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# Binomial Tree, CAPM, and Bond Duration

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1. The current price of IBM stock is \$88. Every year the stock can either go up by 20% with probability 2/3, or go down by 10% with probability 1/3. What is the price of a European Call option on IBM with maturity in 2 years, and a strike price of \$100? The risk free interest rate is 4% a year.

2. Goldseeker Inc. is a company searching for a sunken ship filled with gold at the bottom of the Atlantic. The company will have a single cashflow, two years from today. This cashflow depends only on whether the company finds the ship or not, and does not depend on any macroeconomic market factors. The probability distribution of the cashflow is given below:

probability cashflow
Ship found 0.2 \$100 million

The annual expected market return is 12%. The annual riskfree rate is 4%. The CAPM holds. What is the value of Goldseeker Inc. today?

3. A pension fund has an obligation to pay \$630 million in 5.5 years. The fund invests money in a bond portfolio in order to secure this future obligation. The bond portfolio includes the following two bonds:

Bond A: Par value \$1,000, coupon rate 8%, yield to maturity 6%, maturity 2 years from now (first coupon one year from now).

Bond B: Par value \$1,000, coupon rate 0%, yield to maturity 6%, maturity T years from now.

The fund calculates that the investment which minimizes exposure to interest rate changes is an investment of \$155 million in Bond A and \$302.25 million in Bond B. What is the time to maturity, T, of Bond B?