Paper: arXiv 2603.01344

Authors: Jimmy Risk, Shen-Ning Tung, Tai-Ho Wang

Abstract

This paper develops a robust mathematical framework for Constant Function Market Makers (CFMMs) by transitioning from traditional token reserve analyses to a coordinate system defined by price and intrinsic liquidity. We establish a canonical parametrization of the bonding curve that ensures dimensional consistency across diverse trading functions, such as those employed by Uniswap and Balancer, and demonstrate that asset reserves and value functions exhibit a linear dependence on this intrinsic liquidity. This linear structure facilitates a streamlined approach to arbitrage-free pricing, delta hedging, and systematic risk management. By leveraging the Carr-Madan spanning formula, we characterize Impermanent Loss (IL) as a weighted strip of vanilla options, thereby defining a fine-grained implied volatility structure for liquidity profiles. Furthermore, we provide a path-dependent analysis of IL using the last-passage time. Empirical results from Uniswap v3 ETH/USDC pools and Deribit option markets confirm a volatility smile consistent with crypto-asset dynamics, validating the framework’s utility in characterizing the risk-neutral fair value of liquidity provision.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical foundation for CFMMs by establishing a dimensionally consistent canonical parametrization of bonding curves using price and intrinsic liquidity coordinates. It presents clear analytical proofs for option static replication of Impermanent Loss via Carr-Madan formulas and validates the framework empirically using Uniswap v3 and Deribit data.

Research Flowchart

  flowchart TD
    A[Goal: Dynamic Pricing & Hedging for Liquidity Provision in CFMMs] --> B[Input: Market Price Data & Deribit Options Volatility]
    B --> C[Canonical Parametrization: Price & Intrinsic Liquidity System]
    C --> D[Carr-Madan Spanning: Decompose Impermanent Loss into Vanilla Option Strips]
    D --> E[Path-Dependent Analysis via Last-Passage Time]
    E --> F[Validation: Empirical Testing on Uniswap v3 ETH/USDC Pools]
    F --> G[Outcome: Impermanent Loss Hedging & Volatility Smile Characterization]

Cited References (37)

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  2. Decentralized Finance & Blockchain Technology (2023) — link
  3. The Adoption of Blockchain-based Decentralized Exchanges (2021) — link
  4. Improved Price Oracles: Constant Function Market Makers (2020) — link
  5. The Geometry of Constant Function Market Makers — → 2308.08066
  6. Replicating Market Makers (2023) — link
  7. Uniswap v2 Core (2020) — link
  8. Uniswap v3 Core (2021) — link
and 29 more
  1. Uniswap v4 Core: Hooks and Custom Liquidity (2023) — link
  2. Balancer: A Non-custodial Portfolio Manager, Liquidity Provider, and Price Sensor (2019) — link
  3. Panoptic: The Perpetual, Oracle-free Options Protocol (2022) — link
  4. The Replicating Portfolio of a Constant Product Market (2020) — link
  5. The Replicating Portfolio of a Constant Product Market with Bounded Liquidity (2021) — link
  6. Weighted variance swaps hedge against impermanent loss (2023) — link
  7. Market Neutral Liquidity Provision (2024) — link
  8. Automated Market Making and Loss-Versus-Rebalancing (2022) — link
  9. A Mathematical Framework for Modelling CLMM Dynamics in Continuous Time (2024) — link
  10. Towards a Theory of Volatility Trading (2001)
  11. Unified approach for hedging impermanent loss of liquidity provision — → 2407.05146
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  13. The Price of Liquidity: Implied Volatility of Automated Market Maker Fees (2025) — link
  14. Modeling Loss-Versus-Rebalancing in Automated Market Makers via Continuous-Installment Options — → 2508.02971
  15. Optimal Exiting for Liquidity Provision in Constant Function Market Makers (2025) — link
  16. Spanning and Completeness in Markets with Contingent Claims (1987) — link
  17. A Practical Guide to Splines (1978) — link
  18. Detecting and Repairing Arbitrage in Traded Option Prices (2020) — link
  19. Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints (2015) — link
  20. Cboe European-Style Options Implied Volatility Calculation Methodology (2019) — link
  21. Occupied Processes: Going with the Flow (2023) — link
  22. Controlled Occupied Processes and Viscosity Solutions (2024) — link
  23. Dynamics of Liquidity Surfaces in Uniswap v3 (2025) — link
  24. Dynamics of Liquidity Profile in Concentrated Liquidity Provision: An SPIDE Approach (2026)
  25. A Primer on Uniswap v3 Math: As Easy As 1, 2, v3 (2023) — link
  26. An Arbitrage-Free Interpolation of Class $ C ^2$ for Option Prices (2020) — link
  27. Arbitrage in the Perfect Volatility Surface (2017) — link
  28. On decomposition of the last passage time of diffusions (2025) — link
  29. Option Prices as Probabilities: A New Look at Generalized Black – Scholes Formulae (2010) — link

3 of 37 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.