Itô's Lemma and the Volatility Drag on Geometric Growth

Summary

The document connects Itô's Lemma to volatility drag in a geometric Brownian motion model of an asset price. It states the stochastic differential equation for the price and applies the logarithm transformation to derive the expected log price growth. Under the stated model, this growth rate equals the drift less one half of the variance rate, showing how volatility lowers compound growth relative to arithmetic drift.

The equations offer a concise mathematical explanation of the effect and identify Itô's Lemma as the mechanism behind the correction term. The post frames the relationship as a question and does not discuss alternative return measures, estimation, or empirical evidence. Its result depends on the geometric Brownian motion assumptions and should not be read as a complete model of realized investment returns in markets with changing drift or volatility.

Key ideas

• For a geometric Brownian motion, applying Itô's Lemma to the logarithm of price yields a variance correction.
• Expected log growth is the drift rate minus one half of the variance rate.
• The volatility correction explains why compound growth can trail arithmetic drift.
• The stated relationship relies on the geometric Brownian motion model.

Tags

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


# 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.