= Solution
The <moment-generating function of a normal distribution> gives
$$
\mathbb E e^{\lambda X_1}=\exp\left(-\lambda+\frac{\lambda^2}{2}\right).
$$
Independence of $X_{n+1}$ from $\mathcal F_n$ implies
$$
\mathbb E[e^{\lambda S_{n+1}}\mid\mathcal F_n]=e^{\lambda S_n}\exp\left(-\lambda+\frac{\lambda^2}{2}\right).
$$
Every exponential here is integrable. Thus the process is a <martingale> exactly when $-\lambda+\lambda^2/2=0$. Its two roots are $0$ and $2$, and the unique positive root is
$$
\boxed{\lambda=2.}
$$
The resulting <exponential martingale of a random walk> is nonnegative and has <expectation> $1$ at every finite time.
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