  This paper addresses the optimal scheduling of the liquidation of a portfolio
using a new angle. Instead of focusing only on the scheduling aspect like
Almgren and Chriss, or only on the liquidity-consuming orders like Obizhaeva
and Wang, we link the optimal trade-schedule to the price of the limit orders
that have to be sent to the limit order book to optimally liquidate a
portfolio. Most practitioners address these two issues separately: they compute
an optimal trading curve and they then send orders to the markets to try to
follow it. The results obtained here solve simultaneously the two problems. As
in a previous paper that solved the "intra-day market making problem", the
interactions of limit orders with the market are modeled via a Poisson process
pegged to a diffusive "fair price" and a Hamilton-Jacobi-Bellman equation is
used to solve the problem involving both non-execution risk and price risk.
Backtests are carried out to exemplify the use of our results, both on long
periods of time (for the entire liquidation process) and on slices of 5 minutes
(to follow a given trading curve).
