We deal with chance constrained problems (CCP) with differentiable nonlinear, possibly nonconvex, random functions and finite discrete distribution. First, we reformulate the problem using binary variables and discuss the drawbacks of this approach. By relaxing the binary variables we arrive at a nonlinear programming problem. We approach it as a mathematical program with complementarity constraints and discuss its relations to the original chance constrained problem with a special focus on the stationary points and (local) minima. Then, we regularize the relaxed problem by enlarging the set of feasible solutions using a regularization function. We derive necessary optimality conditions corresponding to the strong stationarity and discuss the convergence issues. We provide a numerical experiment comparing our approach with the integer programming one. We also discuss an application to gas network optimization.