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Subject 3. Pricing futures contracts
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
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 1Consider 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 2Continue 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 3Consider 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 ______.
C. $57.33.Correct Answer: A
F0(T) = S0(1 + r)T = 55 (1.0425) 77/365 = $55.49.
Practice Question 4The 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.
Study notes from a previous year's CFA exam:
c. explain why forward and futures prices differ;