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.

  • $\begingroup$ You may find this link useful: quant.stackexchange.com/questions/79131/… $\endgroup$
    – KaiSqDist
    Commented Jun 11 at 12:10
  • $\begingroup$ 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 $\endgroup$
    – mark leeds
    Commented Jun 11 at 17:17


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