LULD Reopen-Auction Uncertainty Slippage Playbook

Model the Restart as a Two-Stage Stochastic Process (Auction Fill + Post-Reopen Fragility)

Why this note: Many slippage stacks model the pre-trigger chaos well, but still treat the reopen as a single deterministic event. In practice, the reopen is a branching process: uncertain auction participation quality, potential extensions, and a fragile first minute after trading resumes.

1) Failure Mode in One Sentence

If your router treats a volatility-pause reopen as “continuous market restored,” it will systematically underprice tail slippage in the first reopening cycle.

2) Cost Decomposition for Pause→Reopen Events

For a parent order with residual inventory (Q_t) during a pause regime:

[ \mathbb{E}[IS_t] = C_{auction} + C_{post} + C_{delay} + C_{miss} ]

Where:

• (C_{auction}): expected reopen-auction execution cost (if participating)
• (C_{post}): expected post-reopen execution cost for residual quantity
• (C_{delay}): time-value/opportunity decay while paused or extended
• (C_{miss}): completion penalty if auction participation is weak and post-reopen liquidity is thin

A practical branch form:

[ \mathbb{E}[IS_t] = p_{open}\cdot \mathbb{E}[IS\mid \text{reopen now}] + (1-p_{open})\cdot \mathbb{E}[IS\mid \text{extension}] ]

[ \mathbb{E}[IS\mid \text{reopen now}] = p_{fill}^{auc}\cdot C_{auction} + (1-p_{fill}^{auc})\cdot (C_{post}+C_{miss}) ]

This prevents your model from over-crediting optimistic auction participation assumptions.

3) Why the Reopen Is Its Own Regime

In U.S. equities under LULD-style behavior, the transition often follows:

1. limit-state stress (seconds)
2. trading pause window (minutes)
3. reopen auction process (possibly with extension)
4. fragile post-reopen discovery phase

Even when an auction prints, the first post-reopen window can show:

• elevated spread elasticity
• queue re-entry congestion
• unstable fill probabilities for passive child orders
• adverse selection from one-sided urgency carryover

Treating stage (4) as “normal” is a recurring source of p95/p99 slippage drift.

4) Regime State Machine (Production Form)

• V0 NORMAL: no interruption risk signal
• V1 LIMIT-STRESS: close to band / transition hazard elevated
• V2 PAUSED-BUILD: pause active, auction book forming
• V3 REOPEN-AUCTION-LIVE: participation decision and uncross uncertainty
• V4 POST-REOPEN-FRAGILE: first stabilization window after reopen print
• V5 STABLE-REPRICE: liquidity/markout behavior returns to baseline envelope

Use hysteresis + minimum dwell time between V4→V5 to prevent oscillatory policy flips.

5) Feature Set That Matters Most

A) Reopen feasibility + timing

• reopen_now_probability (classifier output)
• auction_extension_count
• time_in_pause_ms

B) Auction quality

• indicative_price_dispersion_bps (rolling)
• imbalance_slope_per_sec
• indicative_volume_stability
• distance_to_collar_bps (or exchange-specific equivalent)

C) Post-reopen fragility

• first_30s_spread_multiple_vs_baseline
• top_of_book_refill_half_life_ms
• queue_reentry_pressure_index
• first_minute_markout_skew

D) Residual urgency

• residual_notional_to_expected_1m_volume
• deadline_slack_sec
• carryover_participation_gap

Without explicit post-reopen features, models often look fine on average and fail exactly in the tails.

6) Two-Stage Modeling Stack

Stage A — Reopen outcome model

Estimate:

• (p_{open}): probability reopen occurs without extension in decision horizon
• (p_{fill}^{auc}): probability and fraction of auction fill

Suitable model families:

• gradient boosted trees for nonlinear interaction terms
• calibrated logistic models when interpretability/governance is priority

Stage B — Conditional cost models

Train separate quantile heads (e.g., q50/q90/q97.5):

• (IS_{auction}) conditional on auction participation
• (IS_{post}) conditional on residual execution after reopen

Final action score:

[ Score(a_t)=\mathbb{E}[IS_t(a_t)] + \lambda\cdot CVaR_{\alpha}(IS_t(a_t)) + \gamma\cdot P(\text{miss deadline}\mid a_t) ]

This avoids one-number predictions that hide completion risk.

7) Control Policy by State

V2 (PAUSED-BUILD)

• Cap discretionary aggression changes until auction-quality confidence is sufficient
• Simulate both branches: participate_auction vs wait_reopen

V3 (REOPEN-AUCTION-LIVE)

• Submit with explicit participation budget tiers (low/medium/high confidence)
• Hard-limit over-concentration in one uncertain uncross

V4 (POST-REOPEN-FRAGILE)

• Enforce burst caps on residual catch-up
• Prefer tactics robust to refill uncertainty over nominal spread capture
• Reduce passive timeout and tighten stale-signal TTL

V5 (STABLE-REPRICE)

• Gradually return to baseline policy (not immediate full reset)

8) Diagnostics & KPIs

1. RER — Reopen Extension Rate
2. RAFR — Reopen Auction Fill Realization (predicted vs realized)
3. RPD95 — Reopen Price Dispersion p95
4. PRF95 — Post-Reopen Fragility cost (first-minute p95 IS)
5. QRT — Queue Rebuild Time to baseline depth/refill
6. MCR — Missed Completion Risk in pause/reopen regimes

If mean IS improves but PRF95 worsens, your policy is likely overfitting auction optimism.

9) Rollout Blueprint

1. Shadow (2–3 weeks): log stage-A probabilities + stage-B quantiles
2. Backtest replay: pause/reopen slices only, compare baseline vs two-stage model
3. Canary: small notional caps on V3/V4 states
4. Promotion gates:
   • improved q95/q97.5 in pause/reopen buckets
   • no material increase in missed-completion probability
   • stable reject/retry/reprice rates

10) Common Anti-Patterns

• Modeling reopen as deterministic once pause ends
• Ignoring extension probability and branch cost
• Treating first-minute post-reopen fills as normal-liquidity fills
• Using only mean IS labels without tail heads
• Allowing residual catch-up bursts without state-specific caps

11) Fast Implementation Checklist

[ ] Add explicit pause/reopen state labels (V0..V5) to execution telemetry
[ ] Build Stage-A models: reopen timing + auction fill probability/fraction
[ ] Build Stage-B quantile models for auction and post-reopen costs
[ ] Add branch-aware action score (mean + CVaR + deadline miss)
[ ] Deploy V3/V4 state controls: participation tiers + burst caps
[ ] Monitor RER/RAFR/RPD95/PRF95/MCR and gate promotion on tails

References

• SEC, “Limit Up-Limit Down” Pilot Plan and Associated Events (DERA white paper).
• SEC, Limit Up-Limit Down Plan and Extraordinary Market Volatility research notes.
• Nasdaq Trader, LULD FAQ (operational examples for pause/reopen behavior).
• Cboe U.S. Equities, Limit Up/Limit Down FAQ (auction extension behavior and collars).
• Cartea, Á., Jaimungal, S., Penalva, J. (2015), Algorithmic and High-Frequency Trading.
• Gould, M. D., et al. (2013), Limit Order Books (Quantitative Finance review).

TL;DR

Pause→reopen should be modeled as a branching regime, not a single event. A two-stage model (reopen/auction outcome + post-reopen conditional cost) with explicit V3/V4 controls typically reduces tail slippage more reliably than tuning one global impact curve.