# How is Itô's Lemma connected to Messmore's Variance Drain?

How does Itô's Lemma explain the concept of volatility drain in investment returns, and how do the associated equations illustrate this effect? I did the following considerations so far:

In financial mathematics, Itô's Lemma for a function $$f(t, S_t)$$ of a stochastic process $$S_t$$ modeled as a geometric Brownian motion is given by:

$$df(t, S_t) = \left( \frac{\partial f}{\partial t} + \mu S_t \frac{\partial f}{\partial S} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 f}{\partial S^2} \right) dt + \sigma S_t \frac{\partial f}{\partial S} dW_t$$

where $$S_t$$ represents the asset price, $$\mu$$ is the drift rate, $$\sigma$$ is the volatility, and $$W_t$$ is a Wiener process. Applying this to the geometric Brownian motion

$$dS_t = \mu S_t dt + \sigma S_t dW_t$$

Thus the expected logarithmic return, which corresponds to the geometric mean return(?), is:

$$\mathbb{E}[\ln(S_t)] = \left( \mu - \frac{1}{2} \sigma^2 \right) t$$

This indicates that the effective growth rate (geometric mean) is reduced by $$\frac{1}{2} \sigma^2$$, illustrating the volatility drain effect. I would therefore assume that Itô's Lemma is kind of a generalization and fundamental concept explaining the more specific variance drain.

• You may find this link useful: quant.stackexchange.com/questions/79131/… Commented Jun 11 at 12:10
• The similar expression does come up in stochastic calculus but, as far as I know, variance drain is only due to taking the log so really nothing to do with stochastic calculus except for the fact that in stochastic calculus, you also have a lognormal rv. See this for lots of gory details. ideas.repec.org/p/inu/caeprp/2012004.html Commented Jun 11 at 17:17