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TL;DR

Itô's Lemma Applied to Geometric Brownian Motion: A canonical quantitative trading interview question at intermediate difficulty. Commonly asked at Two Sigma, DE Shaw, Citadel, Jump Trading.

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

Itô's Lemma Applied to Geometric Brownian Motion

Asked at: Two Sigma, DE Shaw, Citadel, Jump Trading

Problem
Let StS_t follow geometric Brownian motion: dSt=μSt dt+σSt dWtdS_t = \mu S_t \, dt + \sigma S_t \, dW_t. Using Itô's lemma, find the SDE for Yt=ln⁡StY_t = \ln S_t. Hence show that ST=S0exp⁡((μ−σ2/2)T+σWT)S_T = S_0 \exp\left((\mu - \sigma^22\frac{2}{2})T + \sigma W_T\right).
Related concepts
Ito CalculusGeometric Brownian Motion

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