This homework is intended to show how to extend the multi-period binomial tree to price an American option that pays dividends and has constant volatility. You are required to compute a value for the option using the portfolio replication approach. Please turn in a brief answer to all the questions.

Consider International Business Machine Corporation (IBM). The shares of IBM are currently trading at $83. Assume the yearly volatility of IBM is around 30%. Using a two step binomial tree approach price an American call option with 4 months to maturity and strike price equal to $70. The annualized risk-free rate is equal to 5%. In two months the stock pays a dividend of $5.

1. Since the option is American how does the dividend change the exercise policy? Is it possible that it will be optimal to exercise the option before maturity? That is the biggest difference in evaluating an American option versus a European one.

2. Write down the stock tree and be careful to incorporate the dividend payment. Now write down the option tree and compute the final payoffs of the option.

3. Start from the end of the tree and work your way back through the tree nodes. You need to compute: we have not covered this yet, but the volatility of the stock helps you pin down u and d: u = e t , d = e t the () in each node of the tree. is it changing? the present value of the future payoffs of the option in the up and down node of the tree (so to speak, the value of the option after one step).

4. Now here comes the difference relative to evaluation process of a European option: at the node where the dividend payment occurs you need to check whether early exercise is a feasible option. What you have to do is compare the exercise payoff, in the case where you decide to exercise the option, to the present value of the future payoffs. In practical term the value of the option at that node of the tree is just the maximum between the exercise payoff at that point in time and the present value of the future option payoffs.

5. Note that you have to be careful in computing the early exercise payoff: since this is a call option, if we exercise, will you exercise before or after the dividend is paid? Chose the stock price accordingly: if you chose to exercise before the dividend is paid, the relevant stock price is the price before the dividend is paid; if you decide to exercise after the dividend is paid, the relevant stock price will be the stock price after the dividend is paid.

6. Once you have determined the proper option value after one step, you will proceed as normal. What is the price of the option today?

7. Now re-do everything and compute the price of the European put with same maturity and same strike (The portfolio will be delta shares of the stock plus a put option.)