There is an active lease market for gold in which arbitrageurs can short or lend out gold at a lease rate of = 1%. Assume gold has no other costs/benefits of carry.
Consider a three-month forward contract on gold.
(a) If the spot price of gold is $360/oz and the three-month interest rate is 4%, what is
the arbitrage-free forward price of gold?
(b) Suppose the actual forward price is given to be $366/oz. Is there an arbitrage opportunity?
If so, how can it be exploited?
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(a) The arbitrage-free forward price of gold can be calculated using the formula:
F = S * e^(r*t)
Where:
F = Forward price
S = Spot price
e = Base of the natural logarithm (approximately 2.71828)
r = Interest rate
t = Time to maturity
Given that the spot price of gold is $360/oz, the three-month interest rate is 4%, and the time to maturity is 3 months, we can calculate the arbitrage-free forward price as follows:
F = $360 * e^(0.04 * (3/12))
F = $360 * e^(0.04 * 0.25)
F = $360 * e^(0.01)
F ≈ $360 * 1.01005
F ≈ $363.61/oz
Therefore, the arbitrage-free forward price of gold is approximately $363.61/oz.
(b) If the actual forward price is given to be $366/oz, there is an arbitrage opportunity.
To exploit this arbitrage opportunity, an arbitrageur can do the following:
1. Borrow funds at the risk-free rate (in this case, 4%) for 3 months.
2. Use the borrowed funds to purchase gold at the spot price of $360/oz.
3. Enter into a forward contract to sell the gold at the actual forward price of $366/oz, which locks in a profit.
4. Hold the gold until the forward contract matures, then deliver the gold and receive the forward price of $366/oz.
5. Repay the borrowed funds, including the interest.
By following this strategy, the arbitrageur can lock in a risk-free profit of $6/oz ($366/oz - $360/oz).
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