Loss-Versus-Rebalancing under Deterministic and Generalized block-times
Alex Nezlobin, Martin Tassy
TL;DR
This work develops a discrete-time Markov-chain framework to quantify loss-versus-rebalancing (LVR) for AMM liquidity providers under both deterministic and general block-time distributions. By decomposing the long-run arbitrage loss as $\overline{\mathrm{ARB}} = P_{\mathrm{trade}} \times \overline{\mathrm{LVR}}$, the authors separate arbitrage frequency from event magnitude, enabling analytic expressions for key quantities. In the constant block-time case, they derive closed-form, exponentially accurate approximations for $P_{\mathrm{trade}}$, $\overline{\mathrm{LVR}}$, and $\overline{\mathrm{ARB}}$, with universal arbitrage frequency and explicit constants $\kappa$ and $\omega$ tied to ladder-height theory. Extending to general block-time laws, they show the asymptotic arbitrage probability is distribution-invariant to first order, while the per-event LVR and total ARB depend on the distribution via a non-negative constant $C_{\mu}$, minimized (with $C_{\mu}=0$) by the Dirac (constant) block-time. The results provide a rigorous foundation for evaluating LP profitability across modern blockchains and suggest constant block intervals offer the best protection against arbitrage under mean constraints. The work blends random-walk on a strip theory with empirical validation, delivering practical, quasi-exact formulas for LVR in contemporary DeFi environments.
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.