Any Shape of Order Book: Alfonsi, Fruth and Schied on Optimal Execution

Obizhaeva and Wang made a genuine advance by putting an actual order book inside the optimal execution problem. Liquidity is not a mysterious cost function, they said. It is a queue of resting orders that you eat through, and that then refills over time.

But to make the maths work, they made a simplification: they assumed the order book was flat. A uniform slab of liquidity, the same amount of size waiting at every price level, stretching away from the current price.

Real order books are not flat. They are lumpy, sparse near the touch, denser at round numbers, and shaped quite differently for different assets. Aurelien Alfonsi, Antje Fruth and Alexander Schied asked whether the theory survives when you let the book have any shape at all, and the answer turned out to be more interesting than a simple yes.

The problem: shape determines cost

The shape of the order book is not a cosmetic detail. It is the thing that determines what your trade costs.

Think about what happens when you send a large buy order. You consume the resting sell orders, starting at the best ask and walking up. How far up you have to walk depends entirely on how much size is sitting at each level.

• If the book is dense right above the touch, you fill most of your order without moving far. Cheap.
• If the book is thin near the touch and dense further out, your first shares fill immediately but then you punch through a vacuum and the price gaps before you find the next pocket of liquidity. Expensive, and non-linear in a way a flat book cannot capture.

The flat-book assumption effectively says impact is linear in the amount you consume: every additional share costs a little more than the last, by exactly the same increment. That is a strong claim, and it is contradicted by every empirical study of impact ever done.

So the question is whether the elegant Obizhaeva-Wang answer, the famous block-then-drip-then-block strategy, depends on the book being flat, or whether it is robust.

The key idea via analogy: digging through layered ground

Imagine you are digging a trench and the ground is made of layers.

Obizhaeva and Wang assumed uniform soil. Every metre down is the same effort. In uniform soil, a smooth, steady digging pace makes sense, and that is more or less what their solution recommends between the initial and final blocks.

Alfonsi, Fruth and Schied allow any layering you like. Perhaps there is soft topsoil, then a hard band of clay, then loose gravel. Now the optimal digging strategy is obviously not uniform. You should tear through the topsoil quickly, grind carefully through the clay, and speed up again in the gravel. The right pace depends on what you are cutting through.

The paper's contribution is to solve the optimal execution problem for a general shape function, meaning an arbitrary description of how much liquidity sits at each price level, combined with a resilience mechanism describing how the book recovers after you have eaten into it.

Two things come out of this that are worth understanding.

First, the qualitative structure of the Obizhaeva-Wang solution largely survives. You still want an initial discrete trade to harvest the accumulated liquidity, then a period of continuous trading calibrated to the book's recovery, then a final discrete trade to finish. The three-phase shape is not an artefact of the flat book. That is reassuring, and it means the intuition is robust.

Second, and more importantly, the shape of the book determines whether the model is even well-posed. This is the part that connects to the wider literature. Recall from Huberman and Stanzl, and from Gatheral, that badly chosen impact assumptions can permit price manipulation: trading strategies that make free money out of round trips. The same hazard lurks here. Some order book shapes, combined with some resilience mechanisms, produce a model in which manipulation is possible, and any optimiser you point at such a model will chase the manipulation instead of solving your actual execution problem.

So the paper does not only compute optimal strategies. It also establishes conditions on the shape function under which the model is free of manipulation and the optimal strategy is genuinely well-behaved. That is exactly the kind of hygiene the field needed, and it is why the paper is cited alongside the no-arbitrage literature rather than only alongside the execution literature.

Why it mattered

• It removed an assumption everyone knew was false. The flat book was a mathematical convenience, and the field needed to know how much depended on it. The answer, that the three-phase structure is robust but the details and the well-posedness are not, is precisely the kind of thing you want to know.
• It connected execution to the arbitrage-freeness literature. Showing that order book shape governs whether price manipulation exists ties this work directly to Huberman and Stanzl and to Gatheral, and it makes shape a modelling constraint, not just an empirical input.
• It made the model calibratable to real books. If you can specify any shape function, then you can go and measure the actual shape of the book for the asset you trade and plug it in. That is a large practical step up from a flat slab.
• It kept the maths tractable. Generalising a model usually costs you the closed-form solution. The authors managed to preserve enough structure to still say concrete things about the optimal strategy, which is why the paper is used rather than merely admired.

The honest limitations

• The book shape is treated as static and known. In reality the shape changes constantly, it depends on volatility and on time of day, and, awkwardly, it changes in response to your trading. Liquidity providers who suspect a large buyer is present will pull their offers. The model has a book that refills mechanically and never gets scared.
• Resilience is still a mechanical constant. The recovery of the book after you eat into it is modelled as a fixed physical process. Real resilience depends on who is watching, on whether they think you are informed, and on the state of the world.
• There are no other strategic players. As with essentially all of this literature, the market is an object you push against rather than a set of counterparties who react to you. Brunnermeier and Pedersen's predators are absent.
• The block trades remain suspicious. The initial and final discrete trades are a feature of the smooth mathematical setup. Firing a large block into a real book is precisely the behaviour that announces your presence and invites exactly the response the model does not contain.
• Measuring the shape function is harder than it sounds. The visible book is only a fraction of true liquidity, as the latent order book literature argues forcefully. Calibrating a shape function to the displayed book may be calibrating to the wrong object entirely.

The one-line takeaway

Alfonsi, Fruth and Schied generalised optimal execution from a flat, uniform order book to a book of any shape at all, showing that the block-then-drip-then-block structure of the optimal strategy is robust, but that the shape of the book determines whether your model is even coherent, since some shapes secretly permit free money from trading in circles.