Loss-Versus-Rebalancing under Deterministic and Generalized block-times

ArXiv ID: 2505.05113 “View on arXiv”

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.

Keywords: Automated Market Makers (AMMs), Loss-versus-Rebalancing (LVR), Random Walk Theory, Monte Carlo Simulation, Block Time Distributions, Cryptocurrencies / Decentralized Finance (DeFi)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: 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.

  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"]