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TL;DR

Risk-Neutral Pricing: Expectation Under Q: A canonical quantitative trading interview question at olympiad difficulty. Commonly asked at Citadel, Two Sigma, DE Shaw, Point72, Millennium.

By Valenke Exam Prep Team·Last updated 2026-06-01
olympiadGame Theory & Strategy

Risk-Neutral Pricing: Expectation Under Q

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

Problem
A stock follows geometric Brownian motion dSt=μStdt+σStdWtdS_t = \mu S_t \, dt + \sigma S_t \, dW_t with S0=100S_0 = 100, μ=0.12\mu = 0.12, σ=0.30\sigma = 0.30, and risk-free rate r=0.04r = 0.04. A derivative pays ln(ST/S0)\ln(S_T / S_0) at time T=1T = 1. (a) What is the price of this derivative using risk-neutral pricing? (b) Why does μ\mu not appear in the answer? (c) What is EP[ln(ST/S0)]E^{\mathbb{P}}[\ln(S_T/S_0)] under the real-world measure? Why is it different?

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