  We study a mathematical model capturing the support/resistance line method (a
technique in technical analysis) where the underlying stock price transitions
between two states of nature in a path-dependent manner. For optimal stopping
problems with respect to a general class of reward functions and dynamics,
using probabilistic methods, we show that the value function is $C^1$ and
solves a general free boundary problem. Moreover, for a wide range of
utilities, we prove that the best time to buy and sell the stock is obtained by
solving free boundary problems corresponding to two linked optimal stopping
problems. We use this to numerically compute optimal trading strategies for
several types of dynamics and varying degrees of relative risk aversion. We
then compare the strategies with the standard trading rule to investigate the
viability of this form of technical analysis.
