non-dividend-paying stock is trading at S0 = 100, and the interest rate is 6 ln(1.01).
(a) Use a two-period binomial model with up and down factors u = 1.16, d = 0.91
and up probability p = 0.7 to price a European put option with strike K = 110
and maturity T in eight months.
[ 5 marks ]
(b) Price an American put option with the same strike (and maturity). What is the
probability that you will exercise early?
[ 5 marks ]
(c) Suppose the stock has suddenly become 10% more volatile, and the company
announced that it will pay a one-off cash dividend of $5 per share six months
from now. Explain how this new development can be captured in the model
above. Make appropriate adjustment to your model parameters, price the
European and American put option in part (a) and part (b) using an 8 step
binomial tree. Discuss the results.
[ 10 marks ]
(d) Your colleague uses the Black-Scholes model to price the European put option
and gets a different answer. How can you improve your model to try to match
with theirs? Demonstrate in Excel how a closer match can be achieved.
[ 10 marks ]

1. A company’s stock price S0 is $100 today and the interest rate is r = 3%. Suppose
   the company pays a dividend of $2 to its shareholders every quarter (four per year).
   The next dividend is due in one month’s time.
   (a) Find the price of a nine-month forward contract on the stock.
   [ 5 marks ]
   (b) Suppose the forward contract is trading at exactly $100, just like the stock. Is an
   arbitrage possible? If so, how? If not, why not?
   [ 5 marks ]
   (c) Now suppose that S0 = 100 is instead the spot price of crude oil, r = 3% still,
   and storage is paid at a continuous annualized rate of c = 5% of the spot price.
   What are the possible values for the nine-month forward price of oil?
   [ 5 marks ]
   (d) Give an example of a supply or demand effect which could produce an oil
   forward price of $90.