Consider a three-period binomial model with up factor u = 1.2, down factor d = 0.9, and initial stock price S0 = 100. In addition to the stock, a risk-free bond is available with initial price A0 = 1 and periodic return r = 0.1.

a. (5) Calculate the risk-neutral probability that the stock price goes up in one period? In this model, consider an American call option with strike price K = 100 and expira-

tion date T = 3 periods.

First, assume that the stock pays no dividends. Hence, by the result of Q.4, this option is practically equivalent to a European call with the same strike price and expiration date.

b. (5) Calculate the price of this option at time zero. c. (5) Describe a hedging strategy for a short position in this option.

Now, assume that the stock pays a 10% dividend at the end of the second period. Hence, the stock prices at the corresponding nodes of the tree drop down by 10% right after the dividends are paid to the shareholders. Accordingly, the holder of the American call has three choices at such nodes: should she exercise the option right before the dividends are paid, should she exercise it right after the dividends are paid, or should she keep holding the option?

4. (5) Calculate the price of the American call at time zero. Is it ever optimal to exer- cise early this option?

5. (5) Do the prices in parts b and d match? Why?