Cross-Impact Propagator Slippage Model for Basket Execution

Date: 2026-02-27
Category: research (quant execution / slippage modeling)

Why this playbook

Single-name slippage models underestimate cost when orders are executed as a correlated basket. In practice, trading AAPL can move QQQ, and QQQ flow can feed back into AAPL fills. Ignoring this cross-effect causes:

• optimistic cost estimates,
• wrong urgency allocation across names,
• avoidable p95/p99 slippage blowups during macro-flow windows.

This playbook introduces a production-oriented cross-impact propagator design that extends per-symbol models into a basket-level controller.

Core idea

Model execution impact in vector form:

[ \Delta p_t = \sum_{\ell=0}^{L} G_{\ell},u_{t-\ell} + \varepsilon_t ]

Where:

• (u_t \in \mathbb{R}^N): signed trading rate vector across (N) symbols,
• (G_{\ell} \in \mathbb{R}^{N\times N}): lag-(\ell) impact matrix,
• diagonal entries = self-impact, off-diagonal = cross-impact,
• (\varepsilon_t): exogenous return noise.

So child-order sizing is no longer independent by symbol; it is jointly optimized under cross-coupled costs.

What to log (minimum data contract)

Per child-order and per time bucket (e.g., 250ms/1s):

• symbol, side, qty, notional, participation at send,
• decision/sent/ack/fill timestamps,
• spread, top-of-book depth, imbalance, microprice,
• signed own flow + signed peer flow (same sector/index basket),
• short-horizon markouts (1s, 5s, 30s),
• venue/session flags (open, close, auction-adjacent, event windows).

For basket modeling, also persist:

• dynamic correlation block ID (e.g., tech mega-cap, KOSPI theme cluster),
• index-membership and ETF overlap features,
• rolling ADV-normalized flow vector.

Estimation blueprint

1) Universe compression first

Estimate cross-impact in a reduced factor space to avoid noisy (N\times N) overfit.

• Build (K)-factor flow representation (PCA/sector/index factors),
• Estimate factor-space propagator (\tilde{G}_{\ell} \in \mathbb{R}^{K\times K}),
• Map back to symbol space for execution-time costing.

This gives better stability than brute-force pairwise estimation.

2) Constrain kernels for sanity

Practical constraints:

• decay with lag (transient impact),
• non-explosive cumulative response,
• near-symmetry regularization for cross terms (anti-manipulation prior),
• stronger shrinkage on weakly related pairs.

A useful penalty:

[ \mathcal{L}=\mathcal{L}{\text{fit}} + \lambda_1\sum{\ell}\lVert G_{\ell}\rVert_1 + \lambda_2\sum_{\ell}\lVert G_{\ell}-G_{\ell}^{\top}\rVert_F^2 ]

3) Regime-conditional fit

Fit separate kernels for:

• open auction transition,
• continuous day session,
• close auction transition,
• volatility-stress windows.

Cross-impact is regime-sensitive; one global kernel usually drifts quickly.

Slippage forecast decomposition

For a candidate basket schedule (u_{t:t+H}):

[ \widehat{\text{Cost}} = \underbrace{\widehat{C}{\text{spread+fees}}}{\text{instantaneous}} + \underbrace{\widehat{C}{\text{self-impact}}}{\text{diagonal }G} + \underbrace{\widehat{C}{\text{cross-impact}}}{\text{off-diagonal }G} + \underbrace{\widehat{C}{\text{timing risk}}}{\sigma, residual horizon} ]

Track cross-impact share explicitly:

[ \rho_{\text{cross}} = \frac{\widehat{C}_{\text{cross-impact}}}{\widehat{\text{Cost}}} ]

When (\rho_{\text{cross}}) spikes, independent per-name schedulers are unsafe.

Online controller (production policy)

At each control step:

1. Forecast next-horizon cost surface with current (G_{\ell}) regime,
2. Solve constrained optimization for child vector (u_t),
3. Enforce hard limits (participation, residual deadline, per-name max chase),
4. Receding-horizon update every bucket.

A practical objective:

[ \min_{u_{t:t+H}} ; \mathbb{E}[\text{Cost}] + \eta,\text{Var}({\text{Cost}})

• \gamma,\text{ResidualPenalty} ]

subject to inventory completion and risk caps.

Execution heuristics that usually help

• Synchronize correlated legs when cross-impact is strong (do not front-run one leg too far ahead).
• Delay secondary names briefly if lead-name sweep already moved the factor.
• Throttle basket-wide aggression (not just one symbol) in systemic toxicity states.
• Prefer liquidity windows with natural two-way flow for high cross-correlation clusters.

Validation framework

Weekly scorecard:

1. Forecast quality

   • MAPE / calibration by horizon for total and cross-impact components.

2. Execution outcomes

   • p50/p95/p99 implementation shortfall vs diagonal-only baseline,
   • completion delay incidents,
   • underfill rate by basket type.

3. Model stability

   • kernel drift magnitude,
   • symmetry-gap metric (\lVert G-G^\top\rVert),
   • regime-switch frequency.

4. Attribution sanity

   • names where cross-impact term dominates >40%,
   • post-trade markout consistency on those names.

Failure modes and guardrails

• Pairwise overfitting in sparse names

  • Use factor compression + shrinkage.

• Regime leakage (open/close dynamics pollute day model)

  • Fit and serve kernels by session phase.

• False precision in weakly linked pairs

  • Zero out low-confidence off-diagonal entries.

• Controller oscillation

  • Add turnover penalty on child-vector changes.

• “Great IS, terrible completion” pathology

  • Dual-objective with explicit deadline penalty; monitor both.

Minimal rollout plan

Phase 1 — Shadow

• Compute cross-impact forecasts, no execution changes.
• Log counterfactual deltas vs current scheduler.

Phase 2 — Canary (5–10% baskets)

• Enable only synchronization + mild aggression scaling.
• Auto-rollback on p95 degradation or completion SLO breach.

Phase 3 — Scaled control

• Full receding-horizon optimization across correlated basket legs.
• Keep per-basket kill-switch and per-regime override.

Phase 4 — Governance

• Weekly champion/challenger review (cross-impact vs diagonal baseline),
• Drift alerts on kernel instability,
• Monthly stress replay in macro-shock windows.

Practical takeaway

For correlated portfolios, slippage is a network phenomenon, not a sum of independent single-name costs.

A constrained cross-impact propagator framework gives better basket synchronization, fewer tail slippage events, and more reliable completion under real market coupling.

Pointers for deeper reading

• Huang, Lehalle, Rosenbaum — Simulating and analyzing order book data: The queue-reactive model (arXiv:1312.0563).
• Bacry, Mastromatteo, Muzy — Hawkes processes in finance (arXiv:1502.04592).
• Mastromatteo et al. — Trading Lightly: Cross-Impact and Optimal Portfolio Execution (arXiv:1702.03838).
• Schneider & Lillo et al. — Cross-impact and no-dynamic-arbitrage (arXiv:1612.07742).