Question

a) Suppose that gold, which does not pay any dividends and has no convenience value, is currently trading at 1350 USD per ounce in the spot market. The LIBOR rate is 2% per year and the cost of storing and insuring gold is 1.50 USD per ounce for 180 days, payable at the end of the period. Based on these, determine the theoretical price of a forward contract on gold that has 180 days till maturity.

b) Assume that your bank quotes you forward prices of 1360 (bid) and 1362 (ask) for a 180day forward contract on gold. Also assume that you can borrow and lend at a 2% per year interest rate, store gold or receive storage payments for gold at 1.50 dollars per ounce for 180 days, and that you can buy or short gold in the spot market at 1350. Do these set of prices present an arbitrage opportunity? If your answer is yes, describe precisely how you would take advantage of it and calculate the arbitrage profit per ounce.

c) Remark briefly on the fact that the set of transactions you enter into in part (b) are through a bank and not a clearinghouse. Specifically, if one were to hold these arbitrage positions till forward settlement date, what can go wrong in this 180-day holding period that would affect their profitability?

Answer 1

a) Price of commodity futures is given by cost of carry model as under -

F = S + RF + SC - DY -CY

where F = Futures price, S = Spot price, SC = Storage cost

DY = Dividend yield and CY = Convinience Yield.

Since gold do not pay any dividend and has Nil CY, last 2 terms are gone.

RF = 1350 * 2% * 180/360 = 13.50

Therefore F = 1350 + 13.5 + 1.5 = 1365.

ASSUMPTIONS - LIBOR is assumed to be compunded half yearly. We have assumed 360 days in a year for ease.

b) Since both the bid and ask rates are less than theoretical forward price of 1365, there exists arbitrage opportunity. Actual prices are less than theoretical price, so we should buy the futures spot at 1362 per ounce. We should short sell the gold today @ 1350 per ounce. Invest the funds so received 1350 at risk free rate which is Libor for 180 days. It will become 1363.5 after 180 days and we received 1.5 as stoarge costs saved giving us a total of 1365 per ounce. We will cover the short position on gold via long position on futures by paying 1362 per ounce only. This we lock in a profit of 3 per ounce. This is called reverse cash and carry arbitrage.

c) Above calculation is highly subjective and a lot of assumptions are made in this. First of all, we have ignored margin to be paid on futures or forwards and it's interest costs. It can be significant and dilute our profits. Secondly, we have assumed short selling is allowed without any restrictions. This is not true in real life. We may need to pay some charge to lender of gold. Then we have assumed constant rate if borrowing and investment which is never true. They are prone to changes always. Also storage Costa may change invariably depending upon demand and supply conditions. There is basis risk as well. It arises when underlying derivative fails to track the instrument fully.

So any change in these factors in adverse direction can wipe out the profits.