Queue Priority Is the Whole Game: Guilbaud and Pham on Limit and Market Orders

The classical market making models, from Avellaneda and Stoikov onward, let the market maker do something impossible. They let her quote any price she wants, at infinitely fine resolution, and get filled at a rate that depends smoothly on how far her quote sits from the fair price.

Real markets do not work like that. Prices move in ticks, discrete minimum increments. You cannot quote a price halfway between two ticks. And when you post at a tick, you do not get a smooth fill probability. You join a queue behind everyone else who got there first, and whether you get filled depends brutally on where in that queue you are standing.

Fabien Guilbaud and Huyen Pham built a market making model that takes the tick and the queue seriously, and the strategy that comes out of it looks much more like what a real high-frequency market maker actually does.

The problem: the smooth models miss the fight that matters

In a real electronic market with a small tick size relative to volatility, the spread is often just one tick wide. The best bid and the best ask are adjacent. There is nowhere to quote between them.

This changes the market maker's decision completely. It is no longer "how far from the mid should I quote?", because there is no room to be clever about distance. It becomes a much more discrete set of choices:

• Join the queue at the best bid. You are at the back. Everyone in front of you gets filled first. You will be filled only if enough aggressive sellers come through to exhaust the whole queue ahead of you. Safe, but slow, and you might never get there.
• Improve the price by one tick. Now you are at the front of a brand new queue, and you will be filled first. But you have given up a tick of edge, and you are now the one exposed if the price is about to move against you. You have bought priority with money.
• Cross the spread with a market order. Immediate, certain, and you pay the spread and the taker fee.
• Do nothing at all.

That is the real decision set, and it is fundamentally discrete and combinatorial, not smooth. The smooth models cannot see any of it.

The key idea via analogy: the queue at the bakery

Picture a bakery with a queue out the door, selling bread at a fixed price.

If you join the back of the queue, you pay the normal price but you might not get any bread before it sells out. Your fill is uncertain and depends entirely on how many loaves come out of the oven before your turn.

If you offer the person at the front a small payment to swap places, you have bought priority. You will definitely get bread, but you have paid extra for it.

Guilbaud and Pham's market maker is standing in exactly this queue, on both sides at once, making this decision continuously. And her problem has an extra dimension the bakery does not: she does not necessarily want the bread. She wants to buy and sell in roughly equal measure and go home flat, so she has to manage her inventory while playing the queue game.

The model's ingredients are chosen to capture exactly this reality:

• The spread is modelled as a discrete, tick-valued quantity that jumps around according to a Markov chain, rather than as a continuous number. Sometimes the spread is one tick, sometimes two, and the market maker's options depend on which.
• Quoting decisions are discrete choices, including the crucial option to improve by a tick specifically to gain execution priority, which the authors identify as a central issue in high-frequency trading.
• Both limit and market orders are available. The market maker quotes passively to earn the spread, but she can also cross the spread aggressively with a market order when she needs to dump inventory in a hurry. This is important: it separates the "earn the spread" business from the "get flat, now" business, and lets the model trade them off.
• Inventory risk drives everything, as in all market making models. The whole point of the aggressive market orders is to control it.

The resulting optimal strategy is a policy that says, given the current spread, the current inventory, and the state of the market: post here, or improve a tick, or cross and take, or wait.

Why it mattered

• It put the tick size and the queue into the model. These are not details. In modern equity and futures markets they are the dominant features of the market making problem. A model without them is describing a market that does not exist.
• It explains why latency actually matters. The classical models struggle to explain the speed arms race, because in a smooth model being a microsecond faster barely changes anything. In a queue model, being faster means being earlier in the queue, and queue position is the difference between getting filled and not. This paper's framework makes the value of speed structural rather than incidental.
• It combines passive and aggressive orders in one control problem. Real high-frequency market makers do both: they quote to earn the spread and they cross to manage risk. Modelling both, with the trade-off between them solved properly, is closer to the actual job.
• The discrete spread model is a good fidelity upgrade. Treating the spread as a jumping, tick-valued process rather than a continuous variable is a modest change that buys a lot of realism in small-tick markets.

The honest limitations

• Queue position itself is only partially modelled. The paper captures the crucial choice to pay a tick for priority, but a fully realistic model of where you sit in the queue, and how the queue ahead of you evaporates through cancellations rather than fills, is harder still. In practice, cancellations ahead of you are often the main reason you advance.
• The market maker is alone. There are no competing market makers explicitly racing her for queue position, which is odd in a model whose central insight is about queue position. The other participants arrive as a statistical process, not as strategic rivals.
• There is no adverse selection. Orders arrive and fill her according to an intensity process. Nobody is trading against her because they know the price is about to move. This is the same gap that afflicts most of the market making literature, and it means the model understates the danger of being at the front of the queue, which is precisely where informed traders find you.
• The optimal policy is computationally heavy. Solving the discrete control problem is not a closed-form exercise, and a production system has to approximate.
• Calibration is demanding. The model needs the dynamics of the spread, the arrival rates of orders at each level, and the execution probabilities, all of which shift through the day.

The one-line takeaway

Guilbaud and Pham built a market making model around the features that actually dominate modern electronic markets, the discrete tick, the queue, and the choice to pay a tick for execution priority, and by letting the market maker use both passive quotes to earn the spread and aggressive market orders to dump inventory, they produced a strategy that looks far more like the real job than the smooth classical models do.