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, we are going to price 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. Compute the price of the European call option. A few instructions: - 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 , t is the length (in time as a fraction of a year) of one step in the three. - 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. - Start from the end of the tree and work your way back through the tree nodes.

2. Now compute the price of the American call option - 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. - 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.

HOMEWORK 2: Multi-period binomial tree Due before class. This homework is intended to show how to extend the multi-period binomial tree to price an 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, we are going to price 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. Compute the price of the European call option. A few instructions: - 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 , t is the length (in time as a fraction of a year) of one step in the three. - 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. - Start from the end of the tree and work your way back through the tree nodes. 2. Now compute the price of the American call option - 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. - 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. 1