Robust portfolio optimization aims to maximize the worst-case portfolio return given that the asset returns are allowed to vary within a prescribed uncertainty set. If the uncertainty set is not too large, the resulting portfolio performs well under normal market conditions. However, its performance may substantially degrade in the presence of market crashes, that is, if the asset returns materialize far outside of the uncertainty set. We propose a novel robust optimization model for designing portfolios that include European-style options. This model trades off weak and strong guarantees on the worst-case portfolio return. The weak guarantee applies as long as the asset returns are realized within the prescribed uncertainty set, while the strong guarantee applies for all possible asset returns. The resulting model constitutes a convex second-order cone program, which is amenable to efficient numerical solution procedures. We evaluate the model using simulated and empirical backtests and analyze the impact of the insurance guarantees on the portfolio performance.