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

TL;DR

This paper models automated market maker (AMM) positions as exotic options, providing a rigorous option-theoretic framework to estimate adverse-selection costs and optimize liquidity provision over long horizons.

Contribution

It introduces a novel continuous-installment options model for AMMs, linking adverse-selection costs to option time value and deriving practical calibration methods.

Findings

LVR equals the theta of the embedded CI option.

Delta profile and boundaries maintain approximately constant LVR over long periods.

Calibration methods for implied volatility and residual error estimation.

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.