21. (Compare with exercise 40 of chapter 7: Price a two-year European put, with strike price of...

21. (Compare with exercise 40 of chapter 7:Þ Price a two-year European put, with strike price of 100, in the following ways. The stock price is S0 ¼ 100, and based on time steps of Dt ¼ 0:25 years, the quarterly log-ratios have been estimated to have:

mQ ¼ 0:025, and s2Q

¼ ð0:09Þ2. The annual continuous risk-free rate is r ¼ 0:06.

(a) Develop a real world lattice of stock prices, with p ¼ 1 2 and time steps with Dt ¼ 0:05, and price this option using (7.147) with the appropriate value of q.

(b) Evaluate the two prices of this option at time t ¼ 0:05 using the same method as part (a), and construct a replicating portfolio at t ¼ 0 for these prices. Demonstrate that the cost of this replicating portfolio equals the price obtained in part (a).

(c) Price this option using (7.147) with the appropriate value of q based on a lattice for which p ¼ 0:25.

(d) Generate 500 two-year paths in the risk neutral world using the same model as part (a), and estimate the price of this option using (7.150) by counting how many scenarios end in each stock price at time 2 years.