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

TL;DR

This paper derives an analytical approximation for the loss-versus-rebalancing in AMMs under fixed block times, demonstrating that constant block intervals minimize arbitrage risk and improve liquidity provider protection.

Contribution

It provides the first closed-form formula for LVR in blockchains with fixed intervals and extends the analysis to arbitrary block-time distributions.

Findings

The derived formula closely matches Monte Carlo simulations across practical parameters.

Constant block spacing minimizes LVR and offers optimal arbitrage protection.

The probability of arbitrage trades converges to a universal limit under any inter-block law.

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: \[ \overline{\mathrm{ARB}}= \frac{\,\sigma_b^{2}} {\,2+\sqrt{2\pi}\,\gamma/(|\zeta(1/2)|\,\sigma_b)\,}+O\!\bigl(e^{-\mathrm{const}\tfrac{\gamma}{\sigma_b}}\bigr)\;\approx\; \frac{\sigma_b^{2}}{\,2 + 1.7164\,\gamma/\sigma_b}, \] where $\sigma_b$ is the intra-block asset volatility, $\gamma$ the AMM spread and $\zeta$ 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.