  The problem of integrated volatility estimation for the solution X of a
stochastic differential equation with L{\'e}vy-type jumps is considered under
discrete high-frequency observations in both short and long time horizon. We
provide an asymptotic expansion for the integrated volatility that gives us, in
detail, the contribution deriving from the jump part. The knowledge of such a
contribution allows us to build an unbiased version of the truncated quadratic
variation, in which the bias is visibly reduced. In earlier results the
condition $\beta$ > 1 2(2--$\alpha$) on $\beta$ (that is such that (1/n)
$\beta$ is the threshold of the truncated quadratic variation) and on the
degree of jump activity $\alpha$ was needed to have the original truncated
realized volatility well-performed (see [22], [13]). In this paper we
theoretically relax this condition and we show that our unbiased estimator
achieves excellent numerical results for any couple ($\alpha$, $\beta$).
L{\'e}vy-driven SDE, integrated variance, threshold estimator, convergence
speed, high frequency data.
