A calendar spread option is an option to enter into two separate futures positions: one long and one short. A call option can be exercised into a long futures position that is closest to expiration and a short futures position in a more distant month. The put option can be exercised into a short futures position that is closest to expiration and a long futures position in a more distant month. The strike price is the price differential between the long and short futures positions.

A call option with a $0.50 strike price, for example, represents a long futures position priced $0.50 higher than a short futures position of a more distant maturity. A put option with a $0.50 strike price, by contrast, represents a short futures position priced $0.50 higher than a long futures position of a more distant maturity.

The payoff for a long CSO call or put position at maturity T is:

• Call Payoff: CT=max(0,ST−K)
• Put Payoff: PT=max(0,K−ST)

^{Where:
T: Options Expiration
ST= FT(1)-FT(2): Calendar Spread at expiry T
K: Strike Price}

If the futures spread settles higher than $0.50 on the day of the CSO expiration, the owner of the $0.50 call profits on the difference.

CSOs have matured into an important tool in oil trading, allowing traders to manage any structural shifts in the forward curve and inter-month dynamics. 

The term structure in crude oil is largely influenced by supply/demand, storage dynamics and geopolitical events. This results in the market’s expectation of a dynamic evolution in prices. This is in contrast to the natural gas market, where seasonality largely dictates the shape of the forward futures curve.

In crude oil, a backwardated market, where the current price is higher than prices in the forward months, generally reflects potential shortages adding a premium to spot prices. Conversely, a higher price on the more distant futures contracts generally reflects plentiful supply or inventory levels. This is often referred to as contango. Chart 1 provides an example of a backwardated and contango WTI futures markets.

The shape of the forward curve has important implications for inventory management. For example, if the market is backwardated, the current value of inventory is greater than the deferred futures price. Holding inventory in this situation could result in selling at a lower price. Conversely, in a contango market, holding inventory and selling at a deferred date is expected to yield extra revenue net of storage costs.

This convenience yield can be viewed as the embedded optionality attached to holding a physical commodity. It is defined as the difference between the positive gain of holding a commodity minus the cost of storage. Therefore, the convenience yield can be positive or negative depending on the time period and the level of oil inventory.

Chart 2 displays a 1-month calendar spread curve of WTI Crude Oil futures in a backwardated market. If a market participant were holding a long position and needed to roll to the next contract month ahead of expiration, he would earn a profit when selling the position and buying the next month’s futures contract.

A crude oil producer extracts oil during the month of June and intends to store it until a buyer is procured. The producer has an incentive to store the oil because the forward price of oil is currently trading at a premium (contango market). The producer secures a 1-month storage contract requiring him to sell his physical Crude Oil when the July contract expires. The June Crude Oil futures contract is trading at $64.96, while the July futures contract is trading at a price of $65.24, making the June/July calendar spread $-0.28.

The producer is effectively long the June/July calendar spread and is exposed to a narrowing of that spread differential. The producer wants to hedge against a flattening of the term structure, which would reduce his expected profit on the oil sale. He purchases a $-.20 strike call CSO for a premium of $0.15.

Initial June/July spread: $-0.28
CSO Call Strike: $-0.20
Premium paid: $0.15

• Scenario A:
  
  At June futures expiration, the June/July spread has steepened and settled at $-1.50. The producer realized a greater-than-expected profit on the physical position as the final June/July spread widened. The producer protected his original profit target, resulting in a total profit of $1.07 ($1.22 spread difference - $0.15 option premium).
  
  Final June/July spread = $-1.50
  Gain on physical spread position = $-0.28-($-1.50) = $1.22
  Intrinsic value of CSO Max (0, $-1.50-($-0.20)) = 0
  P&L = $1.22 - $.15 = $1.07

• Scenario B:
  
  At June futures expiration, the June/July spread contracted to +$.30. If the producer were to sell, he would now be faced with a $0.58 decline in value on his physical position. However, the CSO call option he purchased is now in-the-money, worth $0.50, helping to offset the physical loss.
  
  Final June/July spread = +$.30
  Loss on physical spread position  = $-0.28-($0.30) = $-0.58
  Intrinsic value of CSO = Max (0, $0.30-($-0.20)) = $0.50
  P&L =  $ -.58 + $.50 - $.15 = $-.23

In example 1, the storage operator benefitted from owning a CSO call. For a relatively small insurance premium of $0.15 to buy the call, he was protected from an undesirable move in the futures spread, and still realized a profit when the spread moved in his favor.

A professional trader has a bearish view on Oil futures and needs to roll his current short futures position when it expires in 30 days to the next calendar month. Depending on the shape of the crude term structure, rolling futures to maintain position can be costly or profitable. If the market is backwardated, he will have to pay a premium each month to roll his short position forward.

