The Bertsimas-Lo Execution Problem

If you need to sell a million shares, dumping them all at once crashes the price you get, but trading too slowly leaves you exposed to the market moving against you for longer than necessary. Bertsimas and Lo were among the first to turn "how should I split this order up over time" into a formal optimisation problem with an actual answer, rather than a rule of thumb.

Their setup: prices follow a random walk that also gets pushed by your own trading (a linear price-impact term), and the goal is to choose how many shares to trade in each of several time periods to minimise the expected total cost of executing the whole order, subject to finishing by a deadline. Because the price-impact cost is convex — trading twice as fast costs more than twice as much in impact — the optimal solution spreads the order out rather than front-loading or back-loading it. In the simplest version of their model, with impact that doesn't depend on time and no extra information, the optimal strategy turns out to be trading equal amounts in each period, which is the theoretical justification behind the naive but common TWAP (time-weighted average price) execution schedule.

The model's real contribution was showing how the optimal schedule changes once you add more realistic ingredients — impact that decays over time, or a linear risk-aversion term penalising the variance of execution cost — at which point the flat TWAP schedule stops being optimal and the algorithm should trade faster when urgency or risk aversion is higher.