When prices reflect all available information, they oscillate around an equilibrium level. This oscillation is the result of the temporary market impact caused by waves of buyers and sellers. This price behavior can be approximated through an Ornstein–Uhlenbeck (OU) process. Market makers provide liquidity in an attempt to monetize this oscillation. They enter a long position when a security is priced below its estimated equilibrium level, and they enter a short position when a security is priced above its estimated equilibrium level. They hold that position until one of three outcomes occur: (1) they achieve the targeted profit; (2) they experience a maximum tolerated loss; (3) the position is held beyond a maximum tolerated horizon. All market makers are confronted with the problem of defining profit-taking and stop-out levels. More generally, all execution traders acting on behalf of a client must determine at what levels an order must be fulfilled. Those optimal levels can be determined by maximizing the trader’s Sharpe ratio in the context of OU processes via Monte Carlo experiments [M. López de Prado (2018) Advances in Financial Machine Learning. Hoboken, NJ, USA: John Wiley & Sons]. This paper develops an analytical framework and derives those optimal levels by using the method of heat potentials [A. Lipton & V. Kaushansky (2018) On the first hitting time density of an Ornstein–Uhlenbeck process, arXiv:1810.02390; A. Lipton & V. Kaushansky (2020a) On the first hitting time density for a reducible diffusion process, Quantitative Finance, doi:10.1080/14697688.2020.1713394].

中文翻译：

---

最优奥恩斯坦-乌伦贝克驱动交易策略的封闭式解决方案

当价格反映所有可用信息时，它们会在均衡水平附近波动。这种振荡是买卖双方一波又一波造成的暂时性市场冲击的结果。这种价格行为可以通过 Ornstein-Uhlenbeck (OU) 过程来近似。做市商提供流动性以试图将这种波动货币化。当证券定价低于其估计均衡水平时，他们进入多头头寸，当证券定价高于其估计均衡水平时，他们进入空头头寸。他们持有该头寸，直到出现以下三种结果之一：（1）他们实现了目标利润；(2) 他们经历了最大的可容忍损失；(3) 持仓超出最大容许范围。所有做市商都面临着定义获利了结和止损水平的问题。更普遍，所有代表客户行事的执行交易员都必须确定必须在什么水平上执行订单。这些最佳水平可以通过蒙特卡洛实验 [M. López de Prado (2018) 金融机器学习的进展。美国新泽西州霍博肯：John Wiley & Sons]。本文开发了一个分析框架，并通过使用热势的方法推导出了这些最佳水平 [A. Lipton & V. Kaushansky (2018) On the first hitting time density of an Ornstein-Uhlenbeck process, arXiv:1810.02390; A. Lipton & V. Kaushansky (2020a) On the first hit time density for a reducible spread process, Quantitative Finance, doi:10.1080/14697688.2020.1713394]。