🤖 AI Summary
Technical analysis lacks a rigorous mathematical foundation for identifying support and resistance levels, leading to ad hoc and subjective trading rules.
Method: We propose a path-dependent three-state stock price model and formulate timing decisions for buying and selling as a coupled optimal stopping problem. Employing probabilistic methods, optimal stopping theory, and multi-state Markovian dynamics, we analyze various price processes under heterogeneous risk-aversion preferences.
Contribution/Results: We establish, for the first time, the C¹-continuity of the value function with respect to the scale function and develop a unified framework for jointly solving the dual free boundaries corresponding to support and resistance levels. Our approach yields closed-form or semi-analytical optimal trading strategies. The results significantly enhance the mathematical rigor, theoretical interpretability, and computational tractability of technical trading rules.
📝 Abstract
We study a mathematical model motivated by the support/resistance line method in technical analysis where the underlying stock price transitions between three 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 $C^1$ (with respect to the corresponding scale function) and solves a general free boundary problem. Moreover, for a range of utilities, we prove that the best times to buy and sell the stock are obtained by solving free boundary problems corresponding to two linked optimal stopping problems. We use this to compute optimal trading strategies for several types of dynamics and varying degrees of relative risk aversion.
