A CLOSED-FORM SOLUTION FOR OPTIMAL ORNSTEIN–UHLENBECK DRIVEN TRADING STRATEGIES

Abstract

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].