微观结构均值回归的最优交易
Optimal Trading of Microstructure Mean Reversion
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中文总结 AI 辅助
针对秒级中间价均值回归特性，构建订单簿模型推导最优交易策略，得出最优阈值与利润率公式，证明等待的期权价值是利润来源。
中文摘要 AI 辅助
在秒级尺度下，观测到的中间价围绕潜在有效价格存在一个平稳的均值回归误差。我们构建了一个订单簿，其自身的订单流会产生该误差，并求解能最大化扣除买卖价差后的长期平均利润率的交易规则。在流动性充足的大价位资产中，价差为1个或2个最小变动价位，且恰好对应半价位网格上中间价的奇偶性：半整数价位时价差较窄，整数价位时价差较宽。因此问题可由一个坐标描述：中间价与有效价格之间的缺口$G$；有效价格是外生布朗鞅，且$G$是可观测的。中间价是纯跳过程，其跳跃强度倾向于向有效价格靠拢。在一个平衡响应条件（该条件使订单簿在不同奇偶性下的修正漂移相等）下，$G$的均值回归是一个定理：其条件均值和平稳协方差恰好与回归速率为$α$、平稳标准差为$s_G$的Ornstein-Uhlenbeck过程一致。但其路径并非如此：中间价是跳跃的。因此，穿越时间通过这两个矩所定义的高斯扩散来估算，我们对收益端的误差进行了界定，而对时机端的误差采用启发式处理。半宽度为$θ$的对称带形策略在缺口达到$-θ$时买入，在$+θ$时卖出，在区间内持有；在替代模型下它在所有可行策略中是最优的，而在跳跃过程本身中这种简化仍是一个猜想。设$ϕ$为窄价差的半价差，最优半宽度及其利润率满足$θ^*(θ^*-ϕ)=s_G^2$和$R^*=αs_G\/{2/π}\,e^{-θ^{*2}/2s_G^2}$。阈值乘以边际等于缺口的平稳方差。缺口一覆盖价差就交易的收益为零：所有利润都是等待的期权价值。
英文摘要
At the scale of seconds the observed mid carries a stationary, mean-reverting error around a latent efficient price. We build an order book whose own flow produces that error and solve for the trading rule that maximises the long-run average profit rate net of the bid-ask spread. In a liquid large-tick asset the spread is one tick or two, and it is exactly the parity of the mid on the half-tick grid: tight at a half-integer, open at an integer. One coordinate therefore carries the problem: the gap $G$ between the mid and the efficient price; the price is an exogenous Brownian martingale, and $G$ is observable. The mid is a pure jump process whose move intensities lean toward the efficient price. Under one balanced-response condition, which equalises the book's corrective drift across parities, mean reversion of $G$ is a theorem: its conditional mean and stationary covariance are exactly those of an Ornstein-Uhlenbeck process with reversion rate $α$ and stationary standard deviation $s_G$. Its paths are not: the mid jumps. Passage times are therefore evaluated on the Gaussian diffusion those two moments define, at an error we bound on the reward side and leave heuristic on the timing side. A symmetric band of half-width $θ$ buys when the gap reaches $-θ$, sells at $+θ$, and holds inside; on the surrogate it is optimal among all admissible strategies, on the jump process itself that reduction remains a conjecture. With $ϕ$ the tight-book half-spread, the optimal half-width and its profit rate are $θ^*(θ^*-ϕ)=s_G^2$ and $R^*=αs_G\sqrt{2/π}\,e^{-θ^{*2}/2s_G^2}$. Threshold times margin equals the stationary variance of the gap. Trading as soon as the gap covers the spread earns zero: all profit is the option value of waiting.
