The midpoint has a dangerous charisma.

It sits exactly between bid and ask, looks mathematically neutral, and invites you to call it “the price.” The spread then becomes a nuisance around the price: two ticks of noise, a commission in disguise, the small toll paid by impatient traders.

That picture is too polite. A spread is not merely a fee. It is a compact insurance premium sold by a liquidity provider who has to answer a hard question before the rest of the market does:

Why is this person willing to trade with me right now?

Sometimes the answer is boring. A fund has an inflow. A hedger reduces risk. A retail order arrives because someone tapped buy. Sometimes the answer is not boring. The trader knows something, or predicts something, or is connected to a larger flow that will move the price after the market maker fills the order.

The spread is the small object where those possibilities get priced.

Microstructure rule: the spread is not outside the price process. It is one of the ways the price process learns who wanted immediacy.

The Dealer Quotes Before Knowing Why

The cleanest starting point is the Glosten-Milgrom model.1 There is a risky asset with unknown value \(V\). Some traders know more about \(V\) than the market maker; others trade for liquidity reasons. A competitive specialist sets quotes before seeing whether the next order comes from information or noise.

The zero-profit logic is brutal:

\[a = E[V \mid \text{buy order}] + c, \qquad b = E[V \mid \text{sell order}] - c,\]

where \(a\) is the ask, \(b\) is the bid, and \(c\) is the order-processing cost. The dealer sells at the ask to someone who wants to buy. If buys are more likely when the asset is underpriced, the dealer’s conditional expectation after seeing a buy is above the unconditional midpoint. Symmetrically, a sell order pushes the conditional expectation downward.

So the spread is

\[a-b = 2c + \left(E[V \mid \text{buy}] - E[V \mid \text{sell}]\right).\]

The first term is mechanical. The second is informational. It is the price of not knowing whether the trader across the screen is uninformed, early, or simply better.

A small binary version is enough to feel the model. Suppose the asset value is either \(+\Delta\) or \(-\Delta\) around the current midpoint. A fraction \(\alpha\) of incoming traders are informed. An informed trader receives a directional signal that is correct with probability \(s\); a noise trader buys with probability \(\beta\). Then

\[\Pr(\text{buy}\mid H) = \alpha s + (1-\alpha)\beta,\]

and

\[\Pr(\text{buy}\mid L) = \alpha(1-s) + (1-\alpha)\beta.\]

Bayes’ rule turns the order into a posterior. When noise flow is balanced (\(\beta=1/2\)), the adverse-selection half-spread in this toy model reduces to

\[\alpha(2s-1)\Delta.\]

That formula is almost too small for what it says. More informed flow, more accurate information, or a larger value jump all widen the quote. The dealer is not being greedy in the model. The dealer is refusing to sell insurance at a price that ignores the accident rate.

Inventory Leans on the Quote

Adverse selection is only one reason the spread exists. A market maker also accumulates inventory.

If the dealer buys from sellers all morning, the position grows. The next quote cannot be the same quote as before. A long-inventory dealer would like to sell, so the quote center may shift down; a short-inventory dealer would like to buy, so the quote center may shift up. Risk aversion, capital limits, and volatility make inventory costly even when every incoming trader is uninformed.

Ho and Stoll framed dealer quotes as an optimal-control problem under transaction and return uncertainty: bid and ask prices depend on the dealer’s state, not only on a single fair value.2 Kyle’s continuous-auction model gives a complementary view from the informed-trading side: market depth is the conversion rate between order flow and price impact when competitive market makers infer information from net demand.3

The practical conclusion is simple:

two markets can have the same midpoint and different executable prices
because they have different information risk and different inventory capacity

That is why “spread” is a liquidity statistic, not just a nuisance number. It is a compressed summary of who is willing to warehouse risk, how much adverse selection they expect, how quickly inventory can be recycled, and how expensive it is to keep the trading machine running.

A Tiny Dealer on the Bench

The lab below is a deliberately small dealer model. It is not a full limit-order book and not a trading system. It has one market maker, binary information, noise flow, order-processing cost, and a crude inventory-risk adjustment. The goal is not realism. The goal is to make the decomposition visible.

The model computes the Bayesian conditional values at the bid and ask, simulates a deterministic pseudo-random tape of trades, marks the dealer’s inventory to the evolving efficient value, and estimates a Roll-style spread from transaction prices. Try to break it. Raise informed flow. Make the signal accurate. Add a buy imbalance. Remove order cost and let adverse selection do the widening.

Order processing Adverse selection Inventory Marked P&L Roll estimate

Deterministic toy model. Values are in basis points around a synthetic midpoint. "Realized spread" uses a short look-ahead in the simulated tape, so it can turn small or negative when adverse selection dominates. The Roll estimate is shown only when transaction-price covariance has the sign its assumptions require.

The first panel is the accounting ledger: order-processing cost, Bayesian adverse selection, and an inventory cushion. The second panel is the posterior belief. If a buy order strongly raises \(P(H)\), the dealer must raise the ask before the trade arrives or lose money after it.

The path panel is intentionally noisy. Even a correctly priced spread does not make every tape profitable. The dealer earns spread from uninformed immediacy traders and loses to informed traders when the efficient value moves after the fill. Inventory can help or hurt depending on the sequence. Market making is not “collecting the spread” in the same way an insurer is not “collecting premiums” without underwriting risk.