In a possible scenario, the oil term structure is backwardated and the Sep/Oct calendar spread is trading at $1.25. This means it would cost the trader $1.25 to roll his short futures position forward and maintain a short position should the term structure remain near this level for 30 days. Given the trader’s desire to roll the short position, he will need to buy the Sep/Oct calendar spread in order to close out the September position and initiate a short position in October.

The trader wants to hedge against the cost of rolling his position. High volatility in the near-month CSOs has increased option premiums, thus, he decides to sell a put against his position. The trader sells the Sep/Oct $1.10 put for a premium of $0.20.

• Scenario A:
  
  At September futures expiration, the Sep/Oct spread has steepened and settled at $1.45. This increases the trader’s cost to roll. However, since he collected $0.20 premium selling the put, which is now out of the money and expires worthless, his roll costs are reduced by $0.20 to $1.25, the original spread cost.
  
  Final Sep/Oct spread = $1.45
  Premium from CSO put sale = $0.20
  Intrinsic value of CSO Max (0, $1.10-$1.45) = 0
  Net cost = $1.45 - $0.20 = $1.25

• Scenario B:
  
  At September futures expiration, the Sep/Oct spread contracted to $0.90. This move was in the trader’s favor, which lessens his rolling cost by $0.35 (from $1.25 to $0.90). However, since he sold a physically settled CSO put at a strike of $1.10, the option expired in-the-money by $ 0.20 ($1.10- $0.90), and he will receive a long September futures position and a short October futures position – the transaction he would have needed to perform to roll.
  
  When he initiated the trade, the calendar spread was $1.25. It finished at $0.90, a move in his favor which reduced his roll costs by $0.35. While the trader benefitted from the spread price contraction of $0.20 and collection of the option premium of $0.20, the move was adverse for the short put position. His net result was a $0 cost for insurance against rising rolling cost ($0.20 loss on the call offset by the $.20 premium collected).
  
  Final Sep/Oct spread = $0.90
  Improvement of roll cost = $1.25-$0.90 = $0.35
  Premium from CSO put sale = $0.20
  Intrinsic value of CSO = Max (0, $1.10-$.90) = $.20 Loss
  Net benefits ($0.35+$0.20-$0.20) = $0.35

In example 2, the trader was able to employ a CSO position to flexibly lower his roll costs. He could also repeat this same transaction each term or depending on how long he wants to maintain his short futures position, he could sell CSO puts each month or as a strip to reduce his overall roll costs.

The natural gas term structure is defined by seasonality. The withdrawal season (winter) – ranges from November to March and is noted for its volatility. The injection season (summer) – ranges from April to October and is generally less volatile. 

During the winter season, gas consumption peaks as a result of increased heating demand from residential, commercial and industrial end-users. During the summer season, gas demand decreases while production continues, resulting in excess natural gas that can be stored. As a result of unpredictable winter demand, the winter Natural Gas futures typically trade at a premium to the summer futures. The winter term structure is in backwardation – defined by the scenario when near-month futures are priced higher than back-month futures – versus the summer term structure in contango, characterized by near-month futures trading at a discount to back-month futures. Chart 3 demonstrates the seasonal term structure of the Henry Hub Natural Gas futures market.

Commercial firms typically produce or purchase natural gas at a lower price during the summer season and inject that natural gas into storage. Those firms then sell the stored natural gas in the winter season at a higher price to profit from the demand premium. The ability to purchase, store and sell natural gas across the winter and summer seasons creates implicit optionality.

As an example, a storage operator has accumulated natural gas inventories late in the summer and intends to sell his inventories early in the winter. This replicates a long summer and short winter futures position, and for the sake of this example, a long October and short November position. This gives the storage operator the optionality to sell inventories in November should the futures price exceed the October price at which the natural gas was stored, after factoring in financing and storage costs. The storage operator’s position is the equivalent of owning a call on the October/November calendar spread.

For simplicity’s sake, there are two possible outcomes as time moves from October to November: the October/ November futures spread either widens or contracts between the time of injection and withdrawal.

The storage operator buys physical natural gas in October at an average price of $2.50 per MMBtu while the price of November Natural Gas is $3.00 per MMBtu (displayed as an Oct/Nov spread price of $-0.50). He wants to ensure he can profit at least $0.50 when he sells the physical gas in November. To do this, he would buy an Oct/Nov $-0.50 call option. Thirty days ahead of the October futures expiration, the Oct/Nov $-0.50 call can be purchased for $0.25.

Initial Oct/Nov spread = $-0.50
CSO Call Strike = $-0.50
Premium paid = $0.25

• Scenario A:
  
  

  At October futures expiration, the Oct/Nov spread settles at $-1.50: October at $2.50 and November at $4.00. The storage operator gains $1.00 on his physical position, but at the same time, he loses $0.25 from the $-0.50 strike call premium which settled out-the-money. The storage operator protected himself from the Oct/Nov spread narrowing with the purchase of a CSO call.
  
