= Solution
Define a new probability measure by the strictly positive <Radon-Nikodym derivative>
$$
\frac{d\mathbb Q}{d\mathbb P}
=\exp\!\left(\widetilde\sigma Z-\frac12\widetilde\sigma^2\right),
\qquad
\widetilde\sigma=-\frac{m+\sigma^2/2}{\sigma}.
$$
The <moment-generating function of a normal distribution> shows that this density has expectation one, so $\mathbb Q$ is a probability measure equivalent to $\mathbb P$. Moreover,
$$
\begin{aligned}
\mathbb E_{\mathbb Q}[S_1^1]
&=\mathbb E_{\mathbb P}\!\left[
e^{\widetilde\sigma Z-\widetilde\sigma^2/2}
e^{\sigma Z+m}\right]\\
&=\exp\!\left(m+\frac{\sigma^2}{2}
+\sigma\widetilde\sigma\right)=1=S_0^1.
\end{aligned}
$$
\b[Thus the discounted risky asset is a one-period $\mathbb Q$-martingale. The <fundamental theorem of asset pricing> implies that there is no arbitrage.]
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