##### 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?

This question evaluates understanding of stochastic processes, simple symmetric random walks, expectation and variance computations, and the discrete-to-continuous connection to Brownian motion.

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This is a medium difficulty Statistics & Math question, commonly asked during Technical Screen rounds at Citadel.

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This question is commonly asked for Data Scientist candidates at Citadel during technical interviews.

Compute variance of trading profits | Citadel Interview Question

Symmetric Random Walk Trading Strategy: Profit Variance and Expectation

Setup

Let S_t be a simple symmetric random walk with S_0 = 0 and increments X_t = S_t − S_{t−1} taking values ±1 with equal probability, independent across t.
Time is discrete: t = 1, 2, ..., T.

Trading Rule

At each time t (after observing the move from t−1 to t):

If X_t = +1, buy 1 share.
If X_t = −1, short 1 share.

You accumulate all positions and mark them to market at the terminal time T.

Tasks

Derive a closed form for the cumulative profit at time T and compute its variance.
What is the expected profit? Explain.
How do these results extend if S_t is replaced by continuous-time standard Brownian motion B_t?

Symmetric Random Walk Trading Strategy: Profit Variance and Expectation

Setup

Let S_t be a simple symmetric random walk with S_0 = 0 and increments X_t = S_t − S_{t−1} taking values ±1 with equal probability, independent across t.
Time is discrete: t = 1, 2, ..., T.

Trading Rule

At each time t (after observing the move from t−1 to t):

If X_t = +1, buy 1 share.
If X_t = −1, short 1 share.

You accumulate all positions and mark them to market at the terminal time T.

Tasks

Derive a closed form for the cumulative profit at time T and compute its variance.
What is the expected profit? Explain.
How do these results extend if S_t is replaced by continuous-time standard Brownian motion B_t?

Compute variance of trading profits

Quick Overview

Symmetric Random Walk Trading Strategy: Profit Variance and Expectation

Setup

Trading Rule

Tasks

Solution

Comments (0)

Compute variance of trading profits

Quick Overview

Symmetric Random Walk Trading Strategy: Profit Variance and Expectation

Setup

Trading Rule

Tasks

Solution

Comments (0)