Looking Forward to Pricing Options from Binomial Trees
  We reconsider the valuation of barrier options by means of binomial trees from a "forward looking" prospective rather than the more conventional "backward induction" one used by standard approaches. This reformulation allows us to write closed-form expressions for the value of European and American put barrier-options on a non-dividend-paying stock. 
Dario Villani and Andrei E. Ruckestein