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TL;DR

Ito's Lemma on Geometric Brownian Motion: A canonical quantitative trading interview question at olympiad difficulty. Commonly asked at Two Sigma, Citadel, DE Shaw, Point72, Millennium.

By Valenke Exam Prep Team·Last updated 2026-06-01
olympiadStochastic Processes & Calculus

Ito's Lemma on Geometric Brownian Motion

Asked at: Two Sigma, Citadel, DE Shaw, Point72, Millennium

Problem
Let WtW_t be a standard Brownian motion. Let St=S0eσWt+(μ−σ2/2)tS_t = S_0 e^{\sigma W_t + (\mu - \sigma^22\frac{2}{2})t}. Using Ito's lemma, show that StS_t satisfies dSt=μSt dt+σSt dWtdS_t = \mu S_t \, dt + \sigma S_t \, dW_t. Why does the −σ2/2-\sigma^22\frac{2}{2} correction appear?
Related concepts
Ito LemmaBrownian Motion

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