  The volume weighted average price (VWAP) execution strategy is well known and
widely used in practice. In this study, we explicitly introduce a trading
volume process into the Almgren-Chriss model, which is a standard model for
optimal execution. We then show that the VWAP strategy is the optimal execution
strategy for a risk-neutral trader. Moreover, we examine the case of a
risk-averse trader and derive the first-order asymptotic expansion of the
optimal strategy for a mean-variance optimization problem.
