Community contribution: a stochastic sizing layer for keel — Dr. Issam Elhattab's Quant Lab note

[eaitbrahim]

Contributed by Dr. Issam Elhattab (LinkedIn) — a 23-page modelling note proposing a stochastic sizing layer for keel, written to be readable before it is formalised.

📄 Read the full note (PDF, French)
🔍 Our cross-verification — findings summarised in the reply below.

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The question it asks

lorsqu'un signal de trading apparaît, il ne suffit pas de demander « combien puis-je gagner ? ». Il faut aussi demander « à quel point suis-je sûr que cet avantage existe réellement ? »

When a signal appears, asking "how much can I win?" is not enough. You also have to ask "how confident am I that this edge exists at all?" The note is about the second question.

That lands squarely on keel's situation. The project's measured result is that no shipped rule family is net-positive at the taker fee actually paid (#304). The note takes that as its starting point rather than working around it, and is explicit that a sizing rule cannot rescue a structurally negative strategy:

Une règle de sizing peut réduire le risque ou améliorer la manière de répartir le capital, mais elle ne peut pas transformer une stratégie structurellement perdante en stratégie rentable. Si l'espérance nette d'une règle est négative, la bonne décision peut tout simplement être de ne pas trader.

What it proposes

Three ideas, each with a formula and a plain-language explanation beside it.

1. Put costs in the same unit as risk. A trade's cost per unit of risk is κ = 2(φ+ψ)/s — taker fee plus half-spread, both sides, divided by the stop distance. From that, the win rate a rule needs merely to break even:

p_be = (1 + κ) / (1 + b)

The consequence is counter-intuitive and important: a tighter stop makes the position larger (f_N = ρ/s), so costs consume a larger share of the risk being taken. Cost, not signal quality, can be what moves a strategy from viable to losing.

2. Say out loud how little 100 trades proves. Detecting an edge of δ above break-even at 80% power needs roughly n ≈ 1.55/δ² observations:

edge to detect	12.4 pts	7.5 pts	5 pts	3 pts
trades required	100	275	618	1,718

keel's promotion gate uses a 100-trade floor. That floor can detect a ~12-point edge — a very large one. A 3–5 point edge, already worth having in systematic trading, needs several times more data.

3. Report the probability that the edge is real. Instead of a point estimate p̂, carry a distribution and publish

π_edge = P(p > p_be | D)

— given the data, how likely is it that this rule is genuinely profitable after costs? The note is careful about the misreading: π_edge = 0.92 means 92% of the posterior sits above break-even, not a 92% chance the next trade wins.

It adds a hierarchical layer so products with thin histories borrow strength from the rest, a CUSUM sequential test for edge erosion, and a multi-asset layer.

The boundary it respects

la modélisation propose, Keel dispose

The layer may only ever shrink a position, never widen one, and never touches the fiqh reasoning, the attested screening, or the guards:

ρ_final = min{ρ_stat, ρ_exposure, ρ_concentration, ρ_drawdown, ρ_liquidity, ρ_Keel}

And §7 states plainly what it cannot do: it does not create an edge, does not remove costs, does not manufacture observations, and depends on statistical assumptions that have to be measured rather than asserted.

Why we are publishing it

Two reasons.

It is rigorous and honest. We re-derived every formula in it and they hold. It offers no alpha, and says so.

And it is useful to a project whose headline result is a non-result. keel declines to trade because it cannot prove an edge. This note supplies the vocabulary for saying how far from provable, how long until it could be settled, and what would have to change — and applying it to keel's real numbers surfaced things we did not know, including a defect in our own evidence pipeline.

Our findings, with numbers, are in the reply below.

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Published with the contributor's work attributed to him. The note is a proposal for discussion, not adopted policy — nothing in it changes keel's rails, its screening, or its measured results.

[eaitbrahim]

Our findings

Thank you, Dr. Elhattab. We re-derived the note rather than read it, checked its claims about keel against the source, and then ran its own methods on our real numbers. Everything below is reproducible from the repository and the cached candle data; where a number is an assumption rather than a measurement, it says so.

