Arbitrage and the Stability of AMM Price Tracking

Technical Summary: Arbitrage and the Stability of AMM Price Tracking

Problem Statement
Automated Market Makers (AMMs) determine prices based on pool state rather than a limit order book. While the prevailing economic intuition suggests that arbitrageurs will correct deviations between AMM prices and external reference prices, existing literature lacks a rigorous proof that this mechanism guarantees stability at the block scale. On blockchains, price correction is not continuous; it is discrete, subject to transaction ordering, delays, fees, and potential failures. Furthermore, in concentrated-liquidity pools, the efficacy of correction is state-dependent. Current research either focuses on continuous-time approximations, isolated arbitrage profits, or specific execution risks (MEV, frontrunning) without unifying these factors into a closed-loop model that proves the tracking error remains stable.

Methodology
The authors model the AMM price tracking process as a stochastic closed-loop system operating at the block level.

1. State Definitions:

   • $P_n$: Reference log-price before block $n$.
   • $\tilde{P}_n$: AMM log-price after block $n$.
   • $x_n = P_n - \tilde{P}_{n-1}$: Pre-execution tracking error (gap seen by arbitrageurs).
   • $z_n = P_n - \tilde{P}_n$: Post-execution tracking error.
   • The dynamics follow $x_{n+1} = z_n + w_{n+1}$, where $w_n$ is an exogenous disturbance.

2. Execution Layer Reduction:
   Instead of modeling the complex game of searchers and transaction ordering, the authors reduce the execution layer to two key quantities:

   • Dead-zone cap ($\bar{\gamma}$): A measurable bound representing the maximum "no-trade" radius due to fees, slippage, and fixed costs.
   • Realized Correction ($C_n$): The total successful reduction in error magnitude delivered by the block.
   • A sequential execution rule determines which transactions in a block succeed based on whether the remaining error exceeds the transaction's specific executable residual band.

3. Theoretical Framework:

   • Assumptions: The model assumes a time-homogeneous stochastic kernel for block environments and a "Large-error correction condition." This condition posits that when the gap is sufficiently large, there is a non-zero probability ($p^*$) that the block will remove at least a fraction ($\lambda^*$) of the excess gap.
   • Proposition 1: Provides an explicit calculation for the "no-trade band" and optimal arbitrage size in a constant-product AMM, demonstrating how fees and fixed costs create a quantitative threshold for profitable correction.
   • Proposition 2: Establishes that the post-execution error is bounded by the pre-execution error minus the realized correction, capped by the dead-zone.

4. Stability Analysis:
   The authors employ Foster-Lyapunov theory to analyze the Markov chain of tracking errors. They construct a Lyapunov function $V(x) = e^{\alpha|x|}$ to prove that, under the large-error correction condition, the system exhibits negative drift outside a compact set.

Key Results

• Theorem 1 (Geometric Ergodicity): Under the stated assumptions, the tracking error process $\{x_n\}$ is a positive Harris recurrent Markov chain that converges to a unique invariant probability measure $\pi_x$ at a geometric rate. This proves that the AMM price tracking is stable in the long run, provided that large deviations trigger reliable arbitrage corrections.
• Explicit Bounds: The paper derives explicit one-step bounds connecting tracking quality to liquidity and execution quality. Specifically, the convergence rate depends on the product of the correction fraction ($\lambda^*$) and the delivery probability ($p^*$).
• Empirical Calibration: Using Ethereum mainnet data (blocks 16800007–16803511), the authors construct empirical proxies for the theorem's parameters:
  • Dead-zone proxy ($\bar{\gamma}$): $\approx 3.82 \times 10^{-4}$.
  • Large-error correction share ($\hat{\lambda}^*$): 0.50.
  • Delivery probability ($\hat{p}^*$): 72.9%.
  • The data confirms that large gaps are corrected more frequently and deeply than small gaps.
• Simulation Validation: Reduced simulations driven by these empirical parameters demonstrate that the system returns to the operating tube quickly after shocks. Comparative statics show that increasing the correction share or delivery probability tightens tracking, while higher fixed costs or lower liquidity widen the tracking error.

Significance and Claims
The paper's primary contribution is transforming the basic economic intuition that "arbitrage aligns AMM prices" into a quantitative stability statement with a tractable calibration interface.

• Theoretical Gap Filled: It provides the first block-scale stochastic closed-loop model that joins AMM dynamics, arbitrage incentives, and execution constraints (fees, ordering, failure) to prove stability.
• Practical Utility: The authors do not claim a sharp empirical validation of the theorem itself. Instead, they demonstrate that the quantities required by the theorem (dead-zone, correction probability) are observable on-chain and that simulations driven by these observed quantities behave consistently with the theoretical predictions.
• Mechanism Insight: Through a constant-product example, the paper explicitly maps how fees, fixed execution costs, and reserve depth determine the economic no-trade band and optimal corrective trade size, offering a clear interface for interpreting how liquidity and friction impact price stability.

In summary, the work establishes that block-scale AMM price tracking is geometrically ergodic if arbitrage incentives are strong enough to ensure that, with sufficient probability, large deviations are partially corrected in every block. This stability is contingent on the interplay between liquidity depth, execution costs, and the reliability of the correction mechanism.