The problem of integrated volatility estimation for the solution X of a stochastic differential equation with Lévy-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 β > 1 2(2−α) on β (that is such that (1/n) β is the threshold of the truncated quadratic variation) and on the degree of jump activity α 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 (α, β). Lévy-driven SDE, integrated variance, threshold estimator, convergence speed, high frequency data.