Portfolio allocation with gross-exposure constraint is an effective method to
increase the efficiency and stability of selected portfolios among a vast pool
of assets, as demonstrated in Fan et al (2008). The required high-dimensional
volatility matrix can be estimated by using high frequency financial data. This
enables us to better adapt to the local volatilities and local correlations
among vast number of assets and to increase significantly the sample size for
estimating the volatility matrix. This paper studies the volatility matrix
estimation using high-dimensional high-frequency data from the perspective of
portfolio selection. Specifically, we propose the use of "pairwise-refresh
time" and "all-refresh time" methods proposed by Barndorff-Nielsen et al (2008)
for estimation of vast covariance matrix and compare their merits in the
portfolio selection. We also establish the concentration inequalities of the
estimates, which guarantee desirable properties of the estimated volatility
matrix in vast asset allocation with gross exposure constraints. Extensive
numerical studies are made via carefully designed simulations. Comparing with
the methods based on low frequency daily data, our methods can capture the most
recent trend of the time varying volatility and correlation, hence provide more
accurate guidance for the portfolio allocation in the next time period. The
advantage of using high-frequency data is significant in our simulation and
empirical studies, which consist of 50 simulated assets and 30 constituent
stocks of Dow Jones Industrial Average index.