The First Real Answer to How Fast Should I Trade: Bertsimas and Lo

Most finance theory quietly assumes you can buy or sell whatever you want at the price on the screen. Anyone who has actually tried to move a large position knows that is a fantasy. The price on the screen is the price for a small trade. Try to sell a hundred times that amount and the price runs away from you.

In 1998, Dimitris Bertsimas and Andrew Lo wrote the paper that took this seriously and asked a very concrete question: if I must trade a fixed number of shares within a fixed number of days, what is the trading schedule that costs me the least? They did not hand-wave. They set it up as a proper optimisation problem, solved it, and in doing so kicked off the entire academic field of optimal execution.

The problem: your own trading moves the price against you

Imagine you manage a fund and you need to unload a large block of stock. You have two bad options, and everything in between.

Dump it all right now. You get certainty. You also get a terrible price, because you are hoovering up every resting buy order in the book and the price sinks as you go. This is called market impact, and it is the single biggest hidden cost in institutional trading.
Trickle it out slowly over weeks. Each small piece barely moves the price, so your impact is tiny. But now you are exposed to the market for weeks. The stock might tank for reasons that have nothing to do with you, and you still hold most of your position.

So there is a genuine tension, and it is not obvious how to resolve it. Before this paper, traders resolved it by feel and by folklore. Bertsimas and Lo asked whether you could actually compute the answer.

The key idea via analogy: getting a car through a snowdrift

Think of a car stuck in deep snow. Slam the accelerator and the wheels spin, you dig yourself in, and you go nowhere. Nudge it gently and repeatedly and you get out, but it takes ages, and meanwhile the blizzard is getting worse.

The paper's insight is that this is a sequential decision problem, not a one-shot guess. You do not commit to a whole plan upfront and then close your eyes. Each period you look at where you are, how many shares you have left, how much time remains, and what the market just did, and then you decide how much to trade in the next slice.

The mathematical tool for this is dynamic programming, which is a fancy name for a simple idea: solve the problem backwards. Ask what you would do on the very last day, when you have no choice but to finish. Then, knowing that, ask what you would do on the second-to-last day. Keep walking backwards to today. Every decision is made knowing exactly how the rest of the game will play out.

Bertsimas and Lo cranked this handle and got an answer that, in its simplest setting, is almost comically plain: trade the same amount every period. If you have a million shares and five days, do two hundred thousand a day. That is it.

Why so boring? Because in the simple version, price impact is proportional to how fast you trade, and the total cost you pay is essentially the square of your trading rate added up across periods. The cheapest way to hit a fixed total with a squared cost is to spread it out perfectly evenly. Any lumpiness costs you extra. The maths formalises what an experienced trader would have told you over a pint, which is exactly what a good first model should do.

The twist: information changes the plan

The equal-slices answer only holds in the plain-vanilla setting. The paper's more interesting sections show what happens when you add realism.

When prices have predictable patterns. If the price has some momentum or mean reversion in it, the optimal schedule stops being flat. You lean into your trading when the price is temporarily in your favour and ease off when it is not.
When there is a useful signal. Suppose you can observe some other variable that helps forecast the stock. Now the strategy becomes genuinely dynamic: you speed up and slow down in response to what that signal says.
When you are trading a whole portfolio. Selling twenty correlated stocks at once is not the same as running twenty separate one-stock problems, because pushing one down drags its neighbours with it. The paper extends the framework to portfolios.

The general lesson is the important one: the more predictability you have, the more your trading schedule should depart from a dumb, even trickle. Without an edge, spread it out. With an edge, be opportunistic.

Why it mattered

Before this paper, execution was a craft. After it, it was a research field.

It made execution cost a thing you optimise, not just a thing you complain about. Once you can write down the cost, you can minimise it, and once you can minimise it, you can build software that does the minimising. Every execution algorithm sold by every broker traces its intellectual lineage back through here.
It set up the framework everyone else built on. The famous Almgren-Chriss model, which added risk aversion and gave traders the ability to say "I want to finish faster because I am nervous," is a direct descendant. Bertsimas and Lo minimised expected cost. Almgren and Chriss said, correctly, that you also care about the variance of that cost.
It legitimised the whole idea that trading is a control problem. Deciding how much to trade right now, based on where you are and how much time is left, is exactly the structure of a control problem. That framing now dominates the field.

The honest limitations

It minimises expected cost and ignores risk. This is the big one. The model finds the schedule with the lowest average cost, but it says nothing about the spread of possible outcomes. A trader who wants to be done early because the world feels fragile gets no help here. Almgren and Chriss fixed exactly this hole three years later.
The linear impact assumption is a simplification. The model leans on the idea that trading twice as fast pushes the price twice as far. Reams of later empirical work suggest the real relationship is closer to a square root, meaning impact grows more slowly than linearly for big orders. That changes the answer.
Real order books do not sit still and wait. The paper's price impact is a fairly abstract object. It does not model the actual mechanics of a limit order book refilling itself after you have eaten through it. Obizhaeva and Wang later showed that when you model that refilling explicitly, the optimal strategy stops being a smooth trickle and starts looking like a big trade, a slow drip, and a final big trade.
You need to know the parameters. The pretty closed-form answers depend on impact coefficients you have to estimate from data, and those estimates are noisy and drift over time. An optimal schedule computed from wrong numbers is not optimal.

The one-line takeaway

Bertsimas and Lo turned "how fast should I trade this block?" from a matter of trader instinct into a solvable control problem, and showed that with no special information, the answer is to spread the order evenly over your window, while any real predictive signal should make you trade opportunistically instead.