Cross-Impact Propagators, Matrix Volterra Equations, and Optimal Multi-Asset Execution

Abi Jaber, Neuman, and Tuschmann develop a rigorous framework for optimal multi-asset portfolio execution in the presence of cross-impact — the phenomenon that trading in one asset shifts the price of other correlated assets — modeled through a matrix-valued Volterra propagator. The single-asset price impact literature (Almgren–Chriss, Gatheral's propagator model) has established that a trader who executes a large order shifts the market price of the traded asset by an amount proportional to the order's signed volume, with the impact decaying over time according to a mean-reverting kernel (the propagator). In the multi-asset case, the impact of trading asset j on the price of asset i is encoded by the off-diagonal entry G_{ij}(t) of a matrix-valued propagator G(t) ∈ R^{d×d}, so that the mid-price vector S(t) ∈ R^d evolves as S_t = S_0 + ∫_0^t G(t-s) dQ_s + M_t where Q_s is the cumulative trading vector at time s (so dQ_s is the vector of instantaneous trading rates), G(t-s) is the propagator matrix evaluated at lag t-s, and M_t is a martingale representing the exogenous price innovations unrelated to the trader's activity. The cross-impact propagator G_{ij}(t) quantifies both the own-impact of trading i on i (the diagonal G_{ii}) and the cross-impact of trading j on i (the off-diagonal G_{ij}), and the key empirical observation motivating the paper is that the off-diagonal elements G_{ij} for highly correlated assets — such as equities in the same sector, or assets linked by index arbitrage — are a significant fraction of the diagonal elements, meaning that ignoring cross-impact leads to substantially suboptimal execution strategies that trade too aggressively in correlated assets and incur more total impact cost than necessary. The optimal portfolio execution problem is to find the trading rate process θ_t (the vector of instantaneous trading rates dQ_t/dt) that minimizes the expected total execution cost C(θ) = E[∫_0^T (θ_t · S_t + (1/2) θ_t · Λ θ_t) dt] — the sum of the price impact cost (the price S_t is pushed away from its initial value by past trades) and the instantaneous execution cost (the bid-ask spread and market impact of the current trade θ_t, modeled by the positive definite matrix Λ) — subject to the inventory constraint ∫_0^T θ_t dt = q_0 (the total quantity q_0 must be executed by time T). The paper establishes that the optimal trading rate θ* satisfies a matrix Fredholm integral equation of the second kind: θ*_t + ∫_t^T K(s-t) θ*_s ds = λ(t) where K is the matrix resolvent kernel of the propagator G and λ(t) is a Lagrange multiplier vector determined by the inventory constraint. This is the direct multi-asset generalization of the scalar Fredholm equation arising in single-asset propagator models (Gatheral's formula for the optimal liquidation rate in the Almgren–Chriss model with power-law impact), and its solution provides the globally optimal trading schedule across all d assets simultaneously. The technical novelty lies in the operator-theoretic analysis of the matrix Fredholm equation: the resolvent kernel K must be constructed from the propagator G via the matrix Volterra equation G + K * G = K (where * denotes matrix convolution), a vector-space generalization of the scalar Volterra resolvent that requires G to satisfy a positive semi-definiteness condition (the no-statistical-arbitrage condition) for the resolvent K to exist and for the trading strategy θ* to be admissible (finite cost). The paper proves that the matrix propagator G satisfies this condition if and only if the spectral measure of the Fourier transform ĝ(ω) = ∫_0^∞ e^{-iωt} G(t) dt satisfies Re[ĝ(ω)] ≥ 0 for all ω — a generalization of the Bochner condition for positive definiteness — and that under this condition the Fredholm equation for θ* has a unique solution in L²([0,T], R^d).

The mid-price vector evolves as S_t = S_0 + ∫_0^t G(t-s) θ_s ds + M_t where G : [0,∞) → R^{d×d} is the matrix propagator with G_{ij}(t) the impact of trading rate θ_j on price i, and M_t is a martingale (exogenous innovations). The execution cost is C(θ) = ∫_0^T (θ_t^⊤ S_t + (1/2) θ_t^⊤ Λ θ_t) dt = ∫_0^T ∫_0^T (1/2) θ_s^⊤ G(|t-s|) θ_t ds dt + (1/2) ∫_0^T θ_t^⊤ Λ θ_t dt + M terms, where Λ ∈ R^{d×d} is positive definite (the temporary impact matrix). The optimal rate θ* satisfies the matrix Fredholm equation of the second kind: θ*_t + ∫_t^T K(s-t) θ*_s ds = (T-t)/(T) · Λ^{-1} q_0 where K(t) is the resolvent kernel of G defined by G(t) + ∫_0^t K(t-s) G(s) ds = K(t). The no-statistical-arbitrage condition on G is: for all square-integrable trading strategies θ with ∫_0^T θ_t dt = 0, one has ∫_0^T ∫_0^T θ_s^⊤ G(|t-s|) θ_t ds dt ≥ 0, equivalently Re[ĝ(ω)] is positive semi-definite for all ω where ĝ(ω) = ∫_0^∞ e^{-iωt} G(t) dt is the one-sided Fourier transform of G. The cost reduction from cross-impact-aware execution versus naive single-asset execution is ΔC = (1/2) ∫_0^T ∫_0^T (θ*_s - θ^{naive}_s)^⊤ [G(|t-s|) + G(|t-s|)^⊤] (θ*_t - θ^{naive}_t) ds dt ≥ 0.

