The price of a firms stock is currently $50 per share. The stock price follows a binomial process where it either goes up 20% or down 20% each year. The stock pays no dividends, and the risk-free interest rate is 2% each year. Consider an American put option on the stock with a strike price of $50 and 2 years to expiration.

3a) Suppose that the stock price goes up 20% in the next year to $60 per share in 1 year. What is the value of the put option at that point assuming that the option is not exercised until expiration (i.e., until the end of the following year, when the stock price could go up to $72 or down to $48)?

3b) Suppose instead that the stock price goes down 20% in the next year to $40 per share. What is the value of the put option at that point assuming that the option is not exercised until expiration (i.e., until the end of the following year, when the stock price could go up to $48 or down to $32)?

3c) Is it optimal to exercise the option one year from today if the stock price at that point is $60 (i.e., the scenario in part a)? Is it optimal to exercise the option one year from today if the stock at that point is $40 (i.e., the scenario in part b)?

3d) What is the value of the option today?

3e Now, consider a special type of option. Like the put option in part a, this option is American and has two years to expiration. However, this option has features of both a call option and a put option. Specifically, it entitles the owner to either buy or sell a share of stock at the strike price of $50 in year 1. If the option is not exercised in year 1, then the owner has the right to sell a share of stock at a strike price of $50 in year 2. That is, if the option is not exercised in year 1, it becomes a put option identical to the put option in part a. What is the value of this special option?

The price of a firms stock is currently $50 per share. The stock price follows a binomial process where it either goes up 20% or down 20% each year. The stock pays no dividends, and the risk-free interest rate is 2% each year. Consider an American put option on the stock with a strike price of $50 and 2 years to expiration.

3a) Suppose that the stock price goes up 20% in the next year to $60 per share in 1 year. What is the value of the put option at that point assuming that the option is not exercised until expiration (i.e., until the end of the following year, when the stock price could go up to $72 or down to $48)?

3b) Suppose instead that the stock price goes down 20% in the next year to $40 per share. What is the value of the put option at that point assuming that the option is not exercised until expiration (i.e., until the end of the following year, when the stock price could go up to $48 or down to $32)?

3c) Is it optimal to exercise the option one year from today if the stock price at that point is $60 (i.e., the scenario in part a)? Is it optimal to exercise the option one year from today if the stock at that point is $40 (i.e., the scenario in part b)?

3d) What is the value of the option today?

3e Now, consider a special type of option. Like the put option in part a, this option is American and has two years to expiration. However, this option has features of both a call option and a put option. Specifically, it entitles the owner to either buy or sell a share of stock at the strike price of $50 in year 1. If the option is not exercised in year 1, then the owner has the right to sell a share of stock at a strike price of $50 in year 2. That is, if the option is not exercised in year 1, it becomes a put option identical to the put option in part a. What is the value of this special option?