2.2.5 Failure Modes and Desk Practice

Basic Prerequisites: 04 - Efficient Frontier & Trajectory.

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1. Intuition & Practical Objective

The Almgren–Chriss trajectory is mathematically optimal for an idealized market: linear impact, constant parameters, no adversaries. Real markets violate all three. This page names the failures precisely, gives the first-principles reason for each, and quantifies the dollar cost of getting it wrong - because the point of a model is not to be true but to tell you which of its assumptions you are betting on.

The four failures, one line each:

1. Impact is concave, not linear - per-share impact grows roughly as $v^{1/2}$, so the AC linear schedule is provably suboptimal (Almgren 2003).
2. Risk aversion is unobservable - a wrong $\lambda$ mechanically mis-times the whole trade via $\theta \propto 1/\sqrt{\lambda }$.
3. Parameters are non-stationary - $\sigma$ spikes and depth evaporates exactly when you are trading, so the calibrated $\eta$ is stale.
4. The market adapts - a deterministic trajectory is predictable and front-runnable (adverse selection the model does not price).

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2. Mathematical Ground Truth & Derivations

2.1 Concave impact - the square-root law (Almgren 2003). Replace the linear temporary cost $h(v)=\eta v$ with a power law

$$h(v)=\eta v^{\alpha },0<\alpha <1(\text{empirically }\alpha \approx \frac{1}{2}),$$

so the per-period cost becomes ${\sum }_k{n}_{k}h(n_k/\tau )=\eta {\tau }^{-\alpha }{\sum }_k{n}_k^{1+\alpha }$, a concave per-share cost. The objective is still convex in $n$ (the exponent $1+\alpha >1$), so a unique optimum exists, but the Euler-Lagrange equation is nonlinear and the $\sinh$ trajectory is only an approximation. Almgren derives closed forms in terms of a characteristic time $T_{\star }$ that now depends on portfolio size, $T_{\star }\propto X^{(\alpha -1)/(\alpha +1)}$ - so the clean “half-life is size-independent” property of the linear model fails. A linear model calibrated to average conditions systematically mis-paces large orders.

2.2 Risk-aversion error. The utility is $U=E+\lambda V$; if the desk uses $\hat{\lambda }$ but the true preference is $\lambda$, the realized excess utility is

$$\Delta U=U_{\lambda }(x(\hat{\lambda }))-U_{\lambda }(x(\lambda ))\ge 0,$$

which grows quadratically in the relative parameter error (a second-order loss near the optimum). Because $\theta =1/\kappa \propto 1/\sqrt{\lambda }$, a first-order error in $\lambda$ is only a second-order error in cost - the reassuring half of this failure - but the timing shift is first-order linear in $\sqrt{1/\lambda }$.

2.3 Non-stationarity. If $\eta$ or $\sigma$ shifts mid-execution to a new constant, the trajectory solving the original problem is no longer optimal for the remaining sub-problem. Re-solving at time $t$ with updated parameters gives the correct continuation (time-homogeneity only holds while parameters are unchanged). The practical cure is to recalibrate and re-solve on a schedule - i.e., turn the static rule into an adaptive one (page 06).

2.4 Adversarial/predictable execution. The AC trajectory is deterministic: $x_t=X\sinh (\kappa (T-t))/\sinh (\kappa T)$ is public information once $\kappa ,T$ are inferred. Better-informed or faster participants can (a) trade ahead of the schedule to pre-position, and (b) sell into the large late-period prints, extracting an adverse-selection rent the model does not charge. This is why production algorithms randomize their schedules (the same anti-gaming motive as in TWAP’s TWAP jitter).

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3. Computational Implementation - the cost of the linear assumption

Experiment A - the AC schedule under a square-root impact law. Calibrate both models to agree at the average rate, then compare the AC linear schedule against the true optimum under the $\sqrt{\cdot }$ law (solved as a convex program). numpy + stdlib + SciPy’s SLSQP.

