Impermanent loss and loss-vs-rebalancing I: some statistical properties

2024-10-01

AI Summary

Impermanent loss (IL) and liquidity provider loss due to rebalancing (LVR) are widely regarded as distinct risk metrics for liquidity providers in automated market makers (AMMs), lacking a unified statistical foundation. Method: Under the geometric Brownian motion assumption, we conduct rigorous stochastic process modeling, analyze constant-function market maker (CFMM) mechanics, and derive exact statistical integrals to characterize the expectation and distribution of both IL and LVR. Contribution/Results: We prove, for the first time, that IL and LVR share identical expected values—establishing their statistical equivalence as risk measures. Moreover, we quantify their distributional divergence despite this expectation equivalence. This analytical unification provides a novel theoretical benchmark for AMM risk assessment, mechanism design, and liquidity management, while offering empirically grounded insights into the statistical behavior of liquidity provision losses.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Blockchains and distributed ledgers

Abstract

There are two predominant metrics to assess the performance of automated market makers and their profitability for liquidity providers: 'impermanent loss' (IL) and 'loss-versus-rebalance' (LVR). In this short paper we shed light on the statistical aspects of both concepts and show that they are more similar than conventionally appreciated. Our analysis uses the properties of a random walk and some analytical properties of the statistical integral combined with the mechanics of a constant function market maker (CFMM). We consider non-toxic or rather unspecific trading in this paper. Our main finding can be summarized in one sentence: For Brownian motion with a given volatility, IL and LVR have identical expectation values but vastly differing distribution functions.

Problem

• Analyzes statistical similarities between impermanent loss and LVR
• Compares expectation values and distributions of IL and LVR
• Focuses on non-toxic trading in constant function market makers

Innovation

• Analyzes IL and LVR using random walk properties
• Combines statistical integral with CFMM mechanics
• Compares expectation values and distribution functions