Loss-Versus-Rebalancing under Deterministic and Generalized block-times
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arXiv
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| Format: | Preprint |
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2025
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| author | Nezlobin, Alex Tassy, Martin |
| author_facet | Nezlobin, Alex Tassy, Martin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_05113 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Loss-Versus-Rebalancing under Deterministic and Generalized block-times Nezlobin, Alex Tassy, Martin Mathematical Finance Probability Portfolio Management Pricing of Securities Trading and Market Microstructure 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. |
| title | Loss-Versus-Rebalancing under Deterministic and Generalized block-times |
| topic | Mathematical Finance Probability Portfolio Management Pricing of Securities Trading and Market Microstructure |
| url | https://arxiv.org/abs/2505.05113 |