Ito's Lemma and the Log-Normal Stock

Applying Itô's Lemma to log-normal stock-price dynamics

This medium-difficulty question tests your ability to apply Itô's Lemma—a cornerstone tool in quantitative finance—to transform a stochastic differential equation. It appears regularly in quant trader and researcher interviews because it bridges theoretical probability and practical pricing models.

The question asks you to work with Geometric Brownian Motion, the standard model for stock prices in continuous-time finance, and derive the SDE governing the logarithm of the stock. This requires setting up the lemma correctly, computing partial derivatives, and carefully handling the quadratic-variation term that arises from the nonlinear transformation. The key challenge is remembering that when you differentiate a nonlinear function of a process with both drift and diffusion, you must account for the interaction between the diffusion term and itself—a step that separates clean answers from plausible-looking wrong ones.

• Itô's Lemma for univariate and multivariate processes
• Quadratic variation and the rule
• Drift adjustment under change of variables
• Risk-neutral pricing and the role of the drift coefficient