Riemannian Geometry of Optimal Rebalancing in Dynamic Weight Automated Market Makers

Battle Count: 61

Highly relevant to quantitative trading in DeFi/AMM contexts. The paper provides a rigorous geometric framework for optimal portfolio rebalancing in automated market makers, directly applicable to TFMM protocols, Balancer-style pools, and any dynamic-weight AMM. The results on optimal step count under LVR, fee-adjusted rebalancing, and on-chain implementation via recursive bisection are immediately actionable for DeFi protocol developers and quantitative traders operating in AMM markets. The connection to Almgren-Chriss optimal execution (splitting large orders to reduce market impact) makes it relevant to traditional quantitative trading as well. However, it is specialized to AMM/G3M pool structures rather than general market microstructure.

Moderate complexity. SLERP requires one arccos evaluation, N square roots, and per-step sin calls - comparable to geometric mean interpolation. The recursive bisection algorithm (Algorithm 2) eliminates trigonometric functions entirely, requiring only addition, multiplication, square root, and normalisation at power-of-two step counts. The modified geodesic equation (Proposition 5) and two-token pendulum (Corollary 5) add complexity for stochastic-price scenarios. Computing f* requires estimating average LVR along the SLERP curve (a single integral). On-chain implementation is feasible via the trig-free bisection. The main complexity lies in correctly implementing the Hellinger embedding and ensuring numerical stability near the simplex boundary.

The paper provides complete algorithms (Algorithm 1 for SLERP, Algorithm 2 for trig-free recursive bisection), explicit formulas, and detailed numerical experiments with specific parameter settings (N=3 setup, f=1000 steps, specific weight vectors). However, no code repository is explicitly linked. The mathematical derivations are self-contained with proofs in appendices. Monte Carlo validation parameters are specified (10,000 GBM paths). Historical backtest uses ETH/USD data but source not explicitly stated.