficient relates to volatility but does not resolve the question.

Tags

# Ito's lemma results in negative volatility processes


# 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

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