In this talk I will present quantitative convergence results for a class of mean field games with common noise and controlled volatility. The basic strategy we employ is the one introduced recently by Laurière and myself – roughly speaking, we use a synchronous coupling argument to prove a “forward-backward propagation of chaos” result for the FBSDEs which characterize the (open-loop) equilibria of the N-player and mean field games. Unlike in earlier works which have adopted this strategy, we do not require smallness conditions, and instead rely on monotonicity. In particular, (displacement) monotonicity of the Hamiltonian and the terminal cost allow us to establish a (uniform in N) stability estimate for the N-player FBSDEs, which implies the convergence result. The arguments are relatively simple, and flexible enough to yield similar results in the setting of mean field control and infinite horizon (discounted) mean field games.