Abstract: We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present several applications and extensions of the above results in decision theory, mathematical finance and beyond. We consider in particular robustness of call option pricing and deduce a new Black-Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and examine in detail optimal investment, utility indifference pricing and Davis’ option pricing in a simple one-period model. Based on joint works with Daniel Bartl, Samuel Drapeau and Johannes Wiesel.