Question

A trader can sell gold at $794.00 per ounce and buy it at $800.00 per ounce. The trader can invest money at 5.7% per year and borrow money at 6% per year, each compounded annually.

For what range of two-year forward prices of gold does the trader have no arbitrage opportunity? Assume no storage costs.

Answer 1

Present Buying price of gold = $800 per ounce

To buy gold, trader will take short position in Forward contract and will need borrow money $800 at rate of 6% per year..

Amount owed after 2 years = 800 x (1+ borrowing rate)² = 800 x (1+6%)² = 800 x 1.1236 = 898.88

If Forward price is equal to $898.88, then there will zero profit or no arbitrage opportunity as proceeds from delivering the gold at forward price in short forward contract will equal to amount owed.

For Forward price above $898.88, trader can make profit as proceeds from delivering the gold at forward price are more than the amount owed. Hence there will be arbitrage opportunity

For Forward price less than 898.88 trader will experience a loss as proceeds from delivering the gold will be less than amount owed and hence there will not be any arbitrage opportunity.

Hence when buying gold , then there will be if no arbitrage opportunity if

Forward gold price ≤ $898.88                         [ inequality 1]

Present selling price = $794 per ounce

Now when selling gold, Trader will take long position in forward contract and will sell gold to receive $794 and invest at 5.7% per year

Value of investment after 2 years = 794 (1+ investing rate)² = 794 x (1+5.7%)² = 794 x 1.117249 = 887.0957 = 887.10

If Forward price is equal to $887.10, then there will be zero profit or no arbitrage as Value of investment will be equal forward price to buy asset at forward price.

For forward price above $887.10 then, Value of investment is not enough to buy gold at forward price and there will not be arbitrage opportunity. Trader will experience loss.

For Forward price below $887.10, Value of investment will be greater than forward price, so trader will be able to make profit by buying gold at forward price and there will be arbitrage opportunity.

Hence when selling gold , then there will be if no arbitrage opportunity if

Forward gold price  ≥ $887.10[ inequality 2]

Now from inequality 1 and inequality 2, we get the range for which there will no arbitrage opportunity

898.88 ≥ Forward gold price  ≥ $887.10

Range of price is from $887.10 to $898.88 (both values inclusive)