Impermanent loss and Loss-vs-Rebalancing II

TL;DR

This paper analyzes the statistical relationship between impermanent loss and loss-vs-rebalancing in AMMs, revealing different regimes over time and how fees influence these dynamics, with implications for DeFi risk management.

Contribution

It provides a detailed characterization of IL and LVR over different time regimes and examines the effects of fees and block times on their relationship.

Findings

IL and LVR are identical at very short times

Distinct distribution functions for IL and LVR at intermediate times

Fees and block times significantly influence IL and LVR dynamics

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'.