Full write-up: docs/research/2026-08-20-quant-lab-note-cross-verification.md

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1. The mathematics holds — all of it

result	status
f_N = ρ/s (eq. 1)	✓ loss at stop N·s = ρ·W
κ = 2(φ+ψ)/s (eq. 2)	✓
κ independent of risk_pct (Prop. 1)	✓ and non-obvious
p_be = (1+κ)/(1+b) (eq. 3)	✓
Kelly after costs (eq. 4)	✓ re-derived from g'(ρ)=0
worked example: b=2, κ=0.10, p=45% → ρ*≈12%, notional 240%	✓ exact
κ / p_be table (§3.3)	✓ every cell
n ≈ 1.55/δ² (eq. 6)	✓ (z₀.₉₅+z₀.₈₀)²/4 = 1.546
sample-size table and durations (§4.2)	✓ every cell
g(λρ*) ≈ (2λ−λ²)g(ρ*) → 75% growth at half Kelly	✓

Your claims about keel check out too: four rule families, min_trades = 100, PBO/CSCV in keel/research/cscv.py, eighteen rails, risk_pct a constant 1%.

And a correction on our side. Our first pass reported your 268 / 940 / 49.4 figures as unsourced — we searched docs/ and the databases but not the discussions. They are sourced, in #359. Our error, and we have said so in the write-up.

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2. Substituting our real costs changes the force of your conclusion

Your §3.3 table is illustrative at 25–60 bp per side and says measured costs must replace them. Ours is 120 bp (config.live-sandbox.yaml: taker_pct: 0.012):

stop s	κ	p_be at b=2	p_be at b=6
2%	1.25	75.0%	32.1%
5%	0.50	50.0%	21.4%
10%	0.25	41.7%	17.9%
15%	0.17	38.9%	16.7%

At a 5% stop, break-even is 50%, not the 36.7% in the illustrative table. Costs do not shift our threshold; they dominate it. Our live rules use target_rr: 6, so the b=6 column is operative and far more forgiving — which is itself a finding your framework produced.

A second consequence of the same formulas: at b=6 and any p above ~28%, unconstrained Kelly demands more than 100% notional. Your constraint (5) binds before the Full-vs-Half-Kelly debate begins, exactly as §3.4 argues.

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3. Applying §5.2 to a scheduled decision — and it changes it

We had a pooled forward-trades review scheduled for 2026-09-30 at n = 100 (#359), treating 100 pooled trades as 100 observations. Your §5.2 warns that naive pooling assumes independence. We measured the correction.

Method. Reconstructed our live rule on ONE_HOUR bars with its own parameters — entry_lookback: 40, adx_period: 14, adx_threshold: 25, atr_period: 20, atr_stop_mult: 2, target_rr: 6 — resolving each entry to win/loss by first touch, 25 products, 5 years of cached candles.

Validation first, because everything downstream depends on it:

	reconstruction	published (#359)
signals per asset-year	46.7	49.4
median n per rule-product	241	268

Within 6% and 10%. (Omitting the ADX gate gives 70.6 — the gate accounts for the difference.)

Result:

episodes (UTC days with ≥1 signal):   1,355
size-weighted mean episode size k̄:     8.43
largest episode:                          24 assets firing the same day
ICC of outcomes within an episode:     0.212
design effect  DEFF = 1 + (k̄−1)ρ:      2.58

n_eff at n = 100 pooled:                  39
detectable edge at 80% power:          20.0%    (12.4% if independent)

Signals fire in herds of about eight assets the same day, and those trades then win or lose together. A pooled sample of 100 carries roughly 39 independent observations.

to detect	n_eff	pooled trades
12.4 pts	101	259
7.5 pts	275	708
5 pts	618	1,593
—	100	258 — what "n=100" was assumed to deliver

So the September review would have been powered to see only a ~20-point edge, and a non-result there would have been easy to misreport as "no edge" rather than "we could not have seen one." Raising the floor to ~258 makes n_eff equal what everyone already believed n=100 meant. Tracked as #427.

Caveats we state rather than bury: DEFF 2.58 is a lower bound (clusters are UTC days; these trades span many hours), the ICC comes from backtest outcomes, and k̄ scales with the pool.

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4. The result we did not expect, and it came out of your κ

Our rail 14 caps monthly BUY notional at the venue's fee-free volume allowance. Trades inside it pay no taker fee. Nobody had put κ = 2(φ+ψ)/s on each side of that boundary. At the measured median hourly stop of 2.40% and b = 6:

regime	φ	κ	p_be	vs reconstructed 14.9% win rate
inside allowance	0	0.042	14.88%	+0.02%
inside, ψ = 2 bp	0	0.017	14.52%	+0.38%
inside, ψ = 10 bp	0	0.083	15.48%	−0.58%
outside allowance	120 bp	1.043	29.18%	−14.28%

We treat +0.02% as coincidence — the win rate is a reconstruction and ψ is assumed. What survives the whole plausible spread range is the shape: indistinguishable from break-even inside the fee-free allowance, decisively negative outside it.

