When studying stochastic processes, it is often fruitful to understand several different notions of regularity. One such notion is the optimal H?lder exponent obtainable under reparametrization. In this paper, we show that chordal $mathrm{SLE}_kappa$ in the unit disk for $kappa le 4$ can be reparametrized to be H?lder continuous of any order up to $1/(1+kappa/8)$.From this, we obtain that the Young integral is well defined along such $mathrm{SLE}_kappa$ paths with probability one, and hence that $mathrm{SLE}_kappa$ admits a path-wise notion of integration. This allows us to consider the expected signature of $mathrm{SLE}$, as defined in rough path theory, and to give a precise formula for its first three gradings.The main technical result required is a uniform bound on the probability that an $mathrm{SLE}_kappa$ crosses an annulus $k$-distinct times.
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