Execution Algorithms

Execution algorithms

Every schedule is optimal for something. The failure is not choosing badly — it is not knowing what your schedule was chosen to minimize, and then reporting it against a benchmark that measures something else.

Everything marked ✅ Measured comes from scripts/execution_algos.py — numpy + pandas, seed 20260909, one 390-bin session plus closed forms, 2 s. Everything marked ✅ source-verified was read by this library in the December 2000 Almgren-Chriss preprint; the equation numbers are as printed there and the full transcription is in the script's docstring.

This skill is the schedule. ../../../fin-core/skills/execution-cost-analysis/SKILL.md is the measurement of fills you already have — read it for benchmark choice, the reversion test, and what to log per parent order. §3 below is the one place they touch, and it is a different mechanism, stated as such.

1. The four schedules, and what each one ignores

✅ Measured — 250,000 shares against a 5M-share session drifting +59 bps against the buyer, with Almgren-Chriss linear impact applied:

🚨 POV 3% has the best cost per share and finished 40% of the order. A participation cap does not reduce cost — it converts cost into unfilled shares, which is the one component a fill-based TCA cannot see. §5 prices that conversion.

⚠️ VWAP here uses a perfect forecast of the realized profile. That is not available in advance, and it is the practical difference between a VWAP schedule and a VWAP benchmark. A real VWAP algo tracks a forecast and misses.

2. Almgren & Chriss (2000), reproduced

✅ source-verified — every formula below was read in the paper. η̃ = η − ½γτ is an unnumbered display immediately after eq. (8), and the κ/κ̃ relation is unnumbered too; both are commonly mis-transcribed.

(6)  g(v) = gamma*v                    (7)  h(n_k/tau) = eps*sgn(n_k) + (eta/tau)*n_k
(8)  E(x) = 0.5*gamma*X^2 + eps*sum|n_k| + (eta~/tau)*sum n_k^2 ,  eta~ = eta - 0.5*gamma*tau
(5)  V(x) = sigma^2 * sum_k tau * x_k^2
(16) (1/tau^2)(x_{j-1} - 2 x_j + x_{j+1}) = kappa~^2 * x_j ,  kappa~^2 = lambda*sigma^2/eta~
     and   (2/tau^2)(cosh(kappa*tau) - 1) = kappa~^2
(17) x_j = sinh(kappa*(T - t_j)) / sinh(kappa*T) * X
(18) n_j = 2*sinh(kappa*tau/2)/sinh(kappa*T) * cosh(kappa*(T - t_{j-1/2})) * X
(19) kappa ~ sqrt(lambda*sigma^2/eta)         section 2.3:  half-life theta = 1/kappa

The paper's own Table 1 test case (p. 25, verbatim): S0=50, X=10^6, T=5 days, N=5, σ=0.95 ($/share)/day^½, ε=0.0625, γ=2.5e−7, η=2.5e−6, λ=1e−6.

✅ Measured — the script reproduces the three numbers the paper prints for it:

And four internal checks that the transcription is right:

🔑 If you implement AC, run the third check. eq. (20) is long enough that a transposed T and τ produces a plausible number, and summing eq. (8) over eq. (17)'s own trajectory is a four-line independent oracle.

🚨 η̃, not η, in the cost. ✅ Measured: writing η where eq. (20) writes η̃ gives $949,318 instead of $911,227 — +4.18% at the paper's own parameters, and the gap grows with γτ/η. And ⚠️ eq. (19) uses bare η under the square root while κ̃² uses η̃ — that is not a typo in the paper.

3. 🚨 Trap 1 — the VWAP your own impact moved

✅ Measured — the same volume-proportional schedule at four sizes, with permanent impact (eq. 6) displacing the mid for every print in the session and temporary impact (eq. 7) on the trader's own fills:

"Clean VWAP" is the session VWAP the market would have had with the order absent — the counterfactual, and not a number any desk can compute.

🔑 Both costs rise, so the benchmark is not flat. What grows is the share of the true cost the benchmark absorbs — 5% at 1% participation, 30% at 25%. Permanent impact moved the mid for everyone else's prints too, so the whole benchmark has been pushed toward you.

⚠️ This is a different mechanism from ../../../fin-core/skills/execution-cost-analysis/SKILL.md §2, which holds the fills fixed and shows the benchmark diluted by your own prints. Here the prices move. Both are one-signed the same way and they compound.

⚠️ Do not extrapolate the magnitudes. Linear temporary impact is the paper's own weakest assumption — ✅ source-verified, AC write that in that term "we would expect nonlinear effects to be most important, and the approximation (7) to be most doubtful." Read the direction of the table, and stop at ~25%.

4. 🚨 Trap 2 — "Almgren-Chriss beats TWAP"

✅ Measured on the paper's own Table 1 parameters, at the paper's own λ = 1e−6:

🚨 Almgren-Chriss costs $248,727 MORE in expectation and $437,242 less in standard deviation. It wins only on U = E + λV, by $470,144 — the objective it was derived to minimize. ✅ source-verified: in this model the constant-rate trajectory is the minimum-expected-cost one; the paper titles eq. (9) "Minimum impact".

