Optimal Execution Using Limit Orders

Classic execution models like Almgren-Chriss assume every child order crosses the spread and fills for certain — the only question is how large and how fast to trade. That assumption misses an entire tool traders actually use: resting passive limit orders, which are cheaper when they fill (no spread paid, sometimes even a rebate) but might not fill at all. Optimal execution with limit orders extends the classic problem to include that uncertainty.

Adding a coin flip to the schedule

In the simplest execution models, a trader chooses a quantity to trade at each point in time and pays a market-impact cost proportional to how fast they trade. Once limit orders enter the picture, the trader also chooses a price to rest at, and that choice determines a fill probability rather than a guaranteed execution — an order resting at the current best price might fill soon with high probability, while one resting a few ticks better has a much lower chance of ever being reached (see how deep to place a limit order). The optimization becomes a balance of three things: the price improvement earned if the limit order fills, the risk of not filling and having to catch up later with a more expensive order, and the usual timing risk of the position sitting unexecuted while the market moves.

This is why these models typically end up with a reservation price: at each moment the algorithm has a threshold — quote at or better than this and you're better off waiting, but if the market moves past it, switch to an aggressive order and pay the spread rather than keep waiting for a fill that's becoming less likely to arrive in time.

Worked example: a threshold-based limit order strategy

A trader needs to buy 50,000 shares by end of day and is currently resting a limit order 1 tick behind the best bid. The model estimates fill probability over the next ten minutes at 40%, and if it doesn't fill, plenty of the day remains — so waiting is worthwhile, since the expected saving from a cheaper fill outweighs the modest risk of falling behind this early. Two hours before the close, with 20,000 shares still to buy, the same placement now carries real risk: if it doesn't fill in ten minutes, there may not be enough time to comfortably work the remainder passively. At this later point, the threshold logic switches the recommendation to cross the spread now for a guaranteed but slightly worse fill, rather than gamble on a passive fill running out of time to arrive.

What this means in practice

This class of model is the theoretical foundation behind adaptive execution algorithms that dynamically choose between passive and aggressive child orders rather than committing to one style (see passive versus aggressive placement). It's most valuable for names with meaningful spread to capture — on a one-tick spread stock, the saving from resting passively is small and rarely worth the fill-probability risk, while on a wider-spread name the saving can be substantial.