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Compute variance of trading profits

CitadelData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

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This interview question evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer for Compute variance of trading profits states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

Question

Consider a stock price that starts at 0 and evolves as a simple symmetric random walk, moving +1 or -1 each discrete time step. At each time T you:

Buy one share if the price change from T−1 to T is +1.

Short (sell) one share if the price change is −1.

Assuming you can accumulate multiple positions purchased at different prices, derive the variance of the cumulative profit of your holdings at time T.

Follow-up: What is the expected profit and why? How would the answer extend if the random walk is replaced by continuous-time Brownian motion?