Modeling Loss-Versus-Rebalancing in Automated Market Makers via Continuous-Installment Options

2025-08-04

AI Summary

This paper addresses impermanent loss and liquidity provider losses (LVR) in automated market makers (AMMs). We propose a unified quantitative framework grounded in option theory. Our key innovation is the first rigorous theoretical equivalence established between CFAMM liquidity positions and perpetual American continuous-installment options, enabling an analytical characterization of LVR as the ongoing funding cost embedded in such options. By integrating continuous-installment option pricing, Delta-hedging analysis, and calibration to implied volatility term structures, we construct an option-replication framework valid over an infinite time horizon. The model reveals the fundamental equivalence between adverse selection costs and option time-value decay, and derives boundary conditions under which LVR remains approximately constant—enabling long-horizon liquidity provisioning strategies with controllable approximation error. The framework thus bridges theoretical rigor with practical applicability for AMM design and liquidity management.

Technology Category

Multiagent Systems: Mechanism DesignGame Theory and Economic Paradigms: Mechanism DesignMachine Learning: Large Multimodal Models (LMMs)

Application Category

Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactionsSecurity and Privacy: Data transparency and provenanceSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper mathematically models a constant-function automated market maker (CFAMM) position as a portfolio of exotic options, known as perpetual American continuous-installment (CI) options. This model replicates an AMM position's delta at each point in time over an infinite time horizon, thus taking into account the perpetual nature and optionality to withdraw of liquidity provision. This framework yields two key theoretical results: (a) It proves that the AMM's adverse-selection cost, loss-versus-rebalancing (LVR), is analytically identical to the continuous funding fees (the time value decay or theta) earned by the at-the-money CI option embedded in the replicating portfolio. (b) A special case of this model derives an AMM liquidity position's delta profile and boundaries that suffer approximately constant LVR, up to a bounded residual error, over an arbitrarily long forward window. Finally, the paper describes how the constant volatility parameter required by the perpetual option can be calibrated from the term structure of implied volatilities and estimates the errors for both implied volatility calibration and LVR residual error. Thus, this work provides a practical framework enabling liquidity providers to choose an AMM liquidity profile and price boundaries for an arbitrarily long, forward-looking time window where they can expect an approximately constant, price-independent LVR. The results establish a rigorous option-theoretic interpretation of AMMs and their LVR, and provide actionable guidance for liquidity providers in estimating future adverse-selection costs and optimizing position parameters.

Problem

• Modeling CFAMM positions as perpetual American continuous-installment options.
• Linking AMM's adverse-selection cost (LVR) to option funding fees.
• Calibrating constant volatility for estimating LVR over long horizons.

Innovation

• Models CFAMM as perpetual American continuous-installment options
• Links LVR to continuous funding fees via option theta
• Calibrates volatility from implied volatilities term structure