The Price of Immediacy: Stoll's Theory of What a Dealer Sells

When you hit a bid, someone takes the other side instantly. You do not wait for a natural buyer to show up. You do not negotiate. You click, and it is done. That instant fill feels like a free feature of modern markets, like tap water.

Hans Stoll's 1978 paper is a careful argument that it is not free, and that understanding what the person on the other side is actually giving up is the key to understanding the bid-ask spread.

The problem: why does anyone quote a price at all?

A dealer, or market maker, stands ready to buy from anyone who wants to sell and sell to anyone who wants to buy. That is a strange business to be in. They are not expressing a view. They do not want the stock. They are, in principle, indifferent to whether it goes up or down.

So why does anyone do it, and what determines the price they charge?

The obvious first answer is "costs": exchange fees, back office, staff. Stoll agreed those exist, but he argued the deeper and more interesting cost is something else entirely.

The key idea via analogy: the dealer as a warehouse that does not want inventory

Think of a currency exchange booth at an airport, but with a twist: the booth owner has a carefully chosen personal portfolio and every transaction forces them to hold something they never wanted.

A tourist walks up and sells the booth a large amount of a currency. The booth now holds a big position in that currency. It did not choose that position. It did not have a view on that currency. It was simply doing its job of being available. And now, until it can unload, it is exposed to price risk on a position it never wanted.

That is Stoll's core insight. A dealer's real cost is not paperwork. It is being pushed away from their optimal portfolio. By accommodating your trade, they take on a position whose risk they did not want and are not being paid to want. They are bearing risk on your behalf, in exchange for a fee, and the fee is embedded in the spread.

From that one idea, a whole set of predictions falls out naturally.

• More volatile stocks should have wider spreads. The unwanted position is riskier, so the compensation must be bigger.
• Bigger trades should cost more per share. A bigger unwanted position means more risk, so the dealer needs a bigger cushion.
• More actively traded stocks should have narrower spreads. If a natural buyer will show up in thirty seconds, the dealer holds the unwanted risk for thirty seconds. If the stock trades twice a week, they might be stuck for days. Time held is risk borne.
• The dealer's quotes should depend on what they are already holding. If they are already long, they will be reluctant to buy more, so they will shade their quotes to encourage selling out of that position. This is the seed of what later became the formal inventory models.

Stoll framed the whole thing as a supply and demand problem: investors demand immediacy, dealers supply it, and the spread is the market-clearing price of that service. That framing, deceptively simple, was the paper's most durable gift.

Why it mattered

• It named the product. Before Stoll (building on Demsetz's earlier work on the cost of transacting), it was easy to think of the spread as a fee or a rip-off. After Stoll, the spread has an economic function: it is the price of a real service, immediacy, provided by someone bearing a real cost, unwanted risk. That reframing is why microstructure is an economics discipline and not just plumbing.
• It founded the inventory branch of microstructure. Stoll's argument set the agenda that Ho and Stoll (1981), Amihud and Mendelson, and eventually Avellaneda and Stoikov would formalize into precise mathematical models of how a dealer should quote given the inventory they are sitting on. Every modern market making system that skews its quotes based on its current position is running Stoll's idea.
• It gave a testable theory of spreads. The predictions above (volatility up, spread up; volume up, spread down) became the standard cross-sectional regressions of the field, and they hold up remarkably well in modern data.
• It supplied one half of the great debate. The other half, adverse selection, would arrive in force with Copeland and Galai, Glosten and Milgrom, and Kyle. Microstructure's central argument for the next twenty years, "is the spread about inventory or about information?", only exists because Stoll built the inventory side of it so well.

The honest limitations

• It largely sets information aside. In Stoll's world, order flow is a random nuisance. The dealer does not really worry that the person selling to them knows the stock is about to fall. But that worry, adverse selection, turns out to be a huge part of the story, and a dealer who ignores it goes broke no matter how carefully they manage inventory. The inventory story alone cannot explain why spreads blow out around earnings announcements, for instance.
• Empirically, inventory effects are smaller than the theory suggests. A recurring finding in the decades since is that dealer inventories mean-revert more slowly, and quotes respond to inventory more weakly, than the pure inventory model predicts. Real market makers seem to lay off risk in ways the model does not capture, and information costs seem to dominate in many markets.
• The single-dealer, monopolist framing dates badly. Stoll's world has one dealer with a captive order flow. Today's equity markets have dozens of competing electronic liquidity providers across many venues, none of whom has a monopoly and all of whom can hedge in correlated instruments in milliseconds. The costs are still real, but their magnitude is very different.
• It is a one-period intuition, not a full dynamic solution. The paper reasons about a dealer accommodating a trade. It does not solve the full problem of a dealer continuously quoting over time while their inventory wanders. That harder problem is exactly what Ho and Stoll took up three years later.

The one-line takeaway

Stoll argued that a market maker sells immediacy, not stock, and that the bid-ask spread is the fair price for being shoved into a risky position you never wanted, which is why spreads widen with volatility and trade size and narrow with trading activity.