Question

3. You observe that the current spot price of gold is TL400 per ounce. You also observe that the yield curve is flat and all maturities up to one year have an interest rate of 12 percent. Since gold is a popular underlying asset in the derivatives markets, you are interested in identifying any mispricing that may allow you to earn arbitrage profits. When you look up gold forward contract prices, you see that there is a contract with a one year maturity whose current price is TL410 per ounce. The size of this forward contract is 100 ounces of gold per contract. Assume that there are no transaction costs in the spot or forward markets.

a. Calculate the theoretical fair price for the forward contract.

b. Based on the price you calculated in part (a) and the price you observe in the market, comment on whether it is possible to earn an arbitrage profit. If your answer is yes, then specifically describe the transactions at time 0 and T and calculate the arbitrage profit.

c. Assume that there is also a gold futures contract with a one year maturity and the same contract size that is available on the futures exchange. This contract is marked to market on a daily basis and the initial (at time 0) price of the futures contract is also TL410. Assume that the futures contract price stays constant except for the following four days during the year:

Time in Days	Price in $
0 (initial)	410
100	400
180	375
300	360
350	420

Assume that on day 0, you take a long position in one futures contract at the initial price of TL410 per ounce. Also assume that on the expiration day of the contract (t = 365), the spot price of gold is TL420 per ounce. Assume that there are no initial or maintenance margin requirements but you have to pay for your losses or will get paid whenever you have a gain. During the one year that you hold your long position, you borrow at 12 percent to finance all your losses and you lend at 12 percent whenever you have a gain. Calculate your final profit on this futures transaction. Compare this profit with the profit you have calculated in part (b) and, if the two profits are different from each other, explain the reason for this difference.

Answer 1

A) Theorectical forward price F₀

Forward price, F₀ = S e^RT

= 400 * e^{0.12 * 1}

^{= 450.997 or appro per ounce}

^{B) Abritrage profit}

• Theoretical forward price = 450.997
• Actual forward price = 410

The actual future price is lower than the theoretical price, this clearly means there is an arbitrage opportunity available.

Arbitrage strategy to be used at time 0:

1. Buy forward contract (Initial value = 0 or no cost to be paid to enter the contract)
2. Short 100 ounces of gold (receives TL40,000)
3. Lend money at risk free rate (lend TL 40,000 @ 12%)

Arbitrage strategy to be used at time T:

1. Collect income on loan ( FV of 40,000 = 40,000 (1+12%)¹ = 44,800 or in other words your 40,000 has grown now to 44,800)
2. Take delivery of Forward contract (Buys back gold for 410 *100 = 41,000)
3. Return borrowed commodity

Arbitrage profit = (44,800 - 41,000) = TL 3,800