Reference no: EM133512684

Problem I. Suppose that the current price of gold is $1,765 per oz and that gold may be stored costlessly. Suppose also that the term structure is flat with a continuously compounded rate of interest of 6% for all maturities.

1. Calculate the forward price of gold for delivery in three months.

2. Nowsuppose it costs $1 per oz per month to store gold (payable monthly in advance). What is the new forward price?

3. Assume storage costs are as in part (b). If the forward price is given to be $1,805 per oz, explain whether there is an arbitrage opportunity and how to exploit it.

Problem II. A stock will pay a dividend of $1 in one month and $2 in four months. The risk-free rate of interest for all maturities is 12%. The current price of the stock is $90.

1. Calculate the arbitrage-free price of (a) a three-month forward contract on the stock and (b) a six-month forward contract on the stock.

2. Suppose the six-month forward contract is quoted at 100. Identify the arbitrage opportunities, if any, that exist, and explain how to exploit them.

Problem III. A bond will pay a coupon of $4 in two months' time. The bond's current price is $99.75. The two-month interest rate is 5% and the three-month interest rate is 6%, both in continuously compounded terms.

1. What is the arbitrage-free three-month forward price for the bond?

2. Suppose the forward price is given to be $97. Identify if there is an arbitrage opportunity and, if so, how to exploit it.

Problem IV. Three months ago, an investor entered into a six-month forward contract to sell a stock. The delivery price agreed to was $55. Today, the stock is trading at $45. Suppose the three-month interest rate is 4.80% in continuously compounded terms.

1. Assuming the stock is not expected to pay any dividends over the next three months, what is the current forward price of the stock?

2. What is the value of the contract held by the investor?

3. Suppose the stock is expected to pay a dividend of $2 in one month, and the onemonth rate of interest is 4.70%. What are the current forward price and the value of the contract held by the investor?

Problem V. A stock is trading at $24.50. The market consensus expectation is that it will pay a dividend of $0.50 in two months' time. No other payouts are expected on the stock over the next three months. Assume interest rates are constant at 6% for all maturities. You enter into a long position to buy 10,000 shares of stock in three months' time.

1. What is the arbitrage-free price of the three-month forward contract?

2. After one month, the stock is trading at $23.50. What is the marked-to-market value of your contract?

3. Now suppose that at this point, the company unexpectedly announces that dividends will be $1.00 per share due to larger-than-expected earnings. Buoyed by the good news, the share price jumps up to $24.50. What is now the marked-to-market value of your position?

Problem VI. The spot price of copper is $3.87 per lb, and the forward price for delivery in three months is $3.94 per lb. Suppose you can borrow and lend for three months at an interest rate of 6% (in annualized and continuously compounded terms). Assume you have no copper but can borrow it to short it if you wish.

1. First, suppose there are no holding costs (i.e., no storage costs, no holding benefits). Is there an arbitrage opportunity for you given these prices? If so, provide details of the cash flows. If not, explain why not.

2. Suppose now that the cost of storing copper for three months is $0.03 per lb, payable in advance. (This is a three-month cost, not a per-month cost.) How would your answer to (1) change? (Note that storage costs are asymmetric: you have to pay storage costs if you are long copper, but you do not receive the storage costs if you short copper.)