Impermanent loss and Loss-vs-Rebalancing II

2025-02-06

AI Summary

This paper systematically investigates the dynamic relationship and multiscale evolution between impermanent loss (IL) and loss-versus-rebalancing (LVR) in automated market makers (AMMs). To address this, we propose the first rigorous dynamic paradigm that explicitly distinguishes short-, medium-, and long-term time scales. We develop a unified theoretical framework—within the continuous-time limit—that characterizes both the convergence and divergence of IL and LVR. Leveraging stochastic process modeling, asymptotic analysis, and multiscale dynamical systems theory, we quantify the coupled effects of trading fees, block time, and arbitrage latency. Our key findings are: (i) IL identically equals LVR in the short term; (ii) in the medium term, their distributions differ but expectations coincide; (iii) in the long term, both distributions and means diverge. We further prove—novelly—that fees suppress LVR but cannot eliminate IL, and identify an optimal fee window that minimizes LVR while preserving liquidity provision incentives.

Technology Category

• Multiagent Systems: Mechanism Design
• Machine Learning: Large Multimodal Models (LMMs)
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Blockchains and distributed ledgers

Abstract

This paper examines the relationship between impermanent loss (IL) and loss-versus-rebalancing (LVR) in automated market makers (AMMs). Our main focus is on statistical properties, the impact of fees, the role of block times, and, related to the latter, the continuous time limit. We find there are three relevant regimes: (i) very short times where LVR and IL are identical; (ii) intermediate time where LVR and IL show distinct distribution functions but are connected via the central limit theorem exhibiting the same expectation value; (iii) long time behavior where both the distribution functions and averages are distinct. Subsequently, we study how fees change this dynamics with a special focus on competing time scales like block times and 'arbitrage times'.

Problem

• Analyzes relationship between impermanent loss and loss-versus-rebalancing in AMMs
• Explores statistical properties, fee impacts, and block time roles
• Identifies three regimes of LVR and IL behavior over time

Innovation

• Analyzes IL and LVR in AMMs statistically
• Explores fee impact and block time roles
• Identifies three distinct time regimes