Consider an American put option written on a non-dividend paying stock. The strike price is 50. The put will expire in two years. The stock price follows a two-period binomial as in the graph. At each date, the natural probability that the stock price increases is one half. The stock is currently selling at 50. A two-year annual coupon bond with annual coupon rate of 10% is selling at par.

Imagine a game tree the first option is 50 and then you have to other options which are 80 or 30. 80 has two options you can choose either 128 or 48. Then 30 has two options which are 48 or 18.

a) What is the price of the American put option?

b) What is the price of an American call option written on the same stock with the same strike price and expiration date?

c) Prove or disprove: the put-call parity holds for American puts and American calls.