Optimal Adaptive Market Making: A Theoretical Framework for High-Yield Liquidity Provision in Perpetual Futures Markets

最优自适应做市：永续合约市场高收益流动性供给的理论框架

Abstract: We develop a rigorous theoretical framework for optimal market making in perpetual futures markets with zero maker fees. We model the market maker’s problem as a stochastic optimal control problem on a filtered probability space, where the controls are adaptive bid-ask spreads and inventory hedging decisions across two exchanges.

摘要： 我们为零挂单费（zero maker fees）永续合约市场中的最优做市行为开发了一个严谨的理论框架。我们将做市商的问题建模为过滤概率空间上的随机最优控制问题，其中控制变量为跨两个交易所的自适应买卖价差及库存对冲决策。

Our contributions include: (i) a PnL decomposition theorem separating revenue into spread income, adverse selection loss, inventory carrying cost, hedging friction, and funding rate exposure; (ii) the Hamilton-Jacobi-Bellman equation for the joint spread-inventory-hedging control problem under CARA utility with a verification theorem; (iii) High-APY Regime Theorems characterizing profitable regions via five dimensionless parameters, culminating in a Master APY Formula; (iv) analysis of zero-fee economics on decentralized perpetual exchanges with optimal entry-exit thresholds.

我们的贡献包括：(i) 一个盈亏（PnL）分解定理，将收益拆解为价差收入、逆向选择损失、库存持有成本、对冲摩擦及资金费率敞口；(ii) 在 CARA 效用函数下，针对联合价差-库存-对冲控制问题的 Hamilton-Jacobi-Bellman 方程及其验证定理；(iii) 通过五个无量纲参数刻画盈利区域的高年化收益率（High-APY）机制定理，并最终推导出主年化收益率公式；(iv) 对具有最优进出阈值的去中心化永续合约交易所的零手续费经济学分析。

(v) optimal cross-exchange hedging policies with funding rate dynamics and a hedge regime trichotomy; (vi) a robustness margin quantifying parameter uncertainty tolerance; (vii) exponential drawdown probability bounds and a universal APY-VaR identity; (viii) ergodic inventory distribution under optimal control with Bayesian adaptive estimation; (ix) Kelly-optimal leverage with ruin boundaries; and (x) multi-pair portfolio allocation with diversification saturation results.

(v) 结合资金费率动态和对冲机制三分法的跨交易所最优对冲策略；(vi) 量化参数不确定性容忍度的稳健性边界；(vii) 指数级回撤概率界限及通用的年化收益率-风险价值（APY-VaR）恒等式；(viii) 在贝叶斯自适应估计下的最优控制遍历库存分布；(ix) 带有破产边界的凯利最优杠杆；以及 (x) 具有多样化饱和结果的多币对投资组合分配。

Numerical analysis with twenty-three figures reveals phase transitions between profitable and unprofitable regimes. Our framework unifies and extends the Avellaneda-Stoikov, Gueant-Lehalle-Fernandez-Tapia, and Glosten-Milgrom paradigms for modern decentralized venue microstructure.

包含 23 张图表的数值分析揭示了盈利与非盈利机制之间的相变。我们的框架统一并扩展了 Avellaneda-Stoikov、Gueant-Lehalle-Fernandez-Tapia 以及 Glosten-Milgrom 等范式，适用于现代去中心化交易场所的微观结构。