  We decompose, within an ARCH framework, the daily volatility of stocks into
overnight and intra-day contributions. We find, as perhaps expected, that the
overnight and intra-day returns behave completely differently. For example,
while past intra-day returns affect equally the future intra-day and overnight
volatilities, past overnight returns have a weak effect on future intra-day
volatilities (except for the very next one) but impact substantially future
overnight volatilities. The exogenous component of overnight volatilities is
found to be close to zero, which means that the lion's share of overnight
volatility comes from feedback effects. The residual kurtosis of returns is
small for intra-day returns but infinite for overnight returns. We provide a
plausible interpretation for these findings, and show that our
Intra-Day/Overnight model significantly outperforms the standard ARCH framework
based on daily returns for Out-of-Sample predictions.
