Commodity Futures and Forwards

After completing this reading, you should be able to:

  • Explain the key differences between commodities and financial assets.
  • Define and apply commodity concepts such as storage costs, carry markets, lease rate and
    convenience yield.
  • Identify factors that impact prices on agricultural commodities, metals, energy and weather derivatives.
  • Explain the basic equilibrium formula for pricing commodity forwards.
  • Describe an arbitrage transaction in commodity forwards and compute the potential arbitrage profit.
  • Define the lease rate and explain how it determines the no-arbitrage values for commodity forwards and futures.
  • Describe the cost of carry model and illustrate the impact of storage costs and convenience yields on commodity forward prices and no-arbitrage bounds.
  • Compute the forward price of a commodity with storage costs.
  • Compare the lease rate with the convenience yield.
  • Explain how to create a synthetic commodity position and use it to explain the relationship between the forward price and the expected future spot price.
  • Explain the relationship between current futures prices and expected future spot prices, including the impact of systematic and nonsystematic risk.
  • Define and interpret normal backwardation and contango.

Storage Costs, Carry Markets, Lease Rate, and Convenience Yield

Unlike financial futures and forwards, commodity derivatives have storage costs. For example, a forward contract to exchange 10,000 tonnes of corn six months from today would have warehouse costs. Others do not have such costs, e.g., a forward contract on a perishable good, say, tomatoes. A commodity for which the forward price compensates a commodity owner for costs of storage is called a carry market.

A trader can also lend out a commodity in the short term if they do not plan to use it. The rate of return demanded by the lender is referred to as the lease rate.

Convenience yield is the additional value that comes with holding the asset rather than having a long forward or futures contract on the asset. A good example of a consumption asset that has convenient yield is oil. If you hold oil, you’ll have the convenience of selling it at a higher price during a shortage.

Equilibrium Formula for Pricing Commodity Forwards

The prepaid forward price is by definition the present value of the commodity on the future date:

$$ { F }^{ P }_{ 0,T }={ e }^{ -( \alpha ) T } { E }_{ 0 } ({ S }_{ T }) $$


\({ F }^{ P }_{ 0,T }\)=prepaid forward price

\({ E }_{ 0 } ({ S }_{ T })\)=present value of the expected stock price at time T

\(\alpha\)=discount rate for the commodity.

\(T\)=time between today and the future date at which the transaction will occur, i.e., maturity

The forward price is the future value of the prepaid forward price computed using the risk-free rate:

$$ { F }_{ 0,T }={ e }^{ rT }  { F }^{ P }_{ 0,T } $$


\(r\)=risk-free rate

We can combine these two equations to come up with the forward price:

$$ { F }_{ 0,T }={ E }_{ 0 } ({ S }_{ T }) { e }^{ -( \alpha – r ) T }$$

Which implies that that the commodity forward price is the discounted expected spot price.

Arbitrage in Commodity Forwards

Cash-and-carry Arbitrage

Cash-and-carry arbitrage involves buying a commodity in cash form, holding it, and delivering it into a futures contract. The steps involved are as follows:

At initiation,

  • Borrow money for the term of the contract at market interest rates
  • Buy the underlying commodity at the spot price
  • Sell a futures contract at the current futures price

At expiration,

  • Deliver the commodity, receiving the futures contract price in the process.
  • Repay the loan plus interest.

Reverse Cash-and-carry Arbitrage

A reverse cash-and-carry arbitrage strategy involves combining a short position in a commodity and a long position in the futures for that commodity.

The steps involved are as follows:

At the initiation,

  • Sell commodity short
  • Lend short sale proceeds at market interest rates
  • Buy futures contract at market price.

At expiration,

  • Collect proceeds of the loan
  • Take delivery of the commodity for the futures price and cover the short sale commitment

Backwardation vs. Contango

Backwardation refers to a situation where the futures price is below the spot price. It occurs when the benefits of holding the asset outweigh the opportunity cost of holding the asset as well as any additional holding costs. A backwardation commodity market occurs when the lease rate is greater than the risk-free rate.

Contango refers to a situation where the futures price is above the spot price. It is likely to occur when there are no benefits associated with holding the asset, i.e., zero dividends, zero coupons, or zero convenience yield. A contango commodity market occurs when the lease rate is less than the risk-free rate.

Commodity Forward Price with Storage costs and the Convenience Yield

If the storage cost is a fixed cost \(U\) that’s independent of the value of the underlying asset, then:

$$ { F }_{ 0 }=\left( { S }_{ 0 }+U \right) { e }^{ rT } $$

If the storage cost \(u\) is a percentage of the underlying asset (yield), then:

$$ { F }_{ 0 }={ S }_{ 0 }{ e }^{ \left( r+u \right) T } $$

If a forward contract has a storage cost, \(u\) expressed as a percentage of the underlying, as well as a convenient yield \(y\), then:

$$ { F }_{ 0 }={ S }_{ 0 }{ e }^{ \left( r+u-y \right) T } $$

Exam tips:

  • Both the lease rate and the convenient yield reduce the forward price because they are advantageous to the holder.
  • Storage costs increase the forward price

Factors that Impact Gold, Corn, Electricity, Natural Gas, and Oil Forward Prices


  • Durable, non-perishable good
  • Lease rate: When the lease rate is positive, investors prefer to hold synthetic gold rather than physical gold because the lease rate represents the cost of holding the gold without lending it.
  • Cost of production


  • Seasonality in production, plus storage costs
  • The forward price curve of corn rises to reward storage between harvests, and it falls at harvest


  • Storage is often not possible
  • Most highly volatile commodity on a daily basis because of day-to-day market forces (demand and supply)

Natural gas:

  • Seasonality, transportation costs, and storage costs are important.
  • These factors influence the forward curve.