The matrix Fredholm equation structure for the optimal multi-asset trading rate θ* is the natural operator-theoretic generalization of the Almgren–Chriss optimal schedule: in the scalar single-asset case with exponential propagator G(t) = γ e^{-ρt}, the Fredholm equation reduces to a simple ODE for θ* whose solution is the well-known hyperbolic trading schedule θ*(t) = q_0 · sinh(κ(T-t)) / (T · sinh(κT)) for an effective decay rate κ determined by the impact parameters. In the matrix multi-asset case, the Fredholm equation cannot in general be reduced to an ODE — the matrix resolvent K depends on the full spectral structure of G and may have memory extending over the entire execution horizon [0,T] rather than decaying exponentially — and the solution must be computed numerically via an iterative Fredholm solver or a finite-element discretization of the integral equation on the grid t_0 < t_1 < … < t_N = T. The positive semi-definiteness of Re[ĝ(ω)] — the no-statistical-arbitrage condition on the cross-impact propagator G — has a direct empirical interpretation: it says that a round-trip trading strategy that buys and sells according to any pattern θ with zero net inventory change incurs non-negative expected cost, meaning the market cannot be profitably round-tripped through the market impact mechanism. Violation of this condition would imply the existence of an arbitrage in the price impact model, and the paper establishes that any empirically estimated propagator matrix G must satisfy this condition if the model is to be internally consistent. In practice, empirical estimation of the off-diagonal cross-impact elements G_{ij}(t) from high-frequency order flow and mid-price data requires careful deconvolution of the lagged correlation structure between order flows and price changes across assets, and the estimated G_{ij} may fail the positive semi-definiteness test if the estimation window is too short or the asset universe is too large (more assets than independent risk factors), requiring regularization of the estimated G before the Fredholm equation can be solved. The paper's result also clarifies the relationship between cross-impact execution optimization and portfolio construction: the optimal multi-asset execution schedule θ*(t) depends not only on the target quantities q_0 but also on the cross-impact matrix G, meaning that the execution cost of a multi-asset portfolio is not the sum of the single-asset execution costs — a sum that ignores the cross-asset impact terms G_{ij} — but is instead determined by the quadratic form ∫∫ θ^⊤ G(|t-s|) θ ds dt, which mixes the trading rates of all assets. This cross-coupling means that the optimal execution strategy for a multi-asset portfolio may involve cross-asset offsetting trades — temporarily trading asset j in the opposite direction to reduce the cross-impact cost of trading asset i — that look suboptimal from a single-asset perspective but are globally optimal when cross-impact is accounted for.

For the desk's execution workflow on correlated equity baskets, the cross-impact propagator framework provides a quantitative basis for breaking the single-asset execution decomposition that is typically applied in practice: rather than scheduling the execution of each asset independently according to its own single-asset optimal schedule, the multi-asset Fredholm equation yields a joint schedule θ*(t) that accounts for the cross-impact between assets. In practice the cross-impact terms G_{ij} are most significant for assets in the same sector or index — where the shared order flow and index arbitrage activity generate strong off-diagonal propagator elements — and the cost reduction from using the joint schedule versus the independent schedules can be estimated as the quadratic difference ΔC = ∫∫ (θ^{naive} - θ*)^⊤ [G + G^⊤] (θ^{naive} - θ*)/2 ds dt, which the paper shows can reach 10–30% of total execution cost for baskets of 10–50 assets with pairwise correlations above 0.5. The practical implementation requires estimating the matrix propagator G(t) from historical high-frequency order flow data: one regresses the price impact ΔS_i(t) of asset i on the lagged trading flows dQ_j(t-s) of all assets j across a range of lags s, obtaining the empirical propagator as a matrix of impulse response functions G_{ij}(s) = ∂ΔS_i(t)/∂dQ_j(t-s). The estimation requires intraday data at a frequency of at least one minute for the lags s between 0 and 30 minutes (the typical propagator decay time), and the empirical G_{ij}(s) must be regularized (via ridge regression or a parametric exponential-decay specification) to ensure positive semi-definiteness of Re[ĝ(ω)] before the Fredholm equation is solved. For a universe of d = 20 assets, the propagator matrix has d² = 400 elements, each a function of lag s, requiring substantial data to estimate reliably — typically 6–12 months of intraday data at one-minute resolution, which provides approximately 10⁵ observations.

The matrix Fredholm equation for θ* assumes a linear impact model — price S_t responds linearly to the trading rate θ_t through the convolution with G — but empirical evidence across many asset classes suggests that the impact of large orders is superlinear (concave-power-law) in the trade size, consistent with the square-root impact law dS ∼ √|dQ| observed empirically. The linear propagator model is a good approximation for small-to-medium order sizes (well within the liquidity of each asset) but may significantly underestimate the impact cost of large orders that consume a substantial fraction of daily volume, in which case a nonlinear propagator model would be required. The nonlinear generalization of the Fredholm equation has not been developed at the same level of mathematical rigor, and the optimal execution strategy for nonlinear cross-impact propagators must be computed numerically via dynamic programming or stochastic control methods rather than via the closed-form Fredholm solution. The empirical estimation of the off-diagonal cross-impact elements G_{ij}(s) from order flow data is confounded by the latent common factor structure of equity order flows: in a market where index arbitrageurs simultaneously trade all assets in an index basket in proportion to their index weights, the empirical correlation between the order flows of assets i and j is driven partly by genuine cross-impact (trading i shifts the price of j) and partly by common factor trading (a latent index arbitrage flow simultaneously trades both i and j, creating a spurious correlation in their price impacts). Disentangling the two requires a factor decomposition of the order flow matrix, which introduces additional estimation uncertainty and may be sensitive to the number of factors assumed. If the common factor structure is not properly accounted for, the estimated G_{ij}(s) will be inflated relative to the true cross-impact propagator, leading to over-aggressive cross-asset offsetting trades in the optimal execution schedule.