6.5.2 Quoted, Effective & Realized Spread

Basic Prerequisites: Econometrics & Time Series.

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1. Intuition & Practical Objective

“Spread” sounds like one number, but a trader and a market maker measure different spreads. This page defines the three canonical measures and shows how they nest - quoted $\ge$ effective $\ge$ realized when there is adverse selection, and why that ordering is the whole story of who pays whom.

• Quoted spread $S_q=a_t-b_t$: what the order book advertises. It is the gross cost of crossing the spread, but you almost never pay it in full because trades fill inside the touch or at better mid-market terms.
• Effective spread $S_e=2q_t(p_t-m_t)$: what a trader actually pays, measured by how far the execution price sits from the prevailing midpoint. This is the number a venue or broker reports to a client as “your cost.”
• Realized spread $S_r=2q_t(p_t-m_{t+\Delta })$: what the market maker keeps, measured by the midquote after the trade relative to the price paid. The gap between effective and realized is the price impact $S_e-S_r=2q_t(m_{t+\Delta }-m_t)$ - the adverse-selection loss to informed flow.

The practical objective: given a trade tape and quote data, compute all three and read off who captured the spread. A market with $S_q$ large but $S_e\ll S_q$ is cheap to trade (liquidity hides inside the quote); a market with $S_e\gg S_r$ is one where the liquidity provider is being run over by informed flow.

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2. Mathematical Ground Truth & Derivations

Under the generalized Roll model the three measures relate to two parameters: the half-spread $c$ and the price-impact $\lambda$ (adverse-selection cost per unit order flow).

• Quoted: $S_q=2(c+\lambda )$ - the full bid-ask width (order-processing $c$ plus the adverse-selection component $\lambda$; Hasbrouck eq. 8.3).
• Effective: a trade at the ask ($q_t=+1$) prices at $p_t=m_t+(c+\lambda )$, so $2q_t(p_t-m_t)=2(c+\lambda )$; at the bid likewise. $S_e=2(c+\lambda )$. The effective spread equals the quoted spread when trades print exactly at the touch.
• Realized: after the trade the midquote is revised by the information content ($m_{t+\Delta }=m_t+\lambda q_t$ plus public noise), so

$$S_r=2q_t(p_t-m_{t+\Delta })=2(c+\lambda )-2\lambda =2c,$$

so $S_r=2c$ (the maker keeps the order-processing half-spread). The loss to informed flow is the price impact:

$$S_e-S_r=2\lambda ,$$

and with $\lambda >0$ the ordering $S_q\ge S_e\ge S_r$ holds (the middle equality is approximate when fills are at the touch; with price improvement $S_e<S_q$).

Half-spread of a single trade (per-side). The signed cost of one trade is $q_t(p_t-m_t)$; the effective half-spread is $|p_t-m_t|$. Averaging over many trades gives the standard estimators used in the literature (Hasbrouck 1993; Huang & Stoll 1996): effective $=2q_t(p_t-m_t)$, realized $=2q_t(m_{t+k}-m_t)$ for a short horizon $k$.

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3. Computational Implementation - all three on one tape

Simulate a quote path where the dealer midpoint shifts by $\lambda$ after each trade, then measure all three spreads. Stdlib only.

The maker advertises $0.04$ but keeps only $0.02$ - exactly half is surrendered to informed flow. That is the economic content of spread decomposition in its simplest form.

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4. Failure Modes & First-Principles Breakdowns

1. Assuming effective = quoted. Real markets offer price improvement (fills inside the touch, hidden liquidity, midpoint cross). Always measure $S_e$ from actual execution prices, not from the displayed quote.
2. Wrong sign convention. Effective and realized spreads are signed by trade direction ($q_t$). If you drop the sign, a buy-then-fall and a sell-then-rise cancel and you compute zero - erasing exactly the adverse-selection signal you want.
3. Realized-horizon sensitivity. $S_r$ depends on $k$ (the post-trade horizon). Too short, and you miss the impact; too long, and other trades contaminate it. Standard practice uses a fixed short window (seconds) or a VWAP benchmark; the choice must be stated.
4. Stale quotes. Using a stale midpoint (not updated at trade time) biases $S_e$ upward because the true $m_t$ has drifted. Quote timestamps must be matched to trade time.

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5. References

• Hasbrouck (2007), Empirical Market Microstructure
• Huang & Stoll (1996), Dealer versus auction markets: a paired comparison of execution costs on NASDAQ and the NYSE, Journal of Financial Economics 41(3), 313–357
• Hasbrouck (1993), Assessing the quality of a security market, RFS 6(1)
• Stoll (1989), Inferring the components of the bid-ask spread

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · The Roll Model · 04 · Spread Decomposition
• Base: Econometrics & Time Series · Limit Order Book Mechanics (quoted spread & effective half-spread in the LOB)