6.5.1 Spread Decomposition & the Roll Model from Zero

Basic Prerequisites: Econometrics & Time Series (what an autocovariance is).

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1. Intuition & Practical Objective

This page builds the why of spread measurement with no prior market-microstructure knowledge needed. The objective is one idea: the mere fact that trades alternate between the bid and the ask creates a fake, measurable regularity in prices, and reading that regularity tells you the spread without ever seeing the order book.

Start with the dumbest observation. A market maker stands ready to buy at the bid $b_t$ and sell at the ask $a_t$. The “true” value of the stock sits between them - call it $m_t$. A seller hits the bid and trades at $b_t$; a buyer lifts the ask and trades at $a_t$. Now watch a sequence of trades: Buy, Sell, Buy, Sell. The prices bounce: up, down, up, down. Even though nothing about the company changed, the recorded trade prices zig-zag around the calm center $m_t$.

Three “aha”s, built from nothing:

1. The bounce is real, but it is not news. If the efficient price $m_t$ is a random walk (unpredictable), then the only reason consecutive price changes look negatively correlated is the bounce between bid and ask. A buy-then-sell “reverses” the previous change by roughly the full spread. Roll (1984) realized this fake pattern is a feature, not noise: its strength is exactly proportional to the spread.

2. Autocovariance is the microscope. The negative correlation between consecutive price changes - the first-order autocovariance ${\gamma }_1$ - is small and negative when the spread is small, large and negative when the spread is large. Roll inverted that: ${\gamma }_1=-c^2$, so $c=\sqrt{-{\gamma }_1}$, and the effective spread is $2\sqrt{-{\gamma }_1}$. One number, no order book.

3. The spread is not free money. A liquidity provider earns the quoted spread on every round trip (buy low at bid, sell high at ask) - but keeps only the realized spread after the price has moved. Informed traders, who trade because they know the next move, take back part of the spread. The difference between quoted/effective and realized is the adverse-selection cost. That gap is what spread decomposition measures.

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2. Mathematical Ground Truth & Derivations

The picture in numbers. Suppose the true price is $m_t=100$ and the half-spread is $c=0.025$ (so quoted spread $=0.05$). Bid $=99.975$, ask $=100.025$. A buy prints at $100.025$; the next sell prints at $99.975$; the change between them is $-0.05$. A sell-then-buy prints $+0.05$. Either way, the trade-to-trade change is $\pm$ the full spread (plus a tiny efficient-price wiggle $u_t$).

So write the observed price change:

$$\Delta p_t=u_t+c(q_t-q_{t-1}),$$

where $q_t=+1$ for a buy, $-1$ for a sell, and $u_t$ is the efficient-price innovation (small, random, unpredictable). Now ask: what is the covariance between consecutive changes?

$${\gamma }_1=\mathrm{Cov}(\Delta p_{t-1},\Delta p_t)=-c^2\mathbb{E}[q_{t-1}^2]=-c^2.$$

The minus sign appears because the $q_{t-1}$ term appears with $+c$ in $\Delta p_t$ and $-c$ in $\Delta p_{t-1}$. The bounce forces the autocovariance negative. Invert:

$$c=\sqrt{-{\gamma }_1},S=2\sqrt{-{\gamma }_1}.$$

This is Roll’s formula - a full spread estimator from a single autocovariance.

Why this matters (the audience arc). This is the measurement foundation for everything else in Pillar 6. If you cannot measure the cost of trading - quoted, effective, realized - you cannot tell whether a market maker is profitable, whether adverse selection is bleeding them, or whether a venue is cheap. Every later page (02, 03, 04) refines this one intuition.

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3. Computational Implementation - see the bounce

Simulate a pure Roll process and watch the estimator recover the spread you baked in. Stdlib only.

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4. Failure Modes & First-Principles Breakdowns

1. The “spread is free money” trap. The quoted spread is gross revenue, not profit. The dealer must cover the price impact of informed flow, inventory risk, and operating cost. Realized < effective < quoted when adverse selection is present. Confusing quoted for kept is the beginner error that spread decomposition exists to correct.
2. Reading too much into a single autocovariance. ${\gamma }_1$ mixes the spread with any genuine predictability in $u_t$. If the true value is itself trending (momentum), ${\gamma }_1$ can lose its clean sign and the Roll formula breaks (see 05 · Failure Modes).
3. The bounce is about prices, not values. The efficient price $m_t$ is unobservable; only $p_t$ is. Roll’s cleverness is that $m_t$‘s randomness is orthogonal to the bounce, so it washes out of ${\gamma }_1$. That orthogonality ($\mathrm{corr}(q_t,u_t)=0$) is an assumption, and when informed traders make direction and information correlated it fails.

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5. References

• Roll (1984), A simple implicit measure of the effective bid-ask spread in an efficient market, Journal of Finance 39(4), 1127–1139
• Hasbrouck (2007), Empirical Market Microstructure

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6. Connected Graph Bridges

• Base: Econometrics & Time Series (autocovariance) · Probability & Measure Theory (random walk, martingale)
• Continue: Realized · Index Hub
• Cross-pillar: Market Microstructure & Order Types