Paper: arXiv 2505.05113

Authors: Alex Nezlobin, Martin Tassy

Abstract

Although modern blockchains almost universally produce blocks at fixed intervals, existing models still lack an analytical formula for the loss-versus-rebalancing (LVR) incurred by Automated Market Makers (AMMs) liquidity providers in this setting. Leveraging tools from random walk theory, we derive the following closed-form approximation for the per block per unit of liquidity expected LVR under constant block time: [" \overline{"\mathrm{ARB"}}= \frac{",σ_b^{2"}} {",2+\sqrt{2π"},γ/(|ζ(1/2)|,σ_b),}+O!\bigl(e^{"-\mathrm{const"}\tfracγ{“σ_b”}}\bigr);\approx; \frac{“σ_b^{2”}}{",2 + 1.7164,γ/σ_b"}, “] where $σ_b$ is the intra-block asset volatility, $γ$ the AMM spread and $ζ$ the Riemann Zeta function. Our large Monte Carlo simulations show that this formula is in fact quasi-exact across practical parameter ranges. Extending our analysis to arbitrary block-time distributions as well, we demonstrate both that–under every admissible inter-block law–the probability that a block carries an arbitrage trade converges to a universal limit, and that only constant block spacing attains the asymptotically minimal LVR. This shows that constant block intervals provide the best possible protection against arbitrage for liquidity providers.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents advanced theoretical derivations using random walk theory and special functions (Riemann Zeta), yielding complex closed-form formulas, but also validates them with extensive Monte Carlo simulations showing high accuracy, indicating strong empirical backing.

Research Flowchart

  flowchart TD
  A["Research Goal<br/>Derive analytical LVR for AMMs<br/>with deterministic/variable block times"] --> B["Key Methodology<br/>Random Walk Theory<br/>Monte Carlo Simulations"]
  B --> C["Data/Inputs<br/>Constant vs. Variable Block Time Laws<br/>Intra-block Volatility σb<br/>AMM Spread γ"]
  C --> D["Computational Process<br/>Stochastic calculus &<br/>Closed-form approximation derivation"]
  D --> E["Key Finding 1: Closed-Form LVR<br/>`ARB ≈ σ_b^2 / (2 + 1.7164γ/σ_b)`<br/>(Quasi-exact across parameters)"]
  D --> F["Key Finding 2: Block Time Optimization<br/>Constant intervals attain asymptotically minimal LVR<br/>(Universal limit proved)"]
  E & F --> G["Outcome<br/>Constant block spacing offers<br/>best protection for Liquidity Providers"]