Where Should I Quote? Ho and Stoll's Inventory Model of Market Making

Stoll's 1978 paper explained why a dealer charges a spread: they get pushed into positions they never wanted. That is a compelling story, but it stops one step short of the question a real market maker asks every second of every day.

Given the position I am currently stuck with, and the risk I face, where exactly should I put my bid and my ask right now?

Ho and Stoll answered that question in 1981, and in doing so they wrote the ancestor of every quoting engine running today.

The problem: a moving target with a moving inventory

Picture a dealer's day. Orders arrive at unpredictable times. Sometimes it is a buy, sometimes a sell. Meanwhile the stock's value drifts around on its own, indifferent to the dealer's problems. And the dealer's inventory is a running tally of everything they have been forced to absorb.

Every quote the dealer posts changes the odds of what happens next. Quote a low bid, and fewer people sell to you. Quote a generous ask, and more people buy from you. So the dealer is not just setting a price. They are steering their own future inventory by choosing how attractive their two sides are.

This is a genuinely hard problem because everything feeds back. Your inventory affects the quotes you want to post. Your quotes affect the inventory you end up with. And a random price shock can wipe out the whole plan. Stoll's earlier paper reasoned about a single transaction. Ho and Stoll had to reason about the whole arc.

The key idea via analogy: the fruit stall that leans

Think of a fruit seller who must post one price at which they will buy apples from anyone and another price at which they will sell apples to anyone. They do not care about apples. They just want the margin. But they will get stuck with whatever inventory the crowd hands them, and apples rot, meaning the value of what they hold can move against them.

Now, suppose the crowd has been selling them apples all morning and their crates are overflowing. What do they do?

They lean. They quietly drop both their buy price and their sell price. Not the width of the spread, the center of it. By moving the whole quote down, they make their sell side more tempting (someone will take those apples off their hands) and their buy side less tempting (please, no more apples). They are using price to nudge the flow that arrives.

That is the central result of Ho and Stoll, and the vocabulary the industry adopted for it is worth learning:

• The spread (the width of the quote) is compensation for risk and cost. It depends on how volatile the asset is, how much risk the dealer can stomach, how long they expect to be stuck with a position, and how big trades tend to be. It does not depend on the current inventory.
• The reservation price or quote center is where the dealer skews. It moves away from the position they are holding. Long too much, shade quotes down. Short, shade quotes up.

That separation, "the width prices the risk, the skew manages the position," is the paper's crisp and lasting contribution. It is not obvious in advance that the two decisions cleanly split apart, and the fact that they roughly do is what makes the model useful rather than merely true.

They also produced a result that experienced traders nod along to: the dealer behaves as if they have a preferred inventory level, usually flat, and their quoting continuously pulls them back toward it. Not by force, but by economics, by making one side of the market cheaper to hit than the other.

To get there, Ho and Stoll modelled order arrivals as random events whose rate depends on the quoted prices, modelled the asset's value as drifting randomly, and then solved for the quoting policy that maximizes the dealer's expected satisfaction with their final wealth, using the tools of dynamic optimization.

Why it mattered

• It is the direct ancestor of Avellaneda and Stoikov. The famous 2008 market making model that every quant desk teaches is, structurally, Ho and Stoll rebuilt in continuous time with cleaner mathematics. The reservation price, the inventory skew, the risk-driven spread: all of it is here first.
• It gave market making a control-theory backbone. Before this, a dealer's spread was described. After this, it was solved for, as the output of an optimization problem with an objective, a state variable (inventory) and a control (the quotes). That is the framing every algorithmic liquidity provider uses.
• It made inventory an observable, testable driver of quotes. The prediction "dealers skew quotes against their inventory and their inventory mean-reverts" launched a large empirical literature testing exactly that on real dealer data.
• It explained something ordinary intuition gets wrong. Many people assume a market maker who is long will widen their spread. The model says no: they will shift it. Width is about risk in general, skew is about the specific bag you are holding. Getting that distinction right is the difference between a working quoting engine and a broken one.

The honest limitations

• Information is missing entirely. In Ho and Stoll's world, the customer selling to you is a random arrival, not a person who knows something. But the reason spreads explode before earnings is not inventory risk, it is fear of the informed. The adverse selection tradition (Copeland and Galai, Glosten and Milgrom, Kyle) grew up alongside this one precisely to fill that gap, and a modern market maker needs both halves.
• Empirical support for the inventory channel is weaker than you would hope. Study after study has found that real dealer inventories mean-revert sluggishly, and that quotes respond to inventory less strongly than the theory predicts. Dealers appear to manage risk through channels the model does not see: hedging in correlated names, trading with each other, or simply having deeper pockets than the model assumes.
• It assumes you know the demand curve. The model needs to know how order arrival rates respond to your quotes. In practice this relationship is unstable, competitor-dependent, and changes with market conditions. Estimating it is the hard part of making the model work, and the paper does not solve that.
• One dealer, one asset, one venue. No competition, no fragmentation, no correlated hedges, no queue position. Modern liquidity provision lives and dies on all four.

The one-line takeaway

Ho and Stoll gave market making its first proper solution: the width of your quote prices the risk you are taking, and the position of your quote steers the inventory you are stuck with, so a dealer who is long leans their prices down to attract buyers and repel sellers.