Could you solve (c) (d) & (e)
2. (9 points) Futures on zero coupon bond A 12month zero coupon bond with a face value of $100 sells for $98.7. The 3month zerorate is 1%. (a) (1 point) What is the 12month zero rate? (b) (2 points) What is the 3-month forward rate on 9month loan? ( (c) 3 points) A 3month futures contract on this zero coupon bond species a price K, which a buyer of such contract will pay to buy this zero coupon bond 3- month from now. The buyer or the seller does not pay anything to enter a futures contract. So the predetermined price K has to be \"fair\". Let's see how we can nd this fair value. Denote the price of this zero-coupon bond three month from now by P (spot price of this bond three month from now). Right now we don't know the precise value of P; it could be lower or higher than the current spot price, $98.7, due to changes of interest rate in the future. Three month later (on the maturity date of the futures contract), the buyer receives the difference between the spot price, P, and the agreed price, K. Basically buyers of futures contracts are not really interested in having these bonds. On the maturity date of the futures, they will pay K to buy them then sell them on the spot market to receive a price of P. Sellers of futures contract may not have these bonds. They will buy from the spot market rst then sell at the xed price K. Settlement of futures contract just save both sides the trouble by asking them directly paying each other the difference. Figure out the cash ow of this nancial contract to ll the second collum of the table below. A replicating portfolio for a futures contract is a combination of other assets that generates the same cash ow in the future as a futures contract. Show that the replicating portfolio for this futures contract are the following: (1) buy the 12 month zero coupon bond now and sell it in three month; (2) borrow some money with continuous compounding. In three month you will pay back K to clear your debt. To show the above is a replicating portfolio, you need to show that the cash ow of a futures contract and the cash ow of the this portfolio are the same in three months. futures replicating portfolio contract borrow buy bond total m-- (d) (2 points) Based on the no arbitrage principle, what should the future price K be? (e) (1 point) What is the connection between the futures price K and the forward rate calculated in part (b)
