On pricing american put options

How come we pick the highest between the discounted weighted average (with risk neutral probabilities) and the early exercise value at each node of the binomial tree?

I dont understand why, I can see why it "logical" to pick the greater but not why this would be the fair value.

• Every day the option holder is faced with a choice: should I exercise the option today or not. A simple strategy is to compare the value of immediate exercise and the value of continuing to hold the option, and choose the most profitable one. It can be shown that this myopic strategy is also the optimal strategy. Why? In essence because in an efficient market you can't predict the future so there is no way to improve upon this myopic strategy by considering future developments, they are unknowable. – Alex C Sep 12 at 16:13
• @AlexC I am sorry but what is optimal is not whats "fair" right? – user1 Sep 12 at 16:24

So, how do you price derivatives in general? You build the tree for the stock price and then a second tree via so-called backward induction: You begin with the terminal payoff and work backwards through the tree by computing the discounted expectation'' of the future nodes. In the end, this gives you the price of a European-style option.
• It follows directly from the definition of an American option and a rational investor: If you had the choice a high immediate payoff or holding on which you can expect to yield less? What do you choose? American-style options have more'' optionality as European-style options due to the early exercise features. Hence, we think about optimal stopping problems and a maximum over stopping times. – KeSchn Sep 12 at 17:39