Liquidity-Based Audit of Algorithmic Trading Strategies

Technical Summary: Liquidity-Based Audit of Algorithmic Trading Strategies

Problem Statement
Market microstructure theory traditionally classifies trading activity based on informational roles: informed traders demand liquidity by trading with private signals, while market makers supply liquidity by absorbing order flow (Kyle, 1985; Glosten and Milgrom, 1985). However, applying this classification to algorithmic trading strategies is difficult because the internal logic, signals, and optimization problems of quantitative funds are unobservable. Regulators and auditors typically only observe realized trade trajectories (prices and positions) rather than the underlying order flow or quotes. This paper addresses the problem of identifying a strategy's net liquidity role (consumer vs. provider) and measuring its performance regret using only observable trade and price history, without access to the strategy's internal model.

Methodology
The paper develops a framework based on multi-period stochastic dynamic programming and exact regret decomposition.

1. Covariance Decomposition: The core theoretical result (Theorem 3.6) establishes that the multi-period regret of a policy $\hat{\pi}$ relative to a perfect-foresight benchmark decomposes exactly into the sum of per-period covariances between the realized cost vector $c_t$ and the strategy's decision $\hat{\pi}_t(c_t)$. Under the assumption that the policy is mean-unbiased (i.e., its expected output matches the conditional cost-minimizing allocation), the regret is simply $\sum \text{Cov}(c_t, \hat{\pi}_t)$.
2. Liquidity Classification: The sign of this covariance statistic serves as a model-free classifier. A positive covariance indicates the strategy systematically buys when costs (prices) are rising and sells when they are falling, identifying it as a net liquidity consumer. A negative covariance indicates the reverse behavior (buying dips, selling peaks), identifying it as a net liquidity provider.
3. AR(1) Cost Process and Roll Spread: Assuming costs follow an AR(1) process (standard for short-horizon return reversal), the paper derives a closed-form correction to the regret sum. This correction is shown to equal $\alpha s^2$, where $\alpha$ is the strategy size and $s$ is the Roll (1984) implied bid-ask spread. This links the performance audit metric directly to market microstructure liquidity.
4. Endogenous Price Impact: The framework is extended to include linear temporary price impact (Almgren and Chriss, 2001). The total regret is shown to be additively separable into a "covariance regret" (the audit metric) and a "price-impact regret," allowing for independent estimation of both channels from trajectory data.
5. Systemic Extension (N-Agent): The paper aggregates $N$ correlated strategies under a common-factor cost structure. It derives an aggregate liquidity-balance condition where the welfare loss scales as $N^2$ in the strategy-correlation coefficient. This provides a closed-form expression for the "fire-sale" externality, where correlated deleveraging overwhelms market liquidity.

Key Results

• Exact Classification: The sign of $\text{Cov}(c_t, \hat{\pi}_t)$ exactly recovers the Kyle (1985) informed-trader/market-maker dichotomy from trajectory data alone. Momentum strategies are proven to be liquidity consumers, while contrarian strategies are liquidity providers.
• Spread Estimation: Under an AR(1) cost process, the auto-covariance correction to the regret sum is exactly $\alpha s^2$. This allows the Roll implied spread to be estimated as a byproduct of the regret audit, requiring no additional data or computational cost beyond the trajectory estimator.
• Empirical Calibration (2016–2025): Using CRSP equity data, the paper tracks implied spreads through distinct market regimes:
  • Normal Markets: The AR(1) correction is small (1–5% of true regret), and contrarian strategies outperform momentum.
  • COVID-19 Crisis (Q2 2020): The implied spread tripled (from ~0.90% to 2.52%), and the AR(1) correction grew significantly, reflecting a breakdown in microstructure depth. The fragility multiplier (systemic risk) rose to 60x for 100 correlated agents.
  • 2022 Rate Shock: The AR(1) coefficient collapsed to near zero, causing the implied spread to hit sample minimums. In this regime, the bias correction vanished, and all mean-reversion strategies underperformed, confirming the estimator's sensitivity to structural regime shifts.
• Systemic Fragility: The paper demonstrates that even if individual agents satisfy audit criteria, aggregate regret can scale quadratically ($N^2$) if strategies are highly correlated. A systemic fragility index is proposed, computable in $O(T \cdot N^2 \cdot nd)$ time, which detects the onset of liquidity spirals before they fully develop.

Significance and Claims
The paper claims to provide a "black-box" audit tool that recovers a strategy's liquidity role and market impact solely from observable inputs (prices and positions), bypassing the need for proprietary signal data.

• Model-Free Liquidity Role: It recovers the fundamental microstructure dichotomy (liquidity consumer vs. provider) using only realized trajectories, making it applicable to opaque algorithmic strategies.
• Direct Proxy for Illiquidity: The regret correction is not merely a statistical adjustment but an exact proxy for the squared Roll spread, offering a real-time measure of market depth that tracks liquidity crises and structural breaks.
• Systemic Risk Quantification: It provides a closed-form statement of the fire-sale mechanism, showing how correlated algorithmic strategies can generate welfare losses scaling as $N^2$. This offers a quantitative basis for systemic regulation (e.g., strategy diversity requirements or aggregate position limits) that complements existing single-firm audits (SR 11-7).
• Computational Efficiency: The proposed estimators are computationally efficient ($O(T \cdot nd)$ for single agents, $O(T \cdot N^2 \cdot nd)$ for systemic), enabling real-time regulatory monitoring without simulating future states or accessing internal optimization engines.

The paper concludes that while the classifier is coarser than order-flow-based measures (as it averages over time rather than capturing intraday dynamics), it is uniquely suited for auditing opaque algorithmic strategies and provides a rigorous, observable bridge between performance attribution and market microstructure theory.