6.5.4 Spread Decomposition

Basic Prerequisites: 03 · The Roll Model and Econometrics & Time Series (regression, autocovariance).

---

1. Intuition & Practical Objective

The Roll model measures the total spread but treats it as one number. Yet a market maker’s revenue is not all profit - the spread pays for three distinct things:

1. Order-processing costs - the fixed, per-trade cost of doing business (clearing, settlement, bookkeeping, the labor of quoting). Modeled in the classic “cost-of-transacting” view (Demsetz 1968; Stoll’s naive order-processing model; Roll’s assumption).
2. Inventory-holding costs - the risk the maker bears by holding a nonzero inventory between trades (Stoll 1978; Ho & Stoll 1981). A dealer with a long inventory faces downside price risk, so it shades quotes down to unwind.
3. Adverse-selection / information costs - the expected loss to traders who know the value is about to move (Copeland & Galai 1983; Glosten & Milgrom 1985). The informed trade at the maker’s quote, then the price moves against the maker.

The practical objective: estimate the share of the spread that is each component. This is the single most useful number for a market maker - it tells you how much of your spread revenue is real profit (order-processing + inventory) versus a transfer to informed flow (adverse selection), and hence how aggressively to quote, how wide, and whether your market is worth making.

In the generalized-Roll notation, the total half-spread is $c+\lambda$: $c$ = order-processing/inventory cost (the transitory part), $\lambda$ = adverse-selection cost (the permanent, price-impact part). The full decomposition literature is a refinement of splitting $c$ and $\lambda$, and of further splitting $c$ into order-processing vs inventory.

---

2. Mathematical Ground Truth & Derivations

2.1 Glosten–Harris (1988): transitory vs permanent

Glosten & Harris specify the efficient price and trade price with trade-size-dependent components ($V_t$ = trade volume; ${\lambda }_0+{\lambda }_1{V}_t$ adverse selection, $c_0+c_1{V}_t$ order processing):

$$m_t=m_{t-1}+u_t+q_t({\lambda }_0+{\lambda }_1{V}_t),$$ $$p_t=m_t+q_t(c_0+c_1{V}_t).$$

Taking differences, the observed price change decomposes into a permanent (information) piece $\lambda q_t$ and a transitory (order-processing) piece $c(q_t-q_{t-1})$:

$$\Delta p_t=u_t+q_t({\lambda }_0+{\lambda }_1{V}_t)+\underset{\text{transitory}}{\underbrace{c_0(q_t-q_{t-1})+c_1(q_t{V}_t-q_{t-1}{V}_{t-1})}}.$$

2.2 The generalized Roll (Hasbrouck Ch 8) - the simplest estimable form

With constant $c,\lambda$ and efficient innovation $w_t=\lambda q_t+u_t$:

$$p_t=m_t+c{q}_t,m_t=m_{t-1}+\lambda q_t+u_t,$$ $$\Delta p_t=c(q_t-q_{t-1})+\lambda q_t+u_t,$$ $${\gamma }_0=c^2+(c+\lambda {)}^2+{\sigma }_u^2,{\gamma }_1=-c(c+\lambda ).$$

Total spread is $2(c+\lambda )$. The autocovariance now captures the product of the two costs. Regression on trade direction separates them: $c$ is the coefficient on $q_t-q_{t-1}$ (the order-processing/transitory part), $\lambda$ on $q_t$ (the adverse-selection/permanent part). Note ${\sigma }_w^2={\lambda }^2+{\sigma }_u^2={\gamma }_0+2{\gamma }_1$ is identified, but $\{c,\lambda ,{\sigma }_u^2\}$ individually are not from autocovariances alone (2 moments, 3 unknowns) - hence the need for trade-direction data. This is the identification lesson that motivates the whole decomposition literature.

2.3 Stoll (1989): reversal probability and size

Stoll models the spread through two observable parameters: the probability of a price reversal $\pi$ and the size of a reversal as a fraction of the quoted spread, $1-\delta$ (reversal size $=(1-\delta )S$). Each pure view of the spread pins them differently:

Determinant of quoted spread	$\pi$	$\delta$
Pure order processing (Roll)	$\frac{1}{2}$	$0$ (reversal $=S$)
Pure adverse information (Copeland–Galai, Glosten–Milgrom)	$\frac{1}{2}$	$\frac{1}{2}$ (reversal $=\frac{1}{2}S$)
Pure inventory holding (Ho–Stoll)	$>\frac{1}{2}$	$\frac{1}{2}$

• The order-processing spread simply bounces the full distance $S$ (reversal $=S$, i.e. $\delta =0$) with reversal probability $\frac{1}{2}$.
• The adverse-information view: after a trade conveys news, quotes shift by $\lambda$, so a reversal is only half the quoted spread ($\delta =\frac{1}{2}$).
• The inventory view: the dealer shifts quotes to induce offsetting trades, so reversal probability exceeds $\frac{1}{2}$ ($\pi >\frac{1}{2}$).

