A.

Consider a pension plan that will pay $10,000 once a year for a 5-year period (5 annual payments). The first payment will come in exactly 5 years (at the end of year 5) and the last payment in 9 years (at the end of year 9).

1)What is the duration of the pension obligation? The current interest rate is 9% per year for all maturities.

2)To generate the scheduled pension payments, the pension fund wants to invest the present value of the future payouts in bonds and match the duration of its obligation in part a). If the fund uses 5-year and 10-year zero-coupon bonds to construct its investment position, how much money (dollar amount) ought to be placed in each bond now? What should be the total face value (not current market value) of each zero-coupon bond held?

3)Right after the fund made its investment outlined in part b), market interest rates for all maturities dropped from 9% p.a.to 8% p.a. Show that the investment position constructed in part b) can still fund (approximately) the future payments by showing that the funds net investment is close to 0 at the end of year 9 after making all the scheduled payments. Assume that interest rates will remain at 8% p.a. Any excess cash from the 5-year investment will be reinvested at 8% and any fraction of the 10-year bonds held can be sold at the going market price at any time to fund the annual payments.

B:

Consider a one-year futures contract for 1 share of a dividend paying stock. The current stock price is $50 and the risk-free interest rate is 10% p.a. It is also known that the stock will pay a $4 dividend at the end of year 1. The current settlement price for the futures contract is $51. Set up a strategy for an arbitrage profit. What are the initial and terminal cash flows from the strategy? Assume that investors can short-sell or buy the stock on margin and that they can borrow and lend at the risk-free rate. There are no margin requirements, transactions costs, or taxes.

Consider a stock that pays no dividends on which a futures contract, a call option and a put option trade. The maturity date for all three contracts is T, the exercise price of the put and the call are both X, and the futures price is F. Show that if X = F, then the call price equals the put price assuming that spot-futures parity and put-call parity conditions hold. Assume that interest is continuously compounded (i.e., use the spot-futures parity with continuously compounded interest).

(C)

Find the Black-Scholes value of a put option on the following non-dividend paying stock:

Time to maturity: 6 months (1/2 year)

Standard Deviation: 40% per year

Exercise price:$50

Current stock price:$50

Interest rate:10%

b)

the put option is trading at the fair value you calculated in part a) but the call option with the same exercise price and maturity is currently selling for 9 dollars. Is there an arbitrage opportunity? If so, identify a trading strategy. Explain what actions you would take at the inception and expiry of the strategy to capture an arbitrage profit and calculate the amount of this profit. Show that strategy eliminates all risk irrespective of the value of the underlying stock at the maturity of the put and call options.

c)You decided to establish a position (unrelated to part a and b) by buying a share of the stock for $50, buying a 6-month put option with exercise price $45, and writing a 6-month call option with exercise price $55. Draw a payoff (not profit and loss) graph to illustrate the outcome of the combined position at expiry. Clearly label all axes and important points, showing relevant numbers on the axes.