We propose stochastic programming formulations to enforce mechanical load requirements in wind turbine controller design procedures. The formulations use statistical extrapolation techniques to construct a probabilistic (chance) constraint that controls the long-term probability of exceeding an extreme load threshold (as described by the IEC-61400 standard). This approach is based on the observation that extreme loads follow a generalized extreme value distribution, which enables an explicit algebraic representation of the probabilistic constraint. We illustrate how to use the formulations to find design parameters for pitch angle and torque controllers that maximize power output while constraining long-term extreme loads. We also use the formulation to explore the ability of a hypothetical model predictive controller to mitigate extreme loads. The proposed formulations can be cast as large-scale (but structured) nonlinear programming problems that contain up to 7.5 million variables and constraints. We show that these problems can be solved in less than 1.3 h on a multi-core computer with existing optimization tools.