Partially Active Automated Market Makers

We introduce a new class of automated market maker (AMM), the \emph{partially active automated market maker} (PA-AMM). PA-AMM divides its reserves into two parts, the active and the passive parts, and uses only the active part for trading. At the top of every block, such a division is done again to keep the active reserves always being (λ)-portion of total reserves, where (λ\in (0, 1]) is an activeness parameter. We show that this simple mechanism reduces adverse selection costs, measured by loss-versus-rebalancing (LVR), and thereby improves the wealth of liquidity providers (LPs) relative to plain constant-function market makers (CFMMs). As a trade-off, the asset weights within a PA-AMM pool may deviate from their target weights implied by its invariant curve. Motivated by the optimal index-tracking problem literature, we also propose and solve an optimization problem that balances such deviation and the reduction of LVR.

💡 Research Summary

The paper introduces a novel class of automated market makers called partially active AMMs (PA‑AMMs). In a PA‑AMM the pool’s total reserves are split at the beginning of each block into an active portion and a passive portion. Only the active portion, which is a fixed fraction λ∈(0,1] of the total reserves, is used for trading during that block. At the start of the next block the split is recomputed so that the active share is again λ of the current total reserves. When λ=1 the mechanism collapses to a standard fully‑active constant‑function market maker (CFMM); when λ<1 the pool deliberately limits the amount of liquidity exposed to arbitrageurs in each block.

The authors adopt the standard DeFi model: two assets X (a risky token) and Y (a stablecoin), with X’s true price following a geometric Brownian motion. Arbitrageurs are risk‑neutral, myopic, and face negligible gas costs, so they instantly arbitrage any price discrepancy between the AMM’s marginal price and the external market price. In a conventional CFMM the marginal price is forced to the true price after each arbitrage trade, which leads to a loss‑versus‑rebalancing (LVR) for liquidity providers (LPs). LVR is defined as the difference between the pool’s value before and after arbitrage, and its instantaneous rate for a CFMM with invariant φ is LVR(P)=−σ²P²/2·V″(P) ≥0.

The paper focuses on the weighted geometric‑mean market maker (G3M) where φ(x,y)=x^θ y^{1‑θ}. For this invariant the instantaneous LVR rate simplifies to LVR(P)=σ²·θ(1‑θ)/2·V(P). Using this baseline, the authors analyze how the PA‑AMM changes the dynamics of the price gap g_n = s_n – p_n (the log‑difference between the true price s_n and the AMM’s marginal price p_n). They show that the gap evolves according to a linear AR(1)‑type recursion g_{n+1} = (1‑λ)·g_n + ε_{n+1}, where ε_{n+1} is the Gaussian price shock. Proposition 2 proves that for any λ>0 and block time Δt there exists a unique stationary distribution π_{Δt} for the gap process, with variance E