Before revision:  We also show that in certain situations, the rough path integral can be constructed as a limit of Riemann sums, and not just as a limit of compensated Riemann sums which are usually used to define it. This proves that the rough path integral is really an extension of F\"ollmer's pathwise It\^o integral. Moreover, we construct a "model free It\^o integral" in the spirit of Karandikar .
After revision:  We also show that in certain situations, the rough path integral can be constructed as a limit of Riemann sums, a new result in itself. Compared to the first approach, rough paths have the disadvantage of severely restricting the space of integrands, but the advantage of being a Banach space theory. Both approaches are based entirely on financial arguments and do not require any probabilistic structure .