Model-free portfolio theory and its functional master formula

We use pathwise It\^o calculus to prove two strictly pathwise versions of the master formula in Fernholz' stochastic portfolio theory. Our first version is set within the framework of F\"ollmer's pathwise It\^o calculus and works for portfolios generated from functions that may depend on the current states of the market portfolio and an additional path of finite variation. The second version is formulated within the functional pathwise It\^o calculus of Dupire (2009) and Cont \& Fourni\'e (2010) and allows for portfolio-generating functionals that may depend additionally on the entire path of the market portfolio. Our results are illustrated by several examples and shown to work on empirical market data.