Exponential moment bound for a Gaussian random-walk maximum ($\mathbb{E}{e}^{bZ}\le 2/(2-b),0<b<2$)

For independent and identically distributed random variables with law $N(-1,1)$, $e^{2S_n}$ is a nonnegative martingale, since the increment's moment-generating function equals one at parameter two. The Doob maximal inequality gives $\mathbb{P}(Z>t)\le e^{-2t}$ for the all-time maximum $Z\ge 0$. Integrating the tail gives the displayed bound. For $b\le 0$, $e^{bZ}\le 1$. The strong law of large numbers guarantees finiteness and attainment of the maximum because $S_n\to -\infty$.