Energy arbitrage has the potential to make electric grids more efficient and reliable. Batteries hold great promise for energy storage in arbitrage but can degrade rapidly with use. In this paper, we analyze the impact of storage degradation on the structure of optimal policies in energy arbitrage. We derive properties of the battery degradation response that are sufficient for the existence of optimal threshold policies, which are easy to interpret and compute. Our experimental results suggest that explicitly considering battery degradation in optimizing energy arbitrage significantly improves solution quality.