The Bounce Leaves Fingerprints

Richard Roll noticed a beautiful fingerprint.4 Suppose the efficient price is a random walk, trades occur at either bid or ask, and the spread is fixed. Then transaction prices bounce around the efficient value. A buy at the ask followed by a sell at the bid creates negative serial covariance in price changes even if the efficient price itself is unpredictable.

Under the model’s assumptions, the effective spread can be inferred from

\[S = 2\sqrt{-\operatorname{Cov}(\Delta P_t,\Delta P_{t-1})}.\]

This is why the lab reports a Roll estimate. When the simulated transaction price changes have negative first-order covariance, the estimator sees a bounce. When information shocks dominate, the covariance can flip sign and the estimator refuses to speak. That failure is useful. It says the simple bid-ask bounce story is not the whole data-generating process.

Modern microstructure measurement usually separates at least three quantities:

  • quoted spread: the displayed ask minus displayed bid;
  • effective spread: how far the execution price is from the midpoint at trade time, signed by trade direction;
  • realized spread: how much of that execution revenue remains after the midpoint has had time to move.

The last quantity is where adverse selection shows its teeth. If a customer buys and the midpoint quickly rises, the dealer’s apparent spread capture was partly compensation for selling too low. Hasbrouck’s VAR approach measures a trade’s information content through eventual quote revisions, and reports that trade impact arrives with lag and varies with trade size, spread, and firm characteristics.5 Madhavan, Richardson, and Roomans similarly model transaction-level price changes as a mixture of public information, asymmetric information, inventory, and order-processing effects.6

Huang and Stoll’s spread-decomposition paper is a useful warning label here: empirical decomposition is model-dependent. They reconcile existing models in a time-series framework and find a large order-processing component plus smaller but significant adverse-selection and inventory components, with estimates sensitive to trade size and assumptions about how orders relate to trades.7 The word “component” sounds physical, as if the spread were a sandwich to be peeled apart. In real data, the layers are inferred through a model.

What Midpoint Backtests Miss

A backtest often assumes fills at midpoint plus a fixed spread cost. That is a reasonable first penalty for slow, small, liquid trades. It becomes dangerous when the strategy itself changes the order-flow state.

If the signal buys immediately before price increases, your fills are not random draws from the quote. You are more likely to demand liquidity when the dealer should widen. If the strategy trades names with stale quotes, the observed spread may understate executable cost. If the strategy leans on a venue with rebates, queue priority matters: the economics of posting a limit order are not the economics of crossing the spread. If the strategy repeatedly buys after the same public event, realized spread and market impact are the relevant objects, not quoted spread.

The audit I would want for a serious strategy is a little ledger:

  1. measured quoted spread at decision time;
  2. effective spread paid at execution;
  3. midpoint move after 1 second, 10 seconds, 1 minute, and 5 minutes;
  4. realized spread by venue, order type, size bucket, and volatility regime;
  5. inventory concentration created by the strategy’s own flow;
  6. performance after replacing midpoint fills with a conditional spread model.

The last step is the important one. Costs should be conditional on the reasons the strategy trades. A model that pays the same spread when alpha is quiet and when alpha is screaming is assuming away adverse selection precisely when it matters most.

The Next Messy Experiment

The next useful blog-sized experiment would take public trade-and-quote data and try to reproduce a decomposition like this at small scale:

quoted spread
effective spread
realized spread
post-trade midpoint response
Roll covariance estimate
simple adverse-selection proxy

The hard part would not be the formulas. It would be the boring machinery: cleaning timestamps, matching trades to quotes, classifying trade direction, handling locked or crossed markets, filtering odd lots, and deciding which midpoint horizon counts as “realized.” In microstructure, the plumbing is often the result.

I like this direction because it connects three worlds that are often taught separately. Bayes explains why the ask is conditional. Control explains why inventory changes quotes. Time-series econometrics explains how the spread leaves traces in transaction prices. Put them together and the midpoint stops looking like a price. It starts looking like a truce.

The spread is the premium paid to trade now, before the market knows whether now was a harmless convenience or the beginning of a repricing.

Primary Sources

  1. Lawrence R. Glosten and Paul R. Milgrom, “Bid, ask and transaction prices in a specialist market with heterogeneously informed traders”, Journal of Financial Economics, 1985. See also the Columbia Business School record. ↩

  2. Thomas Ho and Hans R. Stoll, “Optimal dealer pricing under transactions and return uncertainty”, Journal of Financial Economics, 1981. ↩

  3. Albert S. Kyle, “Continuous Auctions and Insider Trading”, Econometrica, 1985. The JSTOR record is also available. ↩

  4. Richard Roll, “A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market”, Journal of Finance, 1984. ↩

  5. Joel Hasbrouck, “Measuring the Information Content of Stock Trades”, Journal of Finance, 1991. ↩

  6. Ananth Madhavan, Matthew Richardson, and Mark Roomans, “Why Do Security Prices Change? A Transaction-Level Analysis of NYSE Stocks”, Review of Financial Studies, 1997. A working-paper PDF is available from NYU Stern. ↩

  7. Roger D. Huang and Hans R. Stoll, “The Components of the Bid-Ask Spread: A General Approach”, Review of Financial Studies, 1997. ↩