6.4.5 Failure Modes & Real-World Practice

Basic Prerequisites: 03 · The GM Model and 04 · Spread Decomposition.

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1. Intuition & Practical Objective

The GM/adverse-selection theory is mathematically clean and empirically dangerous in five specific ways. This page names them precisely so a market maker or student knows which assumption is biting and how it shows up in P&L. The objective is not cynicism - it is the discipline of knowing where the model is an approximation so the residual risk can be priced and cut.

The five failures, in one line each:

1. The winner’s curse. You get filled because the order was adverse - every fill is a message that you are on the wrong side.
2. Underestimated $\pi$ (informed fraction). Quote a spread thinner than poison and you lose on both sides - the toxic one directly, and the healthy one you no longer cover.
3. Spread widening as protection. The GM logic predicts spreads expand as uncertainty grows (announcements, opens) - a maker who doesn’t widen in time is a gift to insiders.
4. Toxicity ≠ volatility. Price movement is not the same as price information; flow that moves the efficient price permanently is the poison, and it needs a separate detector (VPIN etc.) - see Toxic Order Flow & VPIN.
5. Belief vs reality. $\pi$ is the maker’s belief about informed probability. It lags and can be wrong - and the model’s machinery is exactly as good as that belief (Hasbrouck Ch 5: “the stock doesn’t know that you own it”).

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2. Mathematical Ground Truth & Derivations

The winner’s curse quantified. A competitive maker’s break-even half-spread equals the adverse-selection half-spread. From the GM model at $\theta =\frac{1}{2}$ the full spread is $\pi (V_H-V_L)$, hence the break-even half-spread is $h^{\star }=\pi \frac{V_H-V_L}{2}$. A maker who quotes a half-spread $h$ while the true informed fraction is $\pi$ earns

$$\mathbb{E}[\text{P\&L per trade}]=h-\pi \frac{V_H-V_L}{2}⟹\text{loss if }h<\pi \frac{V_H-V_L}{2}.$$

With $V_H-V_L=2$ (a $\pm 1$ move about the mid) this collapses to $\mathbb{E}[\text{P\&L}]=h-\pi$. Underestimating $\pi$ by $0.2$ means giving up $0.2$ per share on every trade, permanently. Because informed flow is one-sided, underestimating $\pi$ compounds: the maker keeps seeing adverse fills and (in the naive model) keeps quoting, so losses transfer linearly with trade count.

The spread-widening rule. At belief $\theta$, the ask-side adverse half-spread is (Foucault eq. 3.15)

$$s_a^t=\frac{\pi {\theta }_{t-1}(1-{\theta }_{t-1})}{\pi {\theta }_{t-1}+(1-\pi )\frac{1}{2}}(V_H-V_L),$$

which is maximized at $\theta =\frac{1}{2}$ and vanishes toward $0/1$. So uncertainty in the maker’s belief directly inflates the required spread - the mechanism behind pre-announcement and post-open widening. A maker quoting a flat spread through rising $\pi$ or rising uncertainty is systematically mispricing information.

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3. Computational Implementation - the winner’s curse in numbers (stdlib only)

Simulate the P&L of a maker who quotes a fixed half-spread $h$ against a stream whose true informed fraction is $\pi =0.30$ (break-even $h=0.30$): one maker quotes correctly, one underestimates ($h=0.10$), and one over-covers ($h=0.50$).

The three solid numbers: correct $h$ is break-even ($\approx 0$), the underestimating maker bleeds $-0.203$/trade (≈ $h-\pi =0.10-0.30=-0.20$), and the over-priced maker earns $+0.197$ but will be undercut by any competitor quoting the correct $h=0.30$. This is the winner’s curse in one experiment: the fill itself is the signal, and a spread thinner than the information cost is a guaranteed slow loss. The free correction is to widen as $\pi$ and uncertainty rise - exactly what GM prescribes via the belief-dependent half-spread.

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4. Failure Modes & First-Principles Breakdowns (numbered)

1. The winner’s curse is structural, not bad luck. Every fill is conditional on you being willing to trade - and the informed side is the one that exploits that willingness. In GM terms the quotes are regret-free precisely so that no fill is, on average, a gift; a maker who deviates from the conditional-mean quotes invites exactly the adverse fills the model prices out.
2. Underestimated informed fraction. Too-thin spread $\Rightarrow$ linear, one-sided loss (the $-0.20$/trade above). Death by a thousand adverse fills. The theoretical ceiling is tightness; the practical ceiling is $\pi$ - and $\pi$ is a belief, so it must be re-estimated (see PIN/toxicity, bridges).
3. Flat spreads through rising uncertainty. The required half-spread rises with uncertainty/$\pi$; a flat quote through announcements, halts reopens, or earnings is a standing invitation to insiders. Widen (and thin back) in step with the belief $\theta$.
4. Equating volatility with information. A volatile-but-uninformed stock and a quiet-but-informed stock have entirely different spread requirements. Toxicity is the permanent price impact of the flow, not its variance - measure the permanent mark, not the bounce (04 · Spread Decomposition, and VPIN).
5. Belief lag. $\pi$ and $\theta$ are beliefs, and beliefs update with experience. A maker who never updates (keeps $\pi$ from last year) quotes stale spreads into a market that has changed composition.

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5. References

• Glosten & Milgrom (1985), JFE 14
• Copeland & Galai (1983), J. Finance 38
• Hasbrouck (2007)
• Easley, López de Prado & O’Hara (2012), Flow toxicity and liquidity in a high-frequency world, RFS 25

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6. Connected Graph Bridges

• Back: 04 · Spread Decomposition · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling: Toxic Order Flow & VPIN · Inventory Management & Quote Skewing