Spread Decomposition

Spread Decomposition: Huang-Stoll (1997)

The previous articles developed the three theoretical reasons a spread exists:

1. Inventory risk — the maker bears directional exposure (ho-stoll-inventory-model)
2. Adverse selection — some counterparties are informed (glosten-milgrom-model, kyle-lambda)
3. Order processing — the operational cost of running a market-making business (technology, clearing, regulatory compliance)

Huang and Stoll (1997) unified these into an empirical framework that decomposes the observed spread into its constituent parts. This is where theory meets data.

The Decomposition Framework

Let $s$ be the observed spread. Huang-Stoll model the transaction price $P_t$ as:

$P_t={\mu }_t+\frac{s}{2}\cdot Q_t$

where ${\mu }_t$ is the efficient (true) price and $Q_t\in \{-1,+1\}$ indicates the trade direction (buy or sell).

The efficient price evolves as:

${\mu }_t={\mu }_{t-1}+\alpha \cdot \frac{s}{2}\cdot Q_{t-1}+{ϵ}_t$

where $\alpha$ is the adverse selection component — the fraction of the half-spread that represents permanent information. The innovation ${ϵ}_t$ captures public information arrivals.

The full model decomposes the half-spread into three components with weights summing to one:

$\frac{s}{2}=\underset{\text{adverse selection}}{\underbrace{\alpha \cdot \frac{s}{2}}}+\underset{\text{inventory}}{\underbrace{\beta \cdot \frac{s}{2}}}+\underset{\text{order processing}}{\underbrace{(1-\alpha -\beta )\cdot \frac{s}{2}}}$

Component	Symbol	Economic Content	Observable Signature
Adverse selection	$\alpha$	Permanent price impact of trades — information that does not revert	Trade direction predicts future price level
Inventory	$\beta$	Transient price impact — quote adjustment to manage position	Trade direction predicts next quote change, but effect reverts
Order processing	$1-\alpha -\beta$	Pure cost — no price impact, no quote adjustment	Bid-ask bounce with no predictive content

Estimation

The model generates a testable relationship between successive price changes and trade directions. The autocovariance structure of $\Delta P_t=P_t-P_{t-1}$ as a function of lagged $Q_t$ identifies $\alpha$ and $\beta$.

In practice, estimation proceeds via GMM or OLS on the regression:

$\Delta P_t=\frac{s}{2}(\alpha +\beta )Q_t-\frac{s}{2}\beta Q_{t-1}+{ϵ}_t$

The coefficient on $Q_t$ gives $\alpha +\beta$ (total informative component), and the coefficient on $Q_{t-1}$ identifies $\beta$ (inventory reversion). The residual is order processing.

Typical empirical findings for US equities (pre-decimalization):

• Adverse selection: 30-50% of the spread
• Inventory: 10-30%
• Order processing: 20-40%

Post-decimalization and with electronic trading, order processing costs collapsed. Adverse selection now dominates for most liquid names.

For how the three spread components manifest in decentralized exchanges, see constant-product-amm and impermanent-loss.

Practical Application

Spread decomposition is not just academic. It directly informs:

• Execution quality measurement: regulators and institutional desks use adverse selection estimates to evaluate broker performance
• Market maker strategy: knowing which component dominates tells you whether to improve hedging ($\beta$) or flow selection ($\alpha$)
• Protocol design: any venue designing fee structures should estimate the adverse selection share — a flat fee that covers order processing but not adverse selection will erode market maker capital

Connecting the Module

This article synthesizes the theoretical models into an empirical toolkit:

• ho-stoll-inventory-model provides the theory behind the $\beta$ component
• glosten-milgrom-model and kyle-lambda provide the theory behind the $\alpha$ component
• trading-fundamentals introduced the spread as the price of immediacy — now we know that “immediacy” bundles three distinct costs
• order-books and trading-venues showed where these costs manifest across different venue types

The decomposition closes the loop: we started with “why do spreads exist?” and now have both theoretical models and an empirical method to answer “how much of the spread is due to each cause?”

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Companion notebook: notebook — simulate Huang-Stoll tick data, estimate α/β via OLS, visualize the three-way spread decomposition.

Questions to sit with:

1. If adverse selection dominates the spread for a given asset, what does that imply about the profitability of passive market making on that asset?
2. If the adverse selection component of the spread increases but the order processing component decreases (e.g., due to lower exchange fees), what happens to total quoted spreads? What does this predict about market maker profitability?