  In this paper, we develop a penalized realized variance (PRV) estimator of
the quadratic variation (QV) of a high-dimensional continuous It\^{o}
semimartingale. We adapt the principle idea of regularization from linear
regression to covariance estimation in a continuous-time high-frequency
setting. We show that under a nuclear norm penalization, the PRV is computed by
soft-thresholding the eigenvalues of realized variance (RV). It therefore
encourages sparsity of singular values or, equivalently, low rank of the
solution. We prove our estimator is minimax optimal up to a logarithmic factor.
We derive a concentration inequality, which reveals that the rank of PRV is --
with a high probability -- the number of non-negligible eigenvalues of the QV.
Moreover, we also provide the associated non-asymptotic analysis for the spot
variance. We suggest an intuitive data-driven bootstrap procedure to select the
shrinkage parameter. Our theory is supplemented by a simulation study and an
empirical application. The PRV detects about three-five factors in the equity
market, with a notable rank decrease during times of distress in financial
markets. This is consistent with most standard asset pricing models, where a
limited amount of systematic factors driving the cross-section of stock returns
are perturbed by idiosyncratic errors, rendering the QV -- and also RV -- of
full rank.
