Even in simple models in which the volatility is only known to stay in two bounds, it is quite hard to price and hedge derivatives which are not Markovian. The main reason for this difficulty emanates from the fact that the probability measures are singular to each other. In this talk we will prove a martingale representation theorem for this market. This result provides a complete answer to the questions of hedging and pricing. The main tools are the theory of nonlinear G-expectations as developed by Peng, the quasi-sure stochastic analysis of Denis & Martini and the second order backward stochastic differential equations. This is joint work cole Polytechnique and Jianfeng Zhang from University of Southern California.