6.2.5 Failure Modes & Practice

Basic Prerequisites: 04 · Inventory & Risk Aversion.

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1. Intuition & Practical Objective

The AS model is mathematically clean and empirically incomplete in specific, nameable ways. This page names them precisely so a practitioner knows which assumptions to distrust and how they show up in money terms. The objective is not cynicism - it is knowing exactly where the model is an approximation so the residual risk can be measured, bounded, and hedged by other means.

The four failures, in one line each:

1. Adverse selection is absent - AS prices inventory risk over uninformed Poisson flow; informed traders pick off the stale quote and the model has no term for it.
2. $A$ and $k$ must be estimated - the entire arrival mechanism is inferred from the book, and the estimates are unstable exactly when it matters.
3. No inventory limit - the raw formula allows any $|q|$; a real desk must cap it (see 06 · Advanced Extensions).
4. Terminal skew collapse - the inventory penalty $\propto (T-t)$ vanishes at the close.

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2. Mathematical Ground Truth & Derivations

Where the assumptions live. AS solve for pure inventory risk under three interlocking modelling choices:

• (A1) Uninformed, symmetric Poisson flow. Fills arrive at rates $A{e}^{-k\delta }$ independent of the future price. There is no correlation between a fill and the subsequent mid-price move. This is the assumption adverse selection violates.
• (A2) No drift / no alpha. $dS=\sigma dW$ - the mid-price has no predictable component. Any $\mu$ or signal is outside the model.
• (A3) Unbounded inventory, finite horizon $T$. $q$ is unrestricted and the risk penalty scales with remaining time $T-t$.

Failure (A1) in numbers. Suppose a fraction $p$ of fills are informed: the mid-price subsequently jumps adversely by $J$. Each fill then carries an expected adverse cost $pJ$ that the AS spread does not cover. The break-even condition is

$$\underset{\text{AS half-spread per fill}}{\underbrace{\frac{1}{2}\gamma {\sigma }^2(T-t)+\frac{1}{\gamma }\ln \left(1+\frac{\gamma }{k}\right)}}\ge \underset{\text{adverse cost per fill}}{\underbrace{pJ}},$$

so the model loses money once $pJ$ exceeds the AS half-spread. There is no knob inside AS to widen the spread for information - you need an explicit adverse-selection term (Glosten–Milgrom, VPIN-gated quoting, or the Cartea–Jaimungal–Penalva extension in 06).

Failure (A2) in numbers. If the mid-price has a drift $\mu$, the frozen inventory of size $q$ earns/loses $q\mu \tau$ in expectation - a term entirely absent from the AS reservation price. A persistent alpha means the “fair” mid-price is not $s$ but $s+\mathbb{E}[\text{future move}]$; ignoring it systematically misprices one side of the book.

Failure (A3) in numbers. The skew per unit inventory is $\gamma {\sigma }^2(T-t)$ - it is maximized far from $T$ and shrinks to zero as $t\to T$ (paper Figure 1 shows the indifference price converging onto the mid-price at the horizon). Near the close the dealer behaves like a symmetric quoter with no inventory protection.

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3. Computational Implementation - the adverse-selection failure

We add informed flow to the 04 simulation: after a passive fill, with probability $p_{\text{tox}}$ the mid-price jumps adversely by $2 (a pick-off). The AS strategy prices no compensation for this, so its P&L should bleed as $p_{\text{tox}}$ rises.

This is the failure made concrete. Inventory control keeps the final-position std constant at $3.15$ across all toxicity levels - AS does exactly what it promises for inventory risk. But the mean P&L falls monotonically ($56.9\to 36.7\to 16.3\to -3.9$) and turns negative at $p_{\text{tox}}=0.6$: the model has no defence against informed flow, because it never modelled it. The fix is not a bigger $\gamma$ - it is an adverse-selection-aware spread (see Glosten–Milgrom and VPIN).

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

1. Adverse selection is invisible to AS (A1 fails). The spread covers inventory risk only; informed pick-offs bleed the desk (§3). Detection precedes repair: measure flow toxicity (VPIN), infer trade direction (Lee–Ready), and widen or pull quotes when the book is informed. The AS reservation price cannot do this alone.
2. Parameter estimation is fragile (A2/A1). $A$ and $k$ are fitted to fill data: $k$ is the slope of $\ln (\text{fill rate})$ vs quote distance; $A$ the intercept-scale. Both are non-stationary - $k$ collapses in news, $A$ spikes in open/close auctions. A quote engine that hard-codes them will misquote the moment the regime changes. Refit intraday and shrink toward a prior.
3. $\gamma ,k,A,\sigma$ are not jointly identified. The spread couples $\gamma$ and $k$; the skew couples $\gamma$ and $\sigma$. Fit on both fill rate and realized inventory variance, otherwise one bad parameter hides behind another.
4. No inventory cap (A3 fails). The raw formula permits arbitrarily large $q$; the skew only discourages inventory. Add a hard limit or penalty - Guéant et al. (2013) show the asymptotic quotes are only valid for moderate $|q|$.
5. Terminal-time collapse (A3 fails). As $T-t\to 0$ the skew vanishes and the dealer stops protecting inventory at the close. Use a rolling horizon or an explicit $\alpha q_T^2$ penalty rather than a fixed $T$.
6. Quote crossing and marketable quotes. For large $|q|$ the skewed quote can cross the mid or the opposite side; clamp ${\delta }^a,{\delta }^b\ge 0$ and cap $|q|$ before publishing.

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5. References

• Avellaneda & Stoikov (2008), Quantitative Finance 8(3)
• Guéant, Lehalle & Fernandez-Tapia (2013), Math. & Financial Econ. 7(4)
• Glosten & Milgrom (1985), J. Financial Economics 14(1)
• Easley, López de Prado & O’Hara (2012), Flow toxicity and liquidity in a high-frequency world, RFS 25(5)
• Cartea, Jaimungal & Penalva (2015), Algorithmic and High-Frequency Trading

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6. Connected Graph Bridges

• Back: 04 · Inventory & Risk Aversion · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling: Adverse Selection & Glosten–Milgrom · Toxic Order Flow & VPIN · Quote Skewing