##### 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?