Applying Ito's Lemma to Log-Price Process

This question sits squarely in stochastic calculus and SDEs, a core part of quant prep for pricing models and risk. You are given a diffusion for a price process and asked to write down the dynamics of its logarithm, then summarize them into a simple scalar expression. The setting is textbook quantitative finance: continuous-time modeling, Brownian motion, and transformations of state variables that traders and quants use every day.

It trains comfort with Ito's Lemma, especially recognizing how drift and diffusion change under nonlinear transformations such as logs. It also reinforces intuition for log-price dynamics, variance accumulation, and how volatility affects both the randomness and the deterministic trend. Under time pressure, it checks whether these ideas are internalized or still mechanical.

This matters for quant interviews because continuous-time modeling is foundational to derivatives pricing, portfolio construction, and signal design. Interviewers want to see that you can manipulate SDEs fluently, reason about log returns, and avoid sign or coefficient mistakes without relying on memorized formulas. Strong performance on such questions is a reliable indicator of readiness for more advanced quant interview challenges in stochastic calculus and asset dynamics.

What it tests

Whenever you have a stochastic process defined by a stochastic differential equation (SDE) and you want to find the dynamics of a function of that process, the key tool is Ito's Lemma. Ito's Lemma generalizes the chain rule to stochastic calculus by accounting for the additional variability introduced by the Brownian motion term. The crucial insight is that, unlike in classical calculus, the second derivative term contributes to the drift because the quadratic variation of Brownian motion is nonzero: $(dW_t)^2 = dt$. This means that when transforming variables (e.g., from $S_t$ to $\ln(S_t)$), the drift of the new process is not just the transformed drift, but also includes a correction from the volatility term. This correction is always half the second derivative of the function times the square of the volatility coefficient, and it arises because stochastic processes accumulate variance over time in a way that deterministic processes do not.