This reframes the rail. It is not a budget limit; it is the profitability boundary, and your framework is what made that visible.

The timeframe interaction

Both of your formulas key on s, so shortening the stop hits twice:

bars	median 2×ATR20 stop	f_N at 1%	κ (outside)	p_be at b=6
ONE_DAY	13.86%	0.07×	0.18 R	16.9%
ONE_HOUR	2.40%	0.42×	1.04 R	29.2%

Outside the allowance, every hourly round trip costs more than one full unit of risk. A 6R target nets 4.96R; a stop-out loses 2.04R. We had published the hourly clock as a 23× acceleration in evidence collection "with zero parameter changes" (#359). The signal count is unchanged — the economics are not, by a factor of 5.8.

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5. It found a live defect

Following the arithmetic exposed something we did not know: neither paper profile has ever recorded a trade. Zero rows in trade_outcomes, positions and orders in both ledgers. paper-hourly detected 15 setups in 60 cycles and entered none.

The cause is f_N = ρ/s meeting a real constraint. The profiles size against a synthetic $10k account while inheriting the real account's fee-free allowance of $500/month. At risk_pct 1% and a 2.40% hourly stop, f_N = 0.42 gives ~$4,212 per proposal — 8.4× the entire monthly allowance, and against the $2,500 per-asset cap, 0 of 25 products admissible.

Fixed by seeding from real broker equity (#430), tracked in #426. The throughput consequence is stark and is not a settings problem: at $500/month, 258 pooled trades is roughly eight years away.

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6. Where it overlaps work we had already published

In fairness to both: #368 (2026-08-18) analyses the same ten Keeks articles and reaches the same headline. Your §3.1 stake-versus-risk warning, the OptimalF reservation, the streak-sizing reservation and the CPPI gap-risk caveat each have a counterpart there.

The overlap is not a problem — independent arrival at the same conclusion is corroboration. What we would ask is that a revision cite it, so a reader can tell deliberate divergence from rediscovery. Three places need reconciling:

1. Keel meets Kelly: what ten money-management writeups taught an engine that refuses to bet #368 rejects a Merton γ knob outright as "a guess with a dial on it"; your Layer 3 table keeps Part de Merton. Defensible as a benchmark — worth saying so.
2. π_edge and Keel meets Kelly: what ten money-management writeups taught an engine that refuses to bet #368's planned kelly_f_star now compete for the same promotion-gate slot. Both, or one? This is the decision we would most value your view on.
3. Keel meets Kelly: what ten money-management writeups taught an engine that refuses to bet #368 already spec'd a quarter-Kelly ceiling (dormant) and queued the (1−d/D) drawdown throttle as a paper experiment. Neither appears in your table as our existing position.

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7. Disposition

item	disposition
J1 — κ / p_be at real costs and stops	adopt first. Partly produced above; complete per rule family.
J2 — power curves and time-to-decision	adopted. Already applied to the September review (§3).
π_edge	promising; reconcile with kelly_f_star first.
Hierarchical pooling / n_eff	adopted. §3 is the first application; DEFF 2.58 is the number we carry.
CUSUM erosion detector	defer — nothing to detect erosion in until a rule accrues trades.
Multi-asset portfolio layer	defer — same reason.
OptimalF, streak sizing, Merton γ	benchmarks only, already rejected in #368 with reasons.
ρ_final = min{...}, "la modélisation propose, Keel dispose"	endorsed — it matches our guard architecture exactly.

One scope gap to close

The framework rests on R-multiples, which require a stop. dca has none — our live position shows NO bracket — and it is the only family that has ever traded. A revision should state that it covers turtle_breakout, pullback_continuation and rsi_meanrev, and that DCA needs separate treatment or an explicit exclusion.

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In short

Your §7 says the layer "ne crée pas d'avantage" — it does not create an edge. Correct, and it did something we valued more: it gave us the vocabulary to state precisely how far from provable our position is, how long settling it would take, and which constraint actually binds. Three concrete outcomes — a mis-scoped review, a five-fold cost difference between our two clocks, and a paper pipeline that had silently recorded nothing — came out of applying your arithmetic to our own numbers.

Work arising: #426, #427, #430, #431.

We would welcome a revision addressing §6 and the DCA scope gap, and we would particularly value your view on π_edge versus kelly_f_star.