Minimal Model of Simulating Prices of Financial Securities Using an Iterated Finite Automaton

What is the simplest system that can generate the randomness of financial securities? Imagine a single technical trader looking at a single security. He looks at the past 15 days of price changes but he only has 3 internal states (modeled with an iterated finite automaton) so he doesn't remember much. He decides whether the security is cheap or expensive. The price adjusts to reflect his belief and the next day he sees the new last 15 days of price changes, starting with the most recent change, which was his decision from the previous day. Despite the simplicity of this system, certain rules generate complex behavior.