We propose finite difference methods for degenerate fully nonlinear elliptic equations and prove the convergence of the schemes. Our focus is on the pure equation and a related free boundary problem of transmission type. The cornerstone of our argument is a regularisation procedure. It decouples the degeneracy term from the elliptic operator driving the diffusion process. In the free boundary setting, the absence of degenerate ellipticity entails new, genuine difficulties. To bypass them, we resort to the intrinsic properties of the regularised
problem. We present numerical experiments supporting our theoretical results. Our methods are flexible, and our approach can be extended to a broader class of non-variational problems.