Stoll relates the squared quoted spread to two serial covariances (transaction returns and quote returns) and estimates the components on NASDAQ/NMS data. Key theoretical result: under both the inventory and adverse-information models, the realized spread is less than the quoted spread - the dealer earns the quoted spread only in the pure order-processing world.

2.4 Huang–Stoll (1997): the unified three-way decomposition

Huang & Stoll put order-processing, inventory, and adverse-selection in one framework (the general approach). Their structural model yields the trade-by-trade price change

$$\Delta p_t=\frac{S}{2}(q_t-q_{t-1})+\lambda \frac{S}{2}{q}_{t-1}+u_t,$$

where $\frac{S}{2}(q_t-q_{t-1})$ is the transitory (order-processing) part and $\lambda \frac{S}{2}{q}_{t-1}$ is the adverse-selection + inventory part driven by the previous trade’s direction (the dealer reacts to the last trade). By restricting $\lambda$ and adding inventory terms, Huang–Stoll show how to identify the three components separately. The two-way (order-processing vs information) split is their baseline; the inventory piece is the refinement.

---

3. Computational Implementation - estimate $c$ and $\lambda$ from a trade tape

Simulate a Glosten–Harris/Huang–Stoll tape (trade prints at $m_t+q_{t}c$, then the efficient price absorbs the impact $\lambda q_t$), then run the regression $\Delta p_t=c(q_t-q_{t-1})+\lambda q_{t-1}+u_t$ via the normal equations to recover both components and the total spread. (The boxed generalized-Roll equation above places the impact on the contemporaneous $q_t$; the equivalent Huang–Stoll regression places it on the lagged $q_{t-1}$ with a matching regressor, so both recover the same $c,\lambda$.) Stdlib only.

The regression cleanly separates the transitory (order-processing) half-spread $c=$ $0.02 from the permanent (adverse-selection) half-spread $\lambda =$ $0.015. Almost 43% of the maker’s half-spread is expected to be given back to informed flow - exactly the kind of number a quoting desk needs.

---

4. Failure Modes & First-Principles Breakdowns

1. Under-identification. Two autocovariances cannot pin down three parameters $\{c,\lambda ,{\sigma }_u^2\}$. Use trade-direction (signed) data and the regression form, or the estimate is arbitrary. This is the single most important identification warning.
2. Order-processing vs inventory confusion. The regression above folds inventory into $c$ (the transitory part). Splitting inventory from order-processing needs quote data and/or inventory-constrained models (Stoll 1989, Huang–Stoll 1997’s restricted models). Treat “order-processing” estimates as “transitory” unless the model explicitly separates them.
3. Wrong sign on $q_t$. The Lee–Ready (1991) algorithm signs trades (trade vs prevailing quote) because the aggressor side is what matters. Mis-signing destroys the $\lambda$ estimate (see Toxic Order Flow & VPIN).
4. Price impact ≠ mechanical. $\lambda$ is a signal-extraction / competitive-market quantity, not a causal “impact” of your order alone (Hasbrouck Ch 5, 9). Reading $\lambda q_t$ as a guaranteed move misinterprets the estimate.

---

5. References

• Glosten & Harris (1988), Estimating the components of the bid/ask spread, Journal of Financial Economics 21(1), 123–142
• Stoll (1989), Inferring the components of the bid-ask spread: theory and empirical tests, Journal of Finance 44(1), 115–134
• Huang & Stoll (1997), The components of the bid-ask spread: a general approach, Review of Financial Studies 10(4), 995–1034
• Hasbrouck (2007), Empirical Market Microstructure
• Ho & Stoll (1981), Optimal dealer pricing under transactions and return uncertainty, JFE 9(1), 47–73

---

6. Connected Graph Bridges

• Back: 03 · The Roll Model · Realized
• Forward: 05 · Failure Modes & Practice · 06 · Advanced Extensions
• Base: Adverse Selection & Glosten–Milgrom (why $\lambda >0$) · Inventory Management & Quote Skewing (the inventory half) · Econometrics & Time Series (regression)