Concave Multi-Asset Propagator Models

• The paper introduces concave multi-asset propagator models that capture self- and cross-impact effects with sublinear (square-root law) price responses.
• It employs tensor/matrix-valued propagators and nonparametric, shape-constrained estimation to ensure interpretable, high-dimensional calibration without excessive parameter tuning.
• These models are pivotal for optimal execution, risk management, and market making by integrating nonlinear control methods with robust empirical liquidity analysis.

Concave multi-asset propagator models are mathematical frameworks designed to capture the transient and nonlinear effects of trading activity in a system of multiple risky assets. In these models, the evolution of asset prices is influenced both by self-impact (the effect of trading an asset on its own price) and cross-impact (the effect of trading one asset on the prices of others). The central feature is concavity: market impact grows sublinearly with trading size, typically modeled as a power law or square-root function. These models are highly relevant for optimal execution, risk management, market making, and systemic risk, especially in high-dimensional and constrained settings.

1. Mathematical Structure and Core Principles

Concave multi-asset propagator models generalize classical single-asset transient impact frameworks by introducing tensor-valued or matrix-valued propagators and nonlinear (concave) impact functions. For $d$ assets, the reference model expresses the price change for asset $\ell$ over a time interval as:

$$P_{t_{i+1}}^{\ell} - P_{t_0}^{\ell} = \sum_{k=1}^d \sum_{j=0}^i G_{i,j}^{(\ell,k)} \cdot h_{c_{\ell,k}}(Q_{t_j}^k) + \varepsilon_{t_i}^{\ell}$$

where:

• $G_{i,j}^{(\ell,k)}$ is the entry of the multi-asset propagator kernel specifying impact from asset $k$ at earlier time $j$ on asset $\ell$ at time $i$,
• $Q_{t_j}^k$ is the traded volume of asset $k$ at time $j$,
• $h_c(x) = \operatorname{sign}(x) |x|^c$ is the concave impact function (typically with $c \sim 0.5$ for the square-root law),
• $\varepsilon_{t_i}^{\ell}$ captures noise.

This recursive structure propagates the effect of metaorders through time and across assets, naturally allowing for both self- and asymmetric cross-impact. The concavity in $h_c$ reflects empirically observed sublinear price response, offering realism particularly for large trades.

2. Estimation and Calibration

The estimation of the propagator kernel in the multi-asset, concave setting is technically challenging due to high-dimensional data and the need to avoid parameter explosion. Recent work introduces an offline nonparametric estimator that solves a regularized least-squares regression problem:

$$\widehat{G}_{n, \lambda} = \arg\min_{G \in \mathcal{G}_{\mathrm{ad}}} \left[ \sum_{n=1}^N \|y^{(n)} - U^{(n)} \cdot G \|^2 + \lambda \|G\|^2 \right]$$

where $y^{(n)}$ are normalized price responses, $U^{(n)}$ are block matrices of h-transformed volumes, and $\mathcal{G}_{\mathrm{ad}}$ is the cone of admissible kernels: nonnegative, nonincreasing, and convex in lag (see (Hey et al., 8 Oct 2025)). The unconstrained solution is subsequently projected onto this cone to ensure interpretable and microstructurally consistent kernels.

For sparse datasets, metaorder proxies are constructed by random assignment of synthetic trader IDs to individual trades and grouping by sign, stabilizing calibration and producing smoother decay kernels.

Data sources typically include proprietary metaorder files from institutional platforms (e.g. CFM) and public tick-level order flow datasets (e.g. S&P 500), with normalization of returns and volumes to volatility and daily totals for comparability.

3. Empirical Properties: Self- and Cross-Impact

Empirical evidence demonstrates:

• Self-impact is strongly concave, with peak impact following the square-root law:

$$I_\mathrm{peak} = Y \sigma_D \, \operatorname{sign}(Q) \left|\frac{Q}{V_D}\right|^{\delta}, \quad \delta \approx 0.5$$

and the decay kernel is well-fit by a shifted power-law in lag, rather than an exponential.

• Cross-impact is significant and inherently asymmetric. The impact of trades in asset $k$ on asset $\ell$ is typically stronger when $k$ is more liquid than $\ell$. Concave cross-impact models (i.e. $h_c$ with $c < 1$) outperform linear specifications. This suggests that the square-root law may extend to cross-impact effects.
• Modeling with a nonparametric approach, followed by shape-constrained projection, avoids parameter tuning (which increases as $O(d^2)$ for parametric models) and produces kernels that are not only interpretable (monotonic, convex) but empirically more accurate out of sample than conventional exponential or power-law parametric forms.

4. Optimization and Applications

In practical multi-asset trading and execution scenarios, propagator models with concave kernels are embedded in control problems. For example, (Jaber et al., 6 Mar 2025) formulates optimal execution with alpha signals and power-law decay via a nonlinear stochastic Fredholm equation:

$$\gamma u_t + (\mathcal{A}(u))_t + (H_{\phi,\rho} u)_t + (H_{\phi,\rho}^* u)_t = \alpha_t - X_0 [\phi (T - t) + \rho]$$

where $\mathcal{A}(u)$ captures the nonlinear, concave price impact.

Iterative schemes such as projected gradient ascent, combined with Nyström discretization and least-squares Monte Carlo regression for conditional expectations, are employed to compute optimal controls under constraints for battery storage and financial trading applications (Jaber et al., 18 Sep 2024).

These methods ensure numerical tractability and convergence guarantees, even in high-dimensional and illiquid, constrained environments.

5. Interpretation and Predictive Power

Projection onto the admissible cone of kernels improves both interpretability and predictive accuracy. The shape-constrained estimator (projected nonparametric kernel) slightly outperforms both the raw unconstrained estimator and standard parametric approaches (e.g., one- or two-exponential decay, parametric power law). This improvement is documented across both proprietary and public datasets.

Furthermore, empirical cross-impact asymmetry and concavity are shown to directly reflect underlying liquidity structures, forming a basis for explainable market impact models that account for inter-asset dependencies.

6. Significance and Future Directions

Concave multi-asset propagator models unify several strands in market microstructure: transient impact estimation, nonlinear control, and interpretability in high-dimensional financial systems.

This suggests several plausible implications:

• The square-root law's extension to cross-impact is supported both empirically and algorithmically.
• Nonparametric, shape-constrained estimation should become standard for calibration in high-dimensional execution and market-making systems, especially as traditional parametric models become infeasible for large $d$.
• The framework provides a robust foundation for new optimization approaches involving stochastic signals, constraints, and systemic risk quantification.

Challenges remain in scaling shape-constrained projections and offline reinforcement learning approaches for extremely large asset universes. Additionally, effective use of proxies and data normalization continues to be critical for robust out-of-sample performance and interpretability.

Table: Comparison of Estimation Approaches

Approach	Parameter Tuning	Out-of-Sample Accuracy
1-EXP/2-EXP Parametric	$O(d^2)$	Baseline
Nonparametric Raw (RAW)	None	Improved, but unconstrained
Shape-Projected (PROJ)	None	Highest, interpretable kernel

Parametric models require exhaustive grid search and tuning but are limited in dimension. Nonparametric and shape-constrained approaches achieve interpretability and improved accuracy without parameter explosion, making them preferable for multi-asset propagator calibration (Hey et al., 8 Oct 2025).