🔑 So "we switched to Almgren-Chriss and cut costs" inverts the result. What you bought is variance reduction, and the price is on the invoice. If you report only mean shortfall, an AC schedule will look worse than the TWAP it replaced — correctly.

The frontier, ✅ Measured (λ = 2e−6, 0, −2e−7 are the paper's trajectories A, B, C):

⚠️ λ = 0 is exactly the linear trajectory. λ < 0 is risk-seeking: κ̃² goes negative, κ is imaginary, and the solution turns trigonometric (sin/cos rather than sinh/cosh) — it postpones selling. Most implementations silently return nan there.

🔑 ✅ source-verified — θ = 1/κ does not depend on T. The paper: it is "determined only by the security price dynamics and the market impact factors." Doubling the deadline does not slow an AC trade down; it leaves more of the window unused. If your execution policy is "give it more time when the order is large", AC does not implement it — λ does.

5. Implementation shortfall, and the column your TCA does not have

delay        = filled   * (arrival_px  - decision_px)      the research-to-desk handoff
execution    = filled   * (avg_fill_px - arrival_px)       the algorithm
opportunity  = unfilled * (final_px    - decision_px)      the shares never bought
fees         = explicit
total        = delay + execution + opportunity + fees

🚨 Attribution — this four-way split is not Perold's, and calling it "Perold's decomposition" is wrong. ⚠️ Read in the verbatim reprint of Perold (1988), "The Implementation Shortfall: Paper Versus Reality", J. Portfolio Management 14(3), 4–9, reprinted in Streetwise (Princeton UP, 1998) pp. 106–109. What is actually his:

⚠️ The four-component taxonomy — commission, price impact, timing (the delay term), opportunity — is Wagner & Edwards (1993), "Best Execution", Financial Analysts Journal 49(1), 65–71, p. 67, which cites Perold. The modern decision → arrival → fill algebra is Kissell & Glantz (2003). Cite the right one.

🔑 Arithmetically it does not matter — delay + execution = filled × (avg_fill − decision), so the four-way split is a refinement of Perold's execution term, not a contradiction of it. ✅ Measured: implementation_shortfall() returns identity_error, and it is 0.000e+00 — the components reconstruct paper − real with no residual.

⚠️ Perold's own sentence for why the opportunity term is not optional, and the reason this section exists: "You could not begin to measure opportunity costs without the paper portfolio."

✅ Measured — the POV 3% order from §1, which filled 60%:

🚨 A fill-only TCA reports 80.9 bps. The order cost 108.6, and 23% of it is the 100,000 shares the 3% cap never bought. Delay and opportunity have no fill record to be computed from, so they are not "hard to measure" — they are absent from the input.

🔑 Lowering the participation rate does not lower the cost; it moves the cost into the column your TCA cannot see. Report fill rate beside every cost number, and log decision_ts / decision_px — the field list is in ../../../fin-core/skills/execution-cost-analysis/SKILL.md §6.

6. Calibration is the part that is not in the paper

🚨 This library carries no calibrated η, γ or ε. The script uses the paper's Table 1 values, which are a 1999 US large-cap illustration — ε = half of a 1/8 spread, η from "1% of daily volume costs one spread", γ from "10% of daily volume costs one spread". Those are the paper's stated rules of thumb, not measurements, and the tick regime they assume no longer exists.

Fit them on your own fills. η and γ are separable by the reversion test in ../../../fin-core/skills/execution-cost-analysis/SKILL.md §5: the part that reverts after your last fill is temporary, the part that stays is permanent. Until you have that, an AC trajectory is a shape — and the shape is what §4 is about — not a cost estimate.

⚠️ The square-root law impact ≈ Y·σ·sqrt(Q/V) is the standard empirical form and is not what AC assume; AC's temporary impact is linear in the rate. Do not mix a square-root Y into eq. (20).

7. Scripts and where this sits

scripts/execution_algos.py — the session and the three schedules, the full AC closed forms with their internal checks, the efficient frontier, the VWAP-contamination table and the shortfall decomposition. numpy + pandas, seed 20260909, 2 s.

• Measuring fills you already have, benchmark choice, the reversion test, what to log — ../../../fin-core/skills/execution-cost-analysis/SKILL.md.
• Spreads, trade classification, order-flow imbalance and the bar-timestamp leak — ../../../fin-core/skills/intraday-microstructure/SKILL.md.
• Quoting instead of taking, and the inventory problem — ../market-making-models/SKILL.md.
• Order types, routing and not sending this to a live account — ../../../fin-core/skills/broker-execution-apis/SKILL.md and its ../../../fin-core/skills/broker-execution-apis/scripts/paper_account_guard.py.
• Turning a cost assumption into a breakeven and a capacity limit — ../../../fin-core/skills/backtest-validation/scripts/cost_curve.py.
• The strategies that generate the orders — ../trend-following-models/SKILL.md and ../alpha-combination-and-neutralization/SKILL.md §7, whose turnover this prices.
• How large the parent order should have been — ../position-sizing-kelly/SKILL.md.