  • Transportation costs are important

Commodity Spread

Commodity spread refers to the price differential between a raw material commodity and the finished product created from that commodity. For example, crude oil can be refined to produce heating oil, gasoline, and kerosene.

To trade on the spread, an investor typically combines a long position in raw materials with a short position in a finished product.

Common Commodity spreads include:

Crack spread

Crack spread describes the price differential between crude oil and the various petroleum products extracted from it, particularly gasoline, heating oil, and kerosene. The refining process is called cracking and results in a ratio between the raw material and the final products made.


A trader plans to buy \(crude \quad oil\) in one month to produce heating oil and gasoline in two months. The 1-month futures price for crude oil is currently \($30/barrel\). The 2-month futures prices for heating oil and gasoline are \($43/barrel\) and \($33.5/barrel\), respectively. Calculate the 7-4-3 crack (commodity) spread.


The 7-4-3 spread implies that when refined, 7 barrels of crude oil produce 4 barrels of heating oil and 3 barrels of gasoline.

By going long on crude oil and short in both heating oil and gasoline, the trader can lock in a profit:

Profit for a \(7-4-3\quad spread=\left( 4\times $43+3\times $33.5-7\times $30 \right) =$62.5\)

Thus, \(the \quad profit \quad per \quad barrel \quad of \quad crude \quad oil =\) \({ $62.5 }/{ 7 }=$8.93\)


A crush spread describes the price differential between soybeans and its two main products: soybean meal and soybean oil. A trader takes a long position in soybeans and short positions in soybean meal and soybean oil.

Spark spread

The spark spread uses natural gas as the raw material and electricity as the finished product.

Basis Risk in Commodity Hedging

Just like in financial products, basis risk in commodity futures and forwards occurs when spot price and the futures (forward) price do not converge when the futures (forward) contract expires.

Basis risk manifests in two situations:

  • When the risk attached to a commodity’s price cannot be entirely hedged using the commodity futures contracts available in the market due to a quantity mismatch.
  • In cross-hedging – when futures contracts on a particular commodity are unavailable, hedgers turn to closely related commodities that exhibit similar price movement. A good example would be the use of gasoline futures to hedge exposure to jet fuel if jet fuel futures do not exist in the market. Cross-hedging often results in imperfect hedges.

Strip Hedging vs. Stack-and-roll hedging

A strip hedge happens when a series of futures contracts over many maturities ranges are purchased to hedge the underlying cash positions. For example, suppose a farming enthusiast has entered into a contract to supply a fixed number of bags of corn per month at a fixed price. The farmer could set up a strip hedge by buying futures contracts that match the maturity and quantity for every month of the obligation. A strip hedge strategy has no basis risk because the basis becomes locked and changes cannot impact the risk.

A stack-and-roll hedge involves purchasing futures contracts for a nearby delivery date and on that date rolling the position forward by purchasing a fewer number of contracts. The process continues for future delivery dates until the exposure at each maturity date is hedged. Using the example in the preceding paragraph, a stack hedge would involve entering into a one-month futures contract equaling the total value of the year’s promised deliveries.

At the end of the first month, the farmer rolls into the next one-month contract, and so forth, each month setting the total amount of the contract equal to the remaining promised deliveries.

In normal circumstances, short-term contracts require fewer transaction costs. Thus, stack-and-roll hedges are cheaper than strip hedges.


The current spot price of a bag of \(corn\) is \($10\). There exists an active lending market for corn, where the annual lease rate is equal to \(8\%\), the effective annual risk-free rate is equal to \(10\%\), and the \(1-year\) forward price for corn is \($10.35\) per bag. Does arbitrage exist? What’s the risk-free profit up for grabs if indeed an arbitrage opportunity is available?

  1. No; risk-free profit = $0
  2. Yes; risk-free profit = $0.35
  3. Yes; risk-free profit = $0.08
  4. Yes; risk-free profit = $0.13

The correct answer is D.

An arbitrage position exists if the forward price is not equivalent to the expected spot price.

$$ Expected\quad spot\quad price\quad in\quad 1\quad year=S_{ 0 }{ e }^{ \left( r-δ \right) T } $$


\({ F }_{ 0,T }\)=forward price

\(S_{ 0 }\)=commodity spot price

\(r\)=riskfree rate

\(\delta\)=lease rate

\(T\)=time between today and the future date at which the transaction will occur,i.e,maturity

$$ =10{ e }^{ \left( 0.1-0.08 \right) 1 }=10.20 $$

Since 10.35 is greater than 10.20, arbitrage exists.

To take advantage of this opportunity, an arbitrageur can make the following moves:

At initiation,

  • Borrow \($10\) at the rate of \(10\%\)
  • Buy a bag of corn at \($10\)
  • Go short on a corn futures contract
  • Lend the bag of corn at \(8\%\)

At maturity,

  • Take back the bag of corn plus proceeds from the lease amounting to \($0.83(=10{ e }^{ 0.08\times 1 }-10)\)
  • Deliver the bag of corn; receive \($10.35\)
  • Repay borrowed funds amounting to \($11.05(=10{ e }^{ 0.1\times 1 })\)
  • Net profit = \(10.35 + 0.83 – 11.05 = $0.13\)

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