Before revision: We show that Lyons' rough path integral is a natural tool to use in model free financial mathematics by proving that it is possible to make an arbitrarily large profit by investing in those paths which do not have a rough path associated to them.
After revision: We show that Lyons' rough path integral is a natural tool to use in model free financial mathematics . The first one is in the spirit of It\^o's integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlled rough path integral. We prove that every "typical price path" has a naturally associated It\^o rough path, and justify the application of the controlled rough path associated to them.