Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of non-negative c\`adl\`ag functions. In particular, this implies the existence of quadratic variation in the sense of F\"ollmer quasi surely under all martingale measures. Based on the robust existence of quadratic variation and a certain topology which is induced by Vovk's outer measure, model-free It\^o integration is developed on the space of continuous paths, of non-negative c\`adl\`ag paths and of c\`adl\`ag paths with mildly restricted jumps.
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