  We consider a multidimensional Ito semimartingale regularly sampled on [0,t]
at high frequency $1/\Delta_n$, with $\Delta_n$ going to zero. The goal of this
paper is to provide an estimator for the integral over [0,t] of a given
function of the volatility matrix. To approximate the integral, we simply use a
Riemann sum based on local estimators of the pointwise volatility. We show that
although the accuracy of the pointwise estimation is at most $\Delta_n^{1/4}$,
this procedure reaches the parametric rate $\Delta_n^{1/2}$, as it is usually
the case in integrated functionals estimation. After a suitable bias
correction, we obtain an unbiased central limit theorem for our estimator and
show that it is asymptotically efficient within some classes of sub models.
