A futures price is derived by constructing a combination of a long position in the asset and a short position in the futures. This strategy guarantees that the price received from the sale of the asset is known when the transaction is initiated. The futures price is then derived as the unknown value that eliminates the opportunity to earn an arbitrage profit from the transaction.

To save your valuable time we are not going to provide a concrete example here, as it would be very similar to what we have illustrated when deriving the formula F(0, T) = S₀(1 + r)^T to price a forward contract.

The futures price is

F₀(T) = S₀(1 + r)^T

Note that futures contracts are homogeneous and fungible. The marking-to-market process results in each futures contract being terminated every day and reinitiated. Any contract for delivery of the underlying at T (expiration day) is equivalent to any other contract, regardless of when the contracts were created.

Practice Question 1

Consider a futures contract that has a life of 136 days. The annual interest rate is 4.75%. If the spot price is $98, the futures price would then be ______.

Correct Answer: F₀(T) = S₀(1 + r)^T = 98 (1.0475) ^136/365 = $99.71.

Practice Question 2

Continue with question 1. The futures price should be $99.71 based on our analysis if the spot price is $98, the life of the futures contract is 136 days, and the interest rate is 4.75%. If the future is selling for $100, what should an arbitrageur do to net a riskless, positive return?

Correct Answer: He should borrow money to buy the asset for $98, and sell the futures for $100. He will then hold the asset for 136 days, and deliver it to receive $100 when the contract expires. The cost is 98 (1.0475) 136/365 - $98 = $1.71, the interest on $98 at an annual interest of 4.75%. The riskless profit is $100 - $98 - $1.71 = $0.29.

Practice Question 3

Consider a futures contract that has a life of 77 days. The annual interest rate is 4.25%. If the spot price is $55, the futures price would then be ______.

A. $55.49.
B. $56.28.
C. $57.33.

Correct Answer: A

F₀(T) = S₀(1 + r)^T = 55 (1.0425) ^77/365 = $55.49.

Practice Question 4

The spot price is $72. The life of a futures contract is 233 days, and the interest rate is 7.5%. If the future is selling for $75.4019, what should an arbitrageur do to net a riskless, positive return?

A. Buy the asset for $72 and sell the futures for $75.4019.
B. Sell the asset for $72 and buy the futures for $75.4019.
C. There is no arbitrage opportunity in this case.

Correct Answer: C

F₀(T) = S₀(1 + r)^T = 72 (1.075) ^233/365 = $75.4019. As the price of the futures contract is what it should be, there is no arbitrage opportunity at all.