Convex Modeling of Price Cross-Impact over Time

Technical Summary: Convex Modeling of Price Cross-Impact over Time

Problem Statement
Transaction costs are a critical determinant of profitability in relative-value trading, particularly in commodity and macro markets where edges are measured in basis points. Existing price impact models typically suffer from two significant omissions:

1. Cross-Impact: They fail to account for the fact that trading one contract affects the prices of related contracts. Consequently, they overprice the cost of relative-value trades (e.g., calendar spreads) where correlated legs are built and unwound, ignoring the partial cancellation of impacts.
2. Transient Impact: They often assume impact is permanent or decays exponentially, neglecting the empirically observed power-law decay where price impact dissipates over time. This leads to models that fail to capture the recovery of entry costs during unwinding.

Standard multi-period portfolio construction frameworks model impact costs as separable across contracts and time periods. This separability results in conservative portfolios that forgo potentially profitable trades because they do not recognize that the impact of a "long" leg in a spread trade can be offset by the impact of a simultaneous "short" leg in a related contract.

Methodology
The authors propose a convex quadratic model of transient cross-impact cost that simultaneously captures interactions across contracts and over time.

• Single-Period Cross-Impact: For a given time period $t$, the model defines a positive semidefinite matrix $\Lambda_t$ that couples trades across $n$ contracts. This matrix is constructed as $\Lambda_t = \Gamma_t^{1/2} C_t \Gamma_t^{1/2}$, where:

  • $\Gamma_t$ is a diagonal matrix containing self-impact coefficients $\gamma_{t,i}$, derived from forecasted volatility ($\sigma_{t,i}$) and dollar volume ($v_{t,i}$).
  • $C_t$ is a positive semidefinite impact correlation matrix (proxied by return correlation) with unit diagonal.
  • This construction ensures that self-impact remains consistent with standard linear models while introducing cross-impact without disturbing the diagonal terms.

• Transient (Temporal) Impact: The model couples trades across time periods using a power-law decay kernel $k(h) = (1 + h/\tau)^{-\beta}$, consistent with empirical findings that impact decays slower than an exponential.

  • The total impact cost $\phi_{imp}(u)$ for a sequence of trades $u$ is formulated as a quadratic form $u^\top A u$.
  • The matrix $A$ is constructed by combining the block-diagonal structure of the single-period matrices $\Lambda_t$ with a temporal coupling matrix derived from the symmetric version of the decay kernel.
  • Crucially, the model uses a symmetric coupling ($\Lambda_t^{1/2} \Lambda_s^{1/2}$) rather than a strictly causal one ($G_{ts}\Lambda_s$) to ensure the resulting cost matrix $A$ remains positive semidefinite.

• Convexity and No-Price-Manipulation: The paper proves that the resulting cost matrix $A$ is positive semidefinite for any positive semidefinite $\Lambda_t$. This convexity guarantees that the expected cost of any round-trip trade (buying and selling back to zero inventory) is non-negative, thereby ruling out illegal price manipulation strategies that could arise in non-convex models with time-varying liquidity.

Key Contributions

1. Unified Model: The paper introduces a convex quadratic model that simultaneously incorporates cross-impact (across contracts) and transient impact (across time) with power-law decay.
2. Theoretical Guarantee: It provides a mathematical proof that the proposed symmetric coupling ensures convexity and the absence of negative-cost round trips, even when liquidity forecasts vary over the planning horizon.
3. Empirical Validation: The model is tested on the trading of WTI crude oil futures calendar spreads around the S&P GSCI index roll. The study utilizes public data to simulate liquidity and volatility, comparing the proposed model against a separable baseline (3/2-power cost) and a self-impact-only model.

Results
In the empirical study of index-roll positioning (2004–2011):

• Overpricing by Baselines: Existing separable models significantly overprice the impact of relative-value trades. The baseline model, treating legs independently, yields a highly conservative portfolio that barely trades.
• Netting Effect: The proposed model correctly identifies that the price impacts of the long and short legs of a spread (with an estimated correlation of 0.96) partially cancel each other out.
• Performance: The "transient cross-impact" model generated a net profit of 3.9 basis points (of gross capacity) on average, compared to 2.7 basis points for the self-impact model and 2.3 basis points for the baseline.
• Cost Accuracy: The proposed model predicted an impact cost of 5.9 basis points, which aligned closely with the simulated realized cost (5.9 bp), whereas the baseline model predicted a much higher cost (7.7 bp) that did not reflect the actual market dynamics.

Significance
The paper claims that by accurately modeling the interplay between cross-asset correlations and temporal decay, traders can avoid the overpricing inherent in separable models. This allows for the identification of profitable relative-value trades that would otherwise be discarded due to inflated cost estimates. The model is particularly relevant for macro and commodity markets where liquidity is predictable (e.g., scheduled index rolls, report releases) and where relative-value strategies rely on the temporary dislocation of spreads. The convexity of the model ensures that the optimization problem remains tractable and adheres to no-arbitrage conditions, preventing the suggestion of illegal manipulation strategies.