The Order Book Refills: Obizhaeva and Wang on Supply and Demand Dynamics

The first generation of optimal execution models, from Bertsimas and Lo through Almgren and Chriss, all shared a quiet assumption: the cost of trading depends on how fast you are trading right now. Trade quickly, pay a lot. Trade slowly, pay a little. Simple, intuitive, and, as Anna Obizhaeva and Jiang Wang pointed out, missing something important.

Their paper asks a question the earlier models could not answer: where does liquidity actually come from, and how long does it take to come back after you have used it up?

The problem: liquidity is a stock, not a flow

Picture the order book, the queue of resting buy and sell orders sitting at the exchange. When you send a big sell order, you eat through the resting buy orders, starting with the best-priced ones and working down. The price drops. That is the mechanical part.

Now here is the bit the earlier models glossed over. The book does not instantly reappear. After you have chewed a hole in it, new buyers have to notice, decide the price is now attractive, and send fresh orders. That takes time. Seconds, minutes, sometimes longer.

This means liquidity behaves like water in a well, not water from a tap. A tap gives you a steady flow: your only choice is how wide to open it. A well has a finite amount right now, and it refills at its own pace once you stop drawing from it. If you understand that difference, your strategy for getting water out of a well should look nothing like your strategy for a tap.

The old models effectively assumed a tap. Obizhaeva and Wang said: it is a well.

The key idea via analogy: the well and the spring

So you have a well that refills at some rate, and you need a large volume of water by sundown.

What is the smart way to do it? It has three phases, and once you see it you cannot unsee it.

Take a big scoop right at the start. The well is currently full, brimming with liquidity that has accumulated while you were doing nothing. That water is just sitting there. Grab it. In market terms: your first trade should be a large discrete block, because the order book in front of you right now is the deepest it will ever be.
Then drip continuously, matched to the refill rate. Having drawn the well down, you now want to take water roughly as fast as it seeps back in. Take more and you are drilling into an empty well, paying a lot for very little. Take less and you are wasting time. In market terms: a steady continuous stream of small trades that skims off the new liquidity as fresh buyers arrive, without pushing the price further down.
At the very end, take one final big scoop. Sundown is coming and leftover water in the well is worthless to you. There is no future to preserve liquidity for. So on the last stroke you dump whatever remains of your order into the market and accept the impact. In market terms: a final discrete block trade.

That is the shape of the Obizhaeva-Wang solution: block, drip, block. It is a striking departure from the smooth, gentle curves the earlier literature produced, and it comes entirely from taking seriously the idea that the book takes time to recover.

There is a second, subtler point buried in that first block. Pushing the book away from its resting state is not purely a cost. It is also a signal. A depressed price is an attractive price, and attractive prices draw in new liquidity providers who would not otherwise have shown up. Part of what your opening block is doing is advertising: it tells the market there is size available here, and it pulls in the counterparties you need for the rest of your execution.

Why it mattered

It put the order book back into optimal execution. Earlier models had an abstract, almost metaphorical notion of price impact. This paper grounds it in the actual thing that happens: a book gets consumed and then refills. That made the whole field more physical and more testable.
It introduced transient impact properly. The idea that impact decays over time, that the price partially bounces back after you stop pushing, became central. It is the reason later authors like Gatheral had to go and check which combinations of impact shape and decay shape are even internally consistent.
It showed that the smooth answer is fragile. The comforting result that you should just trickle evenly turns out to depend heavily on your assumptions. Change the model of where liquidity comes from and the optimal strategy changes character entirely, from smooth to lumpy. That was a useful shock to the system.
It matched what good traders already did. Experienced execution traders have always known there is value in taking the size that is sitting there when you arrive, and in accepting a hit at the close rather than carrying a residual overnight. The paper gave that intuition a rigorous backbone.

The honest limitations

The block trades are an artefact of a smooth model. Those instantaneous jumps at the start and end come out of a model with no fixed costs, no minimum tick, and no strategic counterparties. In a real market, firing a huge block into the book is exactly the sort of behaviour that announces "there is a big seller here," inviting other participants to trade ahead of you. The model has no predators in it.
The refill rate is treated as a mechanical constant. Real liquidity resupply is not a steady exponential drip. It speeds up and slows down, it depends on news, on the time of day, and, awkwardly, on whether other participants think you are informed. Modelling it as a fixed physical constant is a large simplification.
Impact is assumed linear in the amount you trade. As with the earlier literature, the model leans on a linear relationship between size and price move. A substantial body of empirical work, from Lillo and colleagues to the square-root law crowd, argues that reality is meaningfully non-linear.
It is a single-asset, no-information world. You are not trading because you know something, you are just trading. Real large orders usually exist because someone has a view, and the market spends its time trying to work out whether you do.

The one-line takeaway

Obizhaeva and Wang showed that liquidity is a resource that regenerates rather than a toll you pay per share, and once you model the refilling honestly, the best way to work a large order is not a smooth trickle but a big opening block, a steady drip matched to the book's recovery, and a final clean-up block at the end.