  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 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 differential games and obtain a verification theorem for the optimal
solution, which is characterised by a fully non-linear path-dependent equation.
For the Asian option, we obtain explicit solution.
