Consider a stock with current price S = 50 whose price process can be represented by a binomial tree with parameters u = 1.221 and d = 0.819. Suppose the per-period gross interest rate is R = 1.005. 

(a) Find the value of a two-period European put option with a strike of K = 50. 

(b) Using backwards induction on the tree, find the value of a forward start put option that comes to life in one period, is European, has a further life of two periods, and will be at-the-money when it comes to life. 

(c) Verify that your answers to parts (a) and (b) coincide. 

(d) Suppose the puts had been American. What are the answers to parts (a) and (b)? 

Do they still coincide?