Let $B(t)$ be a standard BM. Consider the stopping time $T = \inf\{ t > 0: |B(t)| = a\},$ the usual first exit time of the interval $(-a, a).$
We can see that $\mathbb{E} e^{tT} < \infty$ for $t \in (0, \rho)$, for some positive $\rho$, by noticing $\mathbb{P}(T > n) \leq \mathbb{P}(T > n-1)\mathbb{P}(|B(n) - B(n-1)| \leq 2)$ and recursively stepping through $n$. We also can see that $\rho$ can be taken to be $-\ln(\mathbb{P}(|B(t)| \leq 2))$.
Now we know from other theory that $\mathbb{E} e^{tT} = \frac{1}{\cosh(a\sqrt{-2t})}$ for $t <0.$ Is there a nice closed form for positive $t$ as well, since we know it's finite. Is the smallest such $\rho$ known?