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

ArXiv ID: 2508.02971 “View on arXiv”

Authors: Srisht Fateh Singh, Reina Ke Xin Li, Samuel Gaskin, Yuntao Wu, Jeffrey Klinck, Panagiotis Michalopoulos, Zissis Poulos, Andreas Veneris

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.

Keywords: Automated Market Maker (AMM), Perpetual American Options, Loss-versus-Rebalancing (LVR), Delta Hedging, Implied Volatility, Cryptocurrencies

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper develops a highly sophisticated mathematical framework modeling AMM positions as perpetual continuous-installment options, featuring dense derivations, stochastic calculus, and advanced option theory. However, it presents theoretical models and calibration concepts with no actual backtests, code, or data implementation details, focusing on analytical results rather than empirical validation.

  flowchart TD
    subgraph Research Goal
        A["Model AMM liquidity position<br>and LVR using option theory"]
    end

    subgraph Methodology
        B["Represent CFAMM position as<br>portfolio of Perpetual American<br>Continuous-Installment CI Options"]
        C["Derive Replicating Portfolio<br>matching AMM Delta across<br>infinite time horizon"]
    end

    subgraph Computational Processes
        D["Calibrate Constant Volatility<br>from Implied Volatility Term Structure"]
        E["Solve for LVR via<br>Continuous Funding Fees<br>Theta of ATM CI Option"]
    end

    subgraph Key Findings & Outcomes
        F["LVR analytically identical to<br>option time value decay Theta"]
        G["Framework for Constant LVR<br>over arbitrary forward window<br>using calibrated parameters"]
    end

    A --> B
    B --> C
    C --> D
    C --> E
    D --> G
    E --> F
    E --> G