  Final Oct/Nov spread = $-1.50 
  Gain from physical position = $-1.50 - $-.50 = $1.00
  Intrinsic Value of CSO = Max (0, $-1.50−($-0.50))= $0.00
  Net P&L = $1.00 + $0.00 - $0.25 = +$0.75

• Scenario B:
  
  At October futures expiration, the Oct/Nov spread settles at $0.80: October at $3.00 and November at $2.20. The storage operator profited $1.30 on his CSO position, less the $0.25 insurance premium of the call, for a total CSO profit of $1.05 ($1.30-$0.25).
  
  In this case, the call finished in-the-money and offset some of the losses resulting from the futures spread trending against the storage operator’s physical position.
  
  Final Oct/Nov spread = $0.80
  Loss on physical position = $0.80 - $-0.50 = $1.30
  Intrinsic Value of CSO = Max(0, +$0.80−($-0.50)) = $1.30
  Net P&L = $-1.30 + $1.30 - $0.25 = $-.25

In example 1, the storage operator benefitted from owning a CSO call. For a relatively small insurance premium of $0.25 to buy the call, he was protected from an adverse move in the futures spread, but still realized a profit when the spread moved in his favor.

Besides hedging physical storage, CSOs can also be used to capitalize on volatile moves in futures spreads. The most volatile of all Natural Gas futures spreads is the March/April spread, when the transition from withdrawal to injection begins.

A natural gas trader is expecting a colder-than-normal winter and is bullish on the March/April spread (expecting March futures prices to trend much higher relative to April futures prices).

On December 1, the March/April spread is trading at a differential of $0.20 with March at $3.10 and April at $2.90. The trader believes the March/April spread will settle higher than $1.00 when it expires at the end of February.

The trader purchases a CSO call on the March/April spread at a strike price of $0.65 for a cost of $0.13. In order to break even, the March/April spread must reach at least $0.78 upon expiration ($0.65 strike + $0.13 premium).

Initial March/April spread = $0.20
CSO Call Strike = $0.65
Premium paid = $0.13

Purchasing a call limits the trader’s downside risk versus going long the futures spread. The maximum loss on the CSO is the premium paid ($0.13), where the maximum loss on a futures position can be much greater.

At March futures expiration, the March/April spread settles at $1.10: March at $4.20 and April at $3.10. The trader’s bullish sentiment was correct – and his CSO call settled in-the-money by $0.45, netting a profit of $0.32 ($0.45 profit – $0.13 option premium).

Final March/April spread = $1.10
Intrinsic Value of CSO = Max(0, +$1.10−$0.65) = $0.45
CSO P&L = $0.45-$0.13 = $0.32

The seasonality and resulting volatility in natural gas markets provide all participants – from physical storage operators to professional traders – opportunities to utilize CSOs to hedge against or benefit from moves in Natural Gas futures spreads.

Option pricing models and hedging tools that are traditionally utilized for standard vanilla options should not be applied to spread options for a variety of reasons, most notably the possibility of negative strike prices. Spread option pricing varies, but can be classified into two main approaches: numerical models and analytical models. Numerical models include Monte Carlo simulation, fast Fourier transform and numerical integration. The primary analytical model is a closed form solution (CFS), known as the Bachelier^[1] model. The Bachelier model relies on the assumption that the underlying spread follows a symmetrically normal price distribution. Since the price follows a normal distribution, volatility also needs to be treated differently. These models are commonly used across the CSO trader community.

Oil and gas prices are highly elastic with respect to various fundamental factors including weather, geopolitical risk and unanticipated supply/demand. Unpredictable changes from any of these factors can have an impact on forward-curve prices and the correlation between the calendar months. CSOs are flexible products that are sensitive to the slope of the forward-term curve and will rise or fall in value as the shape of that term curve changes with time. A move from contango to flat to backwardation will change the values of term differences all along the curve and CSOs can be leveraged to realize those steepening or flattening changes. 

CME Group has a diverse product offering of calendar spread options across Crude Oil, Natural Gas and Refined Products. Additionally, CME Group offers inter-commodity spread options such as WTI-Brent and Crack Spreads.

<sup>[1] 
Bachelier, “Theory of Speculation (English translation) by Paul Cootner,” Risk, January (2000), pp. 50-55. 

Poitras, “Spread options, exchange options, and arithmetic Brownian motion,” Journal of Futures Markets, 18 (1998), pp. 487-517.

Shimko, “Options on future spreads: Hedging, Speculating, and Valuation,” Journal of Futures Markets, 14, No. 2 (1994), pp. 183-213.

Wilcox, “Energy futures and options: Spread options in energy markets,” Goldman Sachs & Co., New York, 1990.</sup>