 Solution

Define a new probability measure by the strictly positive Radon-Nikodym derivativeThe 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,Thus the discounted risky asset is a one-period $\mathbb{Q}$-martingale. The fundamental theorem of asset pricing implies that there is no arbitrage.