Using the Lamperti Transform to Make Diffusion Volatility Constant

Summary

The document explains how to transform a one-dimensional diffusion so its stochastic term has constant volatility. Starting from an SDE with state-dependent volatility, Itô’s lemma shows that applying a function multiplies the diffusion coefficient by the function’s derivative. Choosing that derivative inversely proportional to the original volatility makes the transformed coefficient constant; integrating this relationship gives the transform, subject to the volatility function and domain.

The named method is the Lamperti transform. The document gives a geometric Brownian motion example: when volatility is proportional to the process level, taking the logarithm produces constant diffusion volatility. It provides the transformation principle and an example, but no derivation of the resulting drift, boundary conditions, or discussion of whether the transform is globally well defined. Those details depend on the model.

Key ideas

• Itô’s lemma shows that transforming a diffusion scales its stochastic term by the derivative of the transformation.
• A constant diffusion coefficient results when that derivative is inversely proportional to the original volatility.
• The resulting transformation is known as the Lamperti transform.
• For volatility proportional to the process level, a logarithmic transformation yields constant volatility.

Tags

# Transformation of local volatility model


# 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?

## Answer by user34971 (score 6, accepted)

https://quant.stackexchange.com/a/70353

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

## Answer by fes (score 5)

https://quant.stackexchange.com/a/70352

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 $f(X_t)=\log(X_t)$.

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