24. Price a 2-year European call, with strike price of 100, in the ways noted below.

The stock has 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 0:02 and s2Q

¼ ð0:07Þ2. The annual continuous risk-free rate is r ¼ 0:048, and so for Dt ¼ 0:25 years, you can assume that rQ ¼ 0:012.

(a) Develop a real world lattice of quarterly stock prices, with p ¼ 1 2 , and price this option using (7.147).

(b) Evaluate the two prices of this option at time t ¼ 0:25 from 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) Using exercise 23, price this option using (7.147) with the appropriate value of q based on a lattice for which p ¼ 0:25.

(d) Generate one hundred 2-year paths in the risk-neutral world, each with quarterly time steps and using the model of part (a). Then estimate the price of this option using (7.150), by counting how many scenarios end in each stock price at time 2 years.