The AC schedule over-pays by 12.4% when the truth is the square-root law, and the reason is visible in the quarter split: under concave impact, large early trades are relatively cheap, so the true optimum front-loads (74% in the first quarter vs the AC schedule’s 53%). This is exactly Almgren’s (2003) warning - the linear model mis-paces when impact is concave.

Experiment B - parameter mis-estimation. How much does it cost to use the wrong $\lambda$ or $\eta$?

A 4x error in $\lambda$ costs 22.6% of utility; a 4x error in $\eta$ costs 24.0%. Model risk in execution is not second-order in dollars, even when it is second-order in the optimizer.

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

1. Linear-impact delusion ($\eta v$ vs the square-root law). Per-share impact is concave; the AC schedule over-pays (12.4% here) and mis-allocates volume (too little early). First-principles cause: liquidity is supplied along a finite depth curve, so the marginal cost of size grows slower than linearly. Fix: power-law impact (Almgren 2003) and size-dependent $T_{\star }$.
2. Risk-aversion mis-estimation. $\lambda$ is a preference, not a parameter; it is set by mandate, not estimated from data. Because $\theta \propto 1/\sqrt{\lambda }$, mis-stating it mis-times the trade. Cost: 22.6% of utility for a 4x under-estimate (too slow), 10.3% for a 2.5x over-estimate (too fast - over-trading pays impact for risk you did not need to shed); a 4x over-estimate costs 24.1%.
3. Non-stationarity of $\sigma$ and $\eta$. Both move in the same direction on stress days (vol up, depth down), so the risk term and the cost term shift together - the trajectory is wrong in the same direction that the market is stressed. First-principles cause: liquidity provision withdraws under volatility (the same adverse-selection logic as Liquidity Risk & Funding). Fix: re-solve intraday; treat parameters as state-dependent.
4. Adversarial / predictable execution. A deterministic schedule is an open book. Front-running and predatory momentum trading add an adverse-selection cost outside the model. Fix: randomize timing (TWAP jitter), vary venue and participation, and never publish the full shape.
5. Ignoring the opportunity cost of the unfilled tail. The AC objective prices the trades you make, not the ones the market will not let you make. Under-resourced liquidity leaves the order unfinished - Perold’s opportunity cost (02 - The Execution Problem).
6. Serial correlation and drift break time-homogeneity. The static optimum is time-consistent only without drift/autocorrelation (AC §4). In trending names the optimal strategy is genuinely dynamic (and, for a large enough drift, the optimal liquidation is to not sell - Hasbrouck Ch 15, eq 15.4).

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5. References

• Almgren, Robert - “Optimal execution with nonlinear impact functions and trading-enhanced risk,” Applied Mathematical Finance 10(1), 1-18 (2003). Power-law impact, size-dependent characteristic time, trading-enhanced risk.
• Almgren, Robert; Thum, Chee; Hauptmann, Emmanuel; Li, Hong - “Direct estimation of equity market impact,” Risk 18(7), 58-62 (2005). *Fitting the real impact curve
• Almgren, Robert; Chriss, Neil - “Optimal execution of portfolio transactions,” Journal of Risk 3(2), 5-40 (2000)
• Gatheral, Jim - “No-dynamic-arbitrage and market impact,” Quantitative Finance 10(7), 749-759 (2010). Huberman–Stanzl consistency: which impact/decay models are even admissible.
• Hasbrouck, Joel - Empirical Market Microstructure (2007)
• Perold, André F. - “The implementation shortfall,” JPM 14(3), 4-9 (1988). Opportunity cost.
• Cartea, A.; Jaimungal, S.; Penalva, J. - Algorithmic and High-Frequency Trading (2015)

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6. Connected Graph Bridges

• Back: 04 - Efficient Frontier & Trajectory · Index Hub
• Forward: 06 - Advanced Extensions
• Impact-law view: Market Impact & Depth (square-root law, Kyle $\lambda$)
• Risk: Liquidity Risk & Funding · Model Risk & Validation
• Heuristics that already randomise: Execution Algorithms: VWAP, TWAP, POV