Expected time to reach plus or minus one

Question

A standard Brownian motion starts at $0$ . What is the expected time until it first reaches either $+1$ or $-1$ ?

Short answer

$1$ . $W_t^2-t$ is a martingale, so optional stopping at the exit time $\tau$ gives $\mathbb{E}[W_\tau^2]=\mathbb{E}[\tau]$ . Since $W_\tau^2=1$ , $\mathbb{E}[\tau]=1$ (in general, exiting $(-a,b)$ takes $ab$ on average).

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 5.4 Brownian Motion and Stochastic Calculus, 'Stopping time A', p.130; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

The Optional Stopping Theorem
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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