Paper: arXiv 2401.01526

Authors: Joseph Najnudel, Shen-Ning Tung, Kazutoshi Yamazaki, Ju-Yi Yen

Abstract

We present a model for price dynamics in the Automated Market Makers (AMM) setting. Within this framework, we propose a reference market price following a geometric Brownian motion. The AMM price is constrained by upper and lower bounds, determined by constant multiplications of the reference price. Through the utilization of local times and excursion-theoretic approaches, we derive several analytical results, including its time-changed representation and limiting behavior.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of AMM price dynamics under fee constraints using advanced stochastic calculus tools like local times and excursion theory. However, it lacks empirical validation, backtesting, or real market data to support the theoretical derivations.

Research Flowchart

  flowchart TD
    A["Research Goal: Model AMM Price Dynamics with Fees & Arbitrage"] --> B["Inputs: Reference Market Price (Geometric Brownian Motion) & AMM Fee Bounds"]
    B --> C["Core Model: Constrain AMM Price within Upper/Lower Bounds via Arbitrage"]
    C --> D["Mathematical Framework: Apply Local Times & Excursion Theory"]
    D --> E["Derive Time-Changed Representation of AMM Price Dynamics"]
    D --> F["Analyze Long-Term Limiting Behavior of the AMM Price"]
    
    E --> G["Key Outcome: Analytical Model for AMM Price Dynamics"]
    F --> G

Cited References (6)

  1. The Adoption of Blockchain-based Decentralized Exchanges (2021) — link
  2. Automated Market Making and Loss-Versus-Rebalancing (2022) — link
  3. Automated Market Making and Arbitrage Profits in the Presence of Fees (2023) — link
  4. Improved Price Oracles: Constant Function Market Makers (2020) — link
  5. Growth rate of a liquidity provider’s wealth in $xy=c$ automated market makers (2020)
  6. Liquidity provider’s wealth in G3Ms (upcoming)

0 of 6 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.