We present a non-probabilistic, pathwise approach to continuous-time finance based on causal functional calculus. We introduce a definition of self-financing, free from any integration concept and show that the value of a self-financing portfolio is a pathwise integral (every self-financing strategy is \emph{always} a gradient!) and that generic domain of functional calculus is inherently arbitrage-free. We then consider the problem of hedging a path-dependent payoff across a generic set of scenarios. We apply the transition principle of Isaacs in optimal control and obtain a verification theorem for the solution, which is characterised by a (fully non-linear) path-dependent equation. For the Asian option, we obtain explicit solution.