Paper: arXiv 2410.00854

Abstract

There are two predominant metrics to assess the performance of automated market makers and their profitability for liquidity providers: ‘impermanent loss’ (IL) and ’loss-versus-rebalance’ (LVR). In this short paper we shed light on the statistical aspects of both concepts and show that they are more similar than conventionally appreciated. Our analysis uses the properties of a random walk and some analytical properties of the statistical integral combined with the mechanics of a constant function market maker (CFMM). We consider non-toxic or rather unspecific trading in this paper. Our main finding can be summarized in one sentence: For Brownian motion with a given volatility, IL and LVR have identical expectation values but vastly differing distribution functions.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is heavily theoretical with advanced mathematical derivations involving stochastic calculus (Brownian motion, differential equations) and statistical analysis of distribution functions. It lacks empirical rigor, showing no real data, backtests, or implementation details beyond simulated random walks for conceptual validation.

Research Flowchart

  flowchart TD
  Start["Research Goal<br>Clarify statistical relationship between<br>Impermanent Loss (IL) and Loss-vs-Rebalance (LVR)"] --> Inputs["Key Inputs & Assumptions<br>- Constant Function Market Maker (CFMM)<br>- Non-toxic (unspecific) trading<br>- Random walk / Brownian motion<br>- Asset price volatility (σ)"]
  Inputs --> Analysis["Methodology<br>Statistical analysis & calculus<br>1. Define IL and LVR metrics<br>2. Apply Brownian motion properties<br>3. Compute analytical expectations"]
  Analysis --> Computation["Computational Process<br>- Derive closed-form expectation values<br>- Analyze distribution functions<br>- Compare statistical properties"]
  Computation --> Outcome["Key Findings<br>1. IL and LVR have identical expectations<br>2. Distribution functions differ vastly<br>3. Statistical similarity > conventional appreciation"]
  Outcome --> End["Conclusion<br>IL and LVR are fundamentally linked by<br>volatility, though their risk distributions differ"]

Cited References (35)

  1. Automated Market Making and Loss-Versus-Rebalancing (2022) — link
  2. An Economic Model of a Decentralized Exchange with Concentrated Liquidity (2023)
  3. Automated Market Making and Arbitrage Profits in the Presence of Fees — → 2305.14604
  4. Automated Market Makers: A Practical Guide to Decentralized Exchanges and Cryptocurrency Trading (2023) — link
  5. Theory and Practice of Revenue Management (2005)
  6. Decentralized exchanges (2021)
  7. The adoption of blockchain-based decentralized exchanges (2021) — link
  8. The need for fees at a dex: How increases in fees can increase dex trading volume (2022)
and 27 more
  1. Uniswap v1 Core (2018) — link
  2. Uniswap v2 Core (2020) — link
  3. Uniswap v3 Core (2021) — link
  4. Uniswap v4 Core [Draft] (2023) — link
  5. JOE v2.1 Liquidity Book (2022) — link
  6. ALGEBRA Ecosystem: Decentralized exchange (2022) — link
  7. Automatic market-making with dynamic peg (2021) — link
  8. Mooniswap by 1inch.exchange (2020) — link
  9. MEV capturing AMM (McAMM) (2022) — link
  10. Censorship Resistance in On-Chain Auctions (2023) — link
  11. Flash boys 2.0: Frontrunning in decentralized exchanges, miner extractable value, and consensus instability (2020)
  12. The Costs of Swapping on the Uniswap Protocol — → 2309.13648
  13. Credible decentralized exchange design via verifiable sequencing rules (2022) — link
  14. High-frequency trading on decentralized on-chain exchanges (2021)
  15. The centralizing effects of private order flow on proposer-builder separation (2023) — link
  16. Logarithmic markets coring rules for modular combinatorial information aggregation (2007)
  17. A practical liquidity-sensitive automated market maker (2013)
  18. Twitter thread (2023) — link
  19. UniswapX (2023) — link
  20. Radical Markets: Uprooting Capitalism and Democracy for a Just Society (2018)
  21. Triangle Fees (2023) — link
  22. Predictable Losses of Liquidity Provision in Constant Function Markets and Concentrated Liquidity Markets (2023) — link
  23. Decentralised Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision (2022)
  24. Automated Market Makers Designs Beyond Constant Functions (2023)
  25. Automated Market Making and Loss-Versus-Rebalancing (2024) — link
  26. Optimal Dynamic Fees for Blockchain Resources (2023) — link
  27. Multidimensional Blockchain Fees are (Essentially) Optimal (2024) — link

2 of 35 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.