Question

If the spot price of gold is $980 per troy ounce, the risk-free rate is 4%, storage and insurance costs are zero, ( a) what should the forward price of gold be for delivery in 1 year? (b) Use an arbitrage argument to prove the answer. Include a numerical example showing how you could make risk-free arbitrage profits if the forward price exceeded its upper bound value.

Answer 1

a.) Forward Price = Spot price X (1 + R_f)^t

R_f = Risk-Free Rate

t = forward contract term in years.

FP = 980 (1.04) = $1019.2

b.) The point at which there are no arbitrage opportunities is going to be the efficient clearing price of the assets.

Derivatives are typically priced by forming hedge involving the underlying asset and a derivative such that the combination pay risk-free rate and do so only for one derivative price.

Let us take a numerical example.

Suppose Forward Price for delivery in 1 year is $1050 instead of $1019.2

Now, there is an opportunity to earn arbitrage profit of $30.8

At T = 0,

1.) Short a forward contract @ $1050. Now you have to deliver 1 troy ounce gold at T = 1.

2.) Borrow $980 at the risk-free rate. Now you have to pay back $1019.2 at T = 1.

3.) Buy 1 troy ounce from the borrowed money.

Now at T = 1,

1.) Forward contract matures. Deliver the gold (you already had) and take the proceeds of $1050.

2.) Payback the loan amount $1019.2 form the proceeds.

3.) Enjoy the arbitrage profits of $30.2 [ 1050 - 1019.2 ]

Hope this helps :)