  We propose new nonparametric estimators of the integrated volatility of an
It\^{o} semimartingale observed at discrete times on a fixed time interval with
mesh of the observation grid shrinking to zero. The proposed estimators achieve
the optimal rate and variance of estimating integrated volatility even in the
presence of infinite variation jumps when the latter are stochastic integrals
with respect to locally "stable" L\'{e}vy processes, that is, processes whose
L\'{e}vy measure around zero behaves like that of a stable process. On a first
step, we estimate locally volatility from the empirical characteristic function
of the increments of the process over blocks of shrinking length and then we
sum these estimates to form initial estimators of the integrated volatility.
The estimators contain bias when jumps of infinite variation are present, and
on a second step we estimate and remove this bias by using integrated
volatility estimators formed from the empirical characteristic function of the
high-frequency increments for different values of its argument. The second step
debiased estimators achieve efficiency and we derive a feasible central limit
theorem for them.
