  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 c\`adl\`ag functions possessing a mild restriction on the jumps
directed downwards. In particular, this result includes the existence of
quadratic variation of "typical price paths" in the space of non-negative
c\`adl\`ag paths and implies the existence of quadratic variation in the sense
of F\"ollmer quasi surely under all martingale measures. Based on the robust
existence of the quadratic variation, a model-free It\^o integration is
developed.
