Question

1. Suppose that the current price of gold is $1,734 per oz and that it costs $5 per oz per month to store gold (payable monthly in advance). Suppose also that the term structure is flat with a continuously compounded rate of interest of 6% for all maturities.
   1. Calculate the forward price of gold for delivery in three months.
   2. If the current forward price for gold is $1,775.36 per oz, is there arbitrage opportunity? If so, please explain why there is an arbitrage opportunity and what position (long or short) you would take on the forward market (you do not need to show how to exploit the arbitrage opportunity if any, just say if you would buy or sell forward).

(can you please put the formula you use before plugging the numbers in so i can get clarity because different approaches is what is getting me confused, thank you.

Answer 1

a). Current price of gold = S₀ = $1734 per oz

Cost per month (payable monthly in advance) = $5 per oz, Time to delivery = t = (3/12) year = 0.25 year

As cost are payable monthly in advance, So Cost of each month will paid at beginning of month

Cost for first month will be paid beginning of first month so t1 = time to beginning of first month = 0 years

Cost for second month will be paid at beginning of second month so t2 = time to beginning of second month = 1 month = (1/12) years

Cost for third month will be paid at beginning of third month so t3 = Time to beginning of third month = 2 months = (2/12) years

Interest rate = 6% per year continuously compounded = r = 0.06

Present value of a cash flow   =Cash flow x e^{-r x time to cash flow}

Present value of costs to store gold = PVC = PV of cost for first month + PV of cost for second month + PV of cost for third month = 5 x e^{-r x t1} + 5 x e^{-r x t2} + 5 x e^{-r x t3} = 5 x e^{-0.06 x 0} + 5 x e^{-0.06 x (1/12)} + 5 x e^{-0.06 x (2/12)} = 5 x 1 + 5 x 0.995012 + 5 x 0.990049 = 5 + 4.9750 +4.9502 = 14.9252

Forward price for delivery in three months = (S₀ + PVC) e^rt = (1734 + 14.9252) e^{0.06 x (3/12)} = 1748.9252 x e^0.015 = 1748.9252 x 1.0151130 = 1775.3567 = 1775.36 per oz (rounded to two decimal places)

Forward price for delivery in three months = $1775.36 per oz

b) Current forward price = $1775.36

No Arbitrage forward price = $1775.36                      (As calculated in part a)

As current forward price of gold contract is equal to no arbitrage forward price of gold. So Forward contract on gold is correctly priced. There is no mispricing of forward contract. Hence there exists no arbitrage opportunity. So no positions can be taken in forward contract.

Answer: No Arbitrage opportunity