Optimal Automated Market Maker Design in Cryptocurrency Markets: Finite Orders and Endogenous Exposure

How should an automated market maker in cryptocurrency markets choose its reserve curve and swap fee when finite token swaps change future arbitrage exposure? We study liquidity design for decentralized exchanges through exact reserve transfers, retaining fees, slippage, partial fills, and arbitrage against an external cryptocurrency price. Price-target orders preserve depth-independent quote dynamics: their net-transfer objective is linear in depth and has a global capital-constrained threshold solution under a depth cap. Fixed-input orders instead make exposure design-dependent. A Poisson-equation comparison identifies the missing exposure-response term in marginal design and supplies sufficient global verification and perturbation bounds. The main extension uses active reserves as the state, removing the singular inverse quote at empty price ranges. Under bounded, i.i.d., state-independent opportunity marks and a uniform extreme-price reset probability, a jointly optimal fee and bounded measurable depth profile exists, allowing zero depth and disconnected active support. Piecewise-constant depth grids and grids of constant-liquidity range positions approximate that optimum. Customer fills are nonexpansive in reserves, yielding Wasserstein contraction and an explicit stationary-value error bound for barycentric state projection. Attainment and reserve-projection consistency extend to bounded Markov external prices with a uniform tail minorization. Joint design and state refinement are therefore consistent when optimization error is controlled. Synthetic experiments demonstrate a fee-direction reversal when exposure is frozen and a candidate that improves reward while leaving outer ranges and part of the capital budget unused. Continuous-state simulations check both effects. These experiments illustrate mechanisms rather than calibrate a live cryptocurrency pool. The conditional local-fee allocation problem remains a tractable benchmark. The framework connects token-swap execution, concentrated-liquidity allocation, and numerical approximation while remaining well-defined when liquidity vanishes.