  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 (O-U) 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 O-U processes via Monte Carlo experiments. This paper develops an
analytical framework and derives those optimal levels by using the method of
heat potentials.
