  Based on a rough path foundation, we develop a model-free approach to
stochastic portfolio theory (SPT). Our approach allows to handle significantly
more general portfolios compared to previous model-free approaches based on
F{\"o}llmer integration. Without the assumption of any underlying probabilistic
model, we prove a pathwise formula for the relative wealth process which
reduces in the special case of functionally generated portfolios to a pathwise
version of the so-called master formula of classical SPT. We show that the
appropriately scaled asymptotic growth rate of a far reaching generalization of
Cover's universal portfolio based on controlled paths coincides with that of
the best retrospectively chosen portfolio within this class. We provide several
novel results concerning rough integration, and highlight the advantages of the
rough path approach by showing that (non-functionally generated) log-optimal
portfolios in an ergodic It{\^o} diffusion setting have the same asymptotic
growth rate as Cover's universal portfolio and the best retrospectively chosen
one.
