  We consider specification and inference for the stochastic scale of
discretely-observed pure-jump semimartingales with locally stable L\'{e}vy
densities in the setting where both the time span of the data set increases,
and the mesh of the observation grid decreases. The estimation is based on
constructing a nonparametric estimate for the empirical Laplace transform of
the stochastic scale over a given interval of time by aggregating
high-frequency increments of the observed process on that time interval into a
statistic we call realized Laplace transform. The realized Laplace transform
depends on the activity of the driving pure-jump martingale, and we consider
both cases when the latter is known or has to be inferred from the data.
