Pricing Futures Contracts Part 1: The Theory
Book: Commodities: Markets, Performance, and Strategies
Editors: H. Kent Baker, Greg Filbeck, Jeffrey H. Harris
Publisher: Oxford University Press, 2018
ISBN: 9780190656010
Why Futures Prices Are What They Are
Chapter 15, written by Timothy Krause, tackles a question that sounds simple but really is not: how do you figure out the “correct” price for a futures contract? The answer, as it turns out, is built on a powerful idea called “no arbitrage.” If a futures price is wrong, somebody with enough capital will immediately profit from the mistake and push the price back to where it should be.
Here is the starting point. A futures contract is an agreement to exchange some asset in the future. The price of that contract has to be connected to the current spot (cash) price of the underlying asset. If those two prices get too far apart, arbitrageurs step in and make risk-free money until the gap closes.
The Basic Pricing Formula
The simplest version of the futures pricing equation looks like this: the futures price equals the spot price multiplied by a factor that accounts for the risk-free interest rate and the time to expiration.
Krause uses gold as an introductory example. Say gold is trading at $1,500 per ounce in the spot market. Interest rates are 1 percent for one year. If a one-year gold futures contract is trading at $1,520, an arbitrageur can borrow $1,500, buy gold, and simultaneously sell the futures at $1,520. At the end of the year, they deliver the gold, collect $1,520, repay the loan of about $1,515 (principal plus interest), and pocket roughly $5 per ounce risk-free. No initial investment needed.
That is the core logic. Any deviation from fair value gets “arbitraged away” by traders doing exactly this kind of trade. The theoretical fair value of that gold futures contract would be $1,515.08, not $1,520.
Adjustments for Different Asset Types
The basic formula works for simple cases, but different assets need different adjustments. Krause walks through several:
Stock index futures need to account for the dividend yield on the index. Owning the index directly gives you dividends, but owning a futures contract does not. So the dividend yield reduces the futures price relative to the spot price. As of late 2016, S&P 500 futures were within pennies of their theoretical fair value. The formula works very well here because of the massive amount of index arbitrage that happens in equity markets.
Single stock futures work similarly, except you subtract the present value of expected dividends rather than a continuous yield.
Foreign currency futures replace the dividend yield with the foreign risk-free rate. The futures price reflects the interest rate differential between the two countries. This is basically covered interest rate parity, and it works well in practice.
The Trickier Case: Consumption Commodities
Here is where things get more interesting. Gold can be borrowed and sold short. But you cannot short a barrel of crude oil or a bushel of corn the same way. These are consumption commodities. People actually use them, and there is no easy way to borrow them for short selling.
Because of this limitation, the no-arbitrage argument only works in one direction. You can buy the physical commodity and sell futures against it (which puts an upper bound on the futures price). But you cannot do the reverse. You cannot short oil and buy futures to put a lower bound on prices.
This creates room for something called the convenience yield. The convenience yield is the benefit of having the physical commodity on hand, ready to sell if prices spike. You cannot measure it directly. It is the invisible value that commodity holders get from having inventory available.
The pricing equation for consumption commodities becomes: futures price equals spot price adjusted for interest rates, storage costs, minus the convenience yield.
What Happened with Crude Oil
Krause provides a real example that shows how messy this gets in practice. Between January 2015 and November 2016, crude oil futures deviated significantly from their theoretical fair value. During the oil price crash of 2015, the deviations were extreme because storage costs went through the roof.
At one point, the deviation between futures and fair value approached $10 per barrel. Oil companies were so reluctant to sell at low prices that they stored oil in tanker ships because land storage was full. The average cost of production in the US was about $36 per barrel, and oil was trading near that level.
Compare this to S&P 500 index futures, where the mean deviation from fair value was less than $1 (about 0.048 percent of the contract value). The stock index market was far more efficient because the arbitrage mechanism works cleanly in both directions.
Cost of Carry: The General Framework
Krause ties all these specific cases together under the umbrella of “cost of carry.” The cost of carry is the net expense of holding a futures position compared to holding the underlying asset in the spot market.
For a simple non-dividend stock: cost of carry is just the risk-free interest rate.
For a commodity with storage costs: cost of carry includes interest rates plus storage.
For a stock index: cost of carry is the interest rate minus the dividend yield.
For a consumption commodity with convenience yield: cost of carry includes interest, plus storage, minus the convenience yield.
This framework is clean and intuitive. The futures price is always the spot price multiplied by the exponential of the cost of carry over time. When the cost of carry is positive, futures prices exceed spot prices (contango). When it is negative, spot prices exceed futures prices (backwardation).
Two Competing Theories
Krause also covers two alternative theories about futures pricing. The rational expectations model says that current futures prices reflect the market’s collective opinion about future spot prices. If prices drift too far from expectations, traders push them back.
The Keynes-Hicks theory of normal backwardation takes a different view. Keynes and Hicks argued that commodity producers are typically net sellers of futures contracts, while speculators are net buyers. Because speculators need compensation for taking on risk, futures prices tend to sit below expected future spot prices.
Most academics and practitioners favor the cost of carry model. Research by Chow, McAleer, and Sequeira (2000) found “substantial evidence in favor of the cost of carry hypothesis.” Gorton, Hayashi, and Rouwenhorst (2012) did a thorough study of convenience yield and concluded it is tied to commodity inventories, which supports the cost of carry framework and rejects the Keynesian version.
My Takeaway
The theoretical pricing of futures contracts is one of those topics that looks dry on paper but is deeply practical. The no-arbitrage framework is elegant and works well for financial assets like stock indices and currencies. For commodities, things get messier because of storage costs, convenience yields, and the inability to short physical goods.
The crude oil example is the most valuable part of this section. It shows that theory and practice can diverge sharply when real-world supply and demand get out of balance. The 2015-2016 oil market was a case where the convenience yield, storage costs, and supply gluts all collided to push futures prices far from their “fair” levels.
Understanding this framework matters because it explains why futures prices can be above or below spot prices, and what that relationship tells you about the underlying market conditions.
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