Arbitrage and the Stability of AMM Price Tracking

TL;DR

This paper provides a quantitative analysis of AMM price stability driven by arbitrage, modeling the tracking error as a stochastic process and deriving explicit bounds and conditions for stability.

Contribution

It introduces a stochastic model of AMM price tracking error incorporating blockchain execution details and offers a quantitative stability theorem with empirical calibration.

Findings

Proves geometric ergodicity of the tracking error under certain conditions.

Derives explicit bounds linking tracking quality to liquidity and execution parameters.

Shows how fees and liquidity influence the no-trade band and optimal corrective trades.

Abstract

Automated market makers (AMMs) quote prices from pool state rather than from a limit order book. AMM pools often stay close to a reference price because arbitrageurs correct profitable mispricing. A large part of decentralized finance therefore relies on a simple economic premise: once the AMM price drifts away from the reference price, arbitrage incentives push it back. This paper studies when that premise is strong enough to guarantee block-scale stability. We model the gap between the reference price and the AMM price as a stochastic tracking error, treat arbitrage as the corrective input, and place blockchain execution inside the loop through fees, discrete blocks, transaction ordering, delays, and transaction failure. The detailed execution layer is reduced to the total successful correction confirmed in each block. Under a block-level correction condition, we prove geometric ergodicity of the tracking error and obtain explicit one-step bounds that connect tracking quality to liquidity and execution quality. We also show in a constant-product example how fees, fixed execution costs, and local liquidity map into the no-trade band and the optimal corrective trade. Finally, we build empirical proxies for the theorem quantities from realized block data and use them to organize reduced and mechanism-focused simulations whose comparative statics are consistent with the theory. The contribution is to turn a basic economic intuition behind decentralized finance into a quantitative stability statement together with a tractable calibration interface.