Before revision:  We encode volatility in an enhancement of the price trajectory and we give pathwise presentations of the fundamental equations of Mathematical Finance. In particular this allows us to assess model misspecification, generalising the so-called fundamental theorem of derivative trading (see Ellersgaard et al. 2017).
After revision:  We encode volatility in an enhancement of the price trajectory , we give a pathwise presentation of the replication of European options. The continuity properties of rough-paths allow us to generalise the so-called fundamental theorem of derivative trading (see Ellersgaard et al. 2017).