Consider a two-period binomial model. The current stock prices is $50 and in each of the next 2 years, the stock price can increase or decrease by 20%. Assume risk free is 5.1%. Suppose that the prices of a European put struck at $52 at each node are given by P0=4.1923; Pd=9.4636; Pu=1.4147; Puu=0; Pud=4 and Pdd=20. Then, at time zero (P0), what is the price of an American Put struck at $52? a) 4.19 b) 4.77 c) 5.09 d) 6.01 e) None of the above. It is clear that it cannot be 4.19 because the price of an American style put has to be higher than the European Put.
I pick C… you just need to compute the probabilities from the up/down prices which comes out to pu=.6283 and pd=.3717 then you compute the option price in year 1, which is 0,12 – compare it to 1.4 and 9.46… you keep the option in 1.4>0 scenario and exercise 12>9.46, so your price is PV(pu*1.4+pd*12) =5.09