# Transformation of local volatility model

Assume we have an SDE $$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$ where $$\sigma>0$$ and $$W_t$$ is a Wiener process. Is there a transformation $$y(X_t)$$ that will make the dynamics of the transformed process $$Y_t=y(X_t)$$ have constant volatility?

• The (almost trivial) case is of course $\mu(X)=\mu\times X$ and $\sigma(X)=\sigma\times X$. Then,$y=\ln(X)$ yields constant vol Commented Mar 31, 2022 at 6:16

Yes it is called the Lamperti transform. This document, in particular Theorem 2, page 7, describes what the Lamperti transform is.

• Nice, I did not know that this topic has been studied - thanks! Commented Mar 31, 2022 at 8:51
• Thanks, had not heard this term before.
– fes
Commented Mar 31, 2022 at 14:49

Consider a function $$f(X_t)$$. Ito's lemma gives:

$$df(X_t)=\text{time terms}+f'(X_t)\sigma(X_t)dW_t$$

Now any $$f$$ satisfying:

$$f'(X_t)\sigma(X_t)=\text{constant}$$

gives a constant volatility for $$f(X_t)$$. Solving $$f$$ requires specifying $$\sigma(X_t)$$. For example, and as pointed out by Kermittfrog in the comments, when $$\sigma(X_t)=\sigma X_t$$, you can set $$f(X_t)=\log(X_t)$$.