Consider an American call. S0 = 40, X = 45, = 30% per year, er1 = 1.05 per year, T = 4 months

= 1 year. The stock will pay no dividends over the four month life of the option. 3

The changes in the stocks price can be approximated by a series of up and down movements and

let there be 5 such movements over the 4 month life of the option; i.e., divide the four months into 5

periods.Letn=5.ThelengthofeachintervalisthenT/n=1/3 = 1 year.Nowweneedauanda 5 15

T 12 d:thebestplacetogetthemisfrom.Letu=e n =e 15 ,andd= 1.

u

1. (a) Assuming a totally flat riskless term structure, what is the gross return on a riskless security over each of the intervals of length 1 of a year? 15
2. (b) For each possible stock price and remaining time to maturity (i.e., at each node in the tree of stock prices), calculate and B (i.e., the components of the replicating portfolio) and the value of the call.
3. (c) Suppose the stock first declines in value and then increases during each of the remaining four periods. Use a table to show at each node the value of your stock and bond portfolio both before and after any trading in stock and borrowing/repayment of debt. Show explicitly how your stock purchases are financed by additional borrowing and how the proceeds of any stock sales are used to repay borrowing.