I'm struggling to see how to calculate the price of an American knock in option on a tree.
The dynamic programming (martingale pricing) method on the normal tree iterates backwards in time, taking expectations of future payoff to calculate current price. The auxiliary variable that indicates whether knock-in has occurred surely needs to 'looks back' in time. Do we need to calculate the sum of the probabilities of each path that has knocked in and reached the up branch of the future payoff, and also the sum of probabilities of each path that has knocked in and reached the down branch of the future payoff to perform the calculation? Or is there an easier way? Do you know any useful references that cover the necessary method?
Any help would be greatly appreciated! Thanks,