The current spot price of gold is $1200 per ounce. The riskless interest rate is 10% per annum. For simplicity, assume there are no storage/security costs of gold. a) What is the arbitrage-free forward price for the delivery of gold in 8 month’s time? b) If you see an 8-month forward price of gold quoted at $1240 per ounce, explain how you would capture an arbitrage profit. c) If you see an 8-month forward price for gold quoted at $1290 per ounce, explain how you would capture an arbitrage profit.
Question
The current spot price of gold is 1240 per ounce, explain how you would capture an arbitrage profit. c) If you see an 8-month forward price for gold quoted at $1290 per ounce, explain how you would capture an arbitrage profit.
Solution
a) The arbitrage-free forward price for the delivery of gold in 8 months can be calculated using the formula:
Forward Price = Spot Price * e^(r*t)
where:
- Spot Price is the current price of gold ($1200)
- r is the riskless interest rate (10% or 0.10)
- t is the time in years (8 months or 0.67 years)
- e is the base of the natural logarithm (approximately 2.71828)
So, the forward price is:
Forward Price = 1200 * e^0.067 = 1283.16
b) If the 8-month forward price of gold is quoted at $1240 per ounce, you could capture an arbitrage profit by borrowing money to buy gold at the current spot price and selling it forward. Here's how:
- Borrow 1200 * e^(0.10*0.67) = $1283.16.
- Buy an ounce of gold at the current spot price of $1200.
- Sell the gold forward for delivery in 8 months at the quoted forward price of $1240.
- At the end of 8 months, deliver the gold and receive $1240.
- Repay the loan of $1283.16.
You would make a profit of 1283.16 = -$43.16. This is a negative profit, meaning you would actually lose money in this scenario. Therefore, there is no arbitrage opportunity here.
c) If the 8-month forward price for gold is quoted at $1290 per ounce, you could capture an arbitrage profit by borrowing money to buy gold at the current spot price and selling it forward. Here's how:
- Borrow 1200 * e^(0.10*0.67) = $1283.16.
- Buy an ounce of gold at the current spot price of $1200.
- Sell the gold forward for delivery in 8 months at the quoted forward price of $1290.
- At the end of 8 months, deliver the gold and receive $1290.
- Repay the loan of $1283.16.
You would make a profit of 1283.16 = $6.84. This is a positive profit, meaning you would make money in this scenario. Therefore, there is an arbitrage opportunity here.
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