Chapter 3 - Pricing of Futures Contracts

Question 1

What is arbitrage?

Answer 1

Academic or pure arbitrage refers
to the simultaneous purchase and
sale of instruments that are perfect
equivalents in the hope of taking
advantage of pricing discrepancies
between them to earn a risk-free
profit. Most real world arbitrage,
however, is not pure. There usually is
some element of risk.

Question 2

What is backwardation?

Answer 2

Backwardation is when the current price of an underlying asset is higher than prices trading in the futures market. Backwardation can occur as a result of a higher demand for an asset currently than the contracts maturing in the coming months through the futures market.

Question 3

What is basis with regards to futures contracts?

Answer 3

The difference between the current
cash price and the futures price.

BASIS = FUTURES PRICE - SPOT PRICE

Question 4

What is cash-and-carry arbitrage?

Answer 4

Arbitrage that involves buying the
underlying asset and selling the
futures contract to take advantage of
a situation where futures are priced
higher than fair value.

Question 5

What is a contango market?

Answer 5

A market where the forward or futures
price is higher than the spot price.
For commodity futures contracts,
contango markets are considered
normal as there is typically a cost to
carrying or holding a commodity.

Question 6

What is convenience yield?

Answer 6

The benefit from owning the physical
commodity. The value of the benefit
is dependent upon the probability
of shortages of the commodity. If
the commodity is currently in short
supply and that shortage is expected
to continue, the convenience yield will
be high.

Question 7

What is the concept of convergence with respect to futures?

Answer 7

The narrowing of the basis as a futures
contract nears expiration.

Question 8

What is cost of carry?

Answer 8

Term associated with the cost of
holding a commodity or financial asset
until it is sold or delivered. The cost of
holding a commodity typically includes
financing, storage and insurance
charges. The cost of holding a financial
asset typically includes financing costs
less income received such as dividends
for stocks and interest for debt
instruments.

Question 9

What is fair value of a futures contract?

Answer 9

If a futures contract is trading at a price
that reflects full carry, it is said to be
trading at fair or theoretical value.

Question 10

What is an inverted market?

Answer 10

An inverted market shows futures prices that decrease over time, while a normal market sees futures prices that increase over time. A futures market is inverted if the spot price is higher for a contract that expires in one month than a contract that expires in four months.

Question 11

What is a normal market?

Answer 11

A normal futures curve, or normal market, demonstrates that the cost to carry increases with time. An inverted futures curve, or inverted market, demonstrates that the prices for further out deliveries are less expensive than the current spot price.

Question 12

What is reverse cash-and-carry arbitrage?

Answer 12

Arbitrage that involves buying the
futures contract and selling the
underlying asset to take advantage of a
situation where the futures contract is
underpriced relative to fair value.

Question 13

What is a spot price?

Answer 13

The price of an asset on the spot
market.

Question 14

Explain why futures differ from spot prices.

Answer 14

Futures prices represent the current price of the underlying asset adjusted for the cost of carry that arises from
delayed delivery of the asset.

Question 15

Describe the basic cost-of-carry model and explain how it works.

Answer 15

• The cost of delayed settlement includes financing costs and storage and insurance costs. As well, any other
  cash flows associated with owning the underlying asset must be considered.
• The cost of carry model starts with the spot price of an asset and calculates the fair value futures price based
  on these costs from the present until the maturity of the futures contract. The difference between a spot price
  and a futures price is referred to as the basis.

Question 16

Explain how arbitrage keeps futures prices in line with the basic cost-of-carry model.

Answer 16

As the only difference between buying in the futures market and buying in the spot market is the delayed
settlement, it stands to reason that futures prices should be the same as spot prices adjusted for the costs of
the delayed settlement.
If futures prices deviate significantly from fair value, market participants will either buy the “cheap” alternative,
sell the “expensive” alternative, or do both simultaneously. When participants buy and sell the same asset
simultaneously, it is referred to as arbitrage.
The collective actions of market participants attempting to buy low and sell high help keep spot and futures
prices in line with one another during a futures contract’s life.

Question 17

Identify an arbitrage opportunity.

Answer 17

If a futures contract is trading at a price other than its fair value, an arbitrage opportunity may exist. In general,
the futures price must differ from its fair value price by more than the transaction costs involved in executing
the arbitrage before arbitrageurs will try to take advantage of the discrepancy.

Question 18

Explain how to exploit an arbitrage opportunity.

Answer 18

• If the futures price is greater than its fair value price, the futures contract should be sold, and the underlying
  asset should be bought in the cash market. This is known as cash-and-carry arbitrage.
• If the futures price is lesser than its fair value, the futures contract should be bought and the underlying asset
  should be sold short in the cash market. This is known as reverse cash-and-carry arbitrage.

Question 19

Describe situations where the cost-of-carry model may not apply.

Answer 19

The cost-of-carry model applies to futures markets that display the following characteristics…
- ease of short selling;
- a large supply of the underlying interest;
- high storability;
- non-seasonal production or consumption (or both).
When these conditions exist, arbitrage is easy to implement, and futures prices will trade on the basis of fair
value as determined by the cost-of-carry model.
When these conditions do not exist, arbitrage may be very difficult or even impossible to implement. Futures
contracts based on these markets will trade more on the basis of expectations of the future spot price of the
underlying asset.

• Financial assets such as bonds and equities and precious metals are relatively easy to arbitrage as they share these characteristics.
• Agricultural and energy commodities are more difficult to arbitrage because they generally do not meet all of these characteristics. As result, the agriculture – and energy – related futures contracts trade more on the basis of expectations than carry.

Question 20

Describe and calculate the basis of a futures contract.

Answer 20

The basis is a numerical measure of the difference between a cash price and a futures price.
As mentioned above, the basis is simply the difference between the spot price and the futures price.

Question 21

Explain the concepts of normal market, inverted market, and convergence.

Answer 21

• A normal commodity futures market (also referred to as a contango market) is one where futures prices trade at
  a premium to spot prices, reflecting all or at least some of the costs of carry.
• An inverted commodity futures market (also referred to as a market in backwardation) is one where spot prices
  are higher than futures prices. An inverted market places an increased value on owning the physical asset. This
  value is known as a convenience yield.
• Convergence is the process whereby futures prices gradually merge with cash prices as expiration of the
  futures contract approaches.

Question 22

Questions i) and ii) below pertain to copper futures and are based on the following information.

Annual financing rate 4%
Spot price of copper $3.60 per pound
Storage costs $0.005 per pound per month
Insurance $0.015 per pound per month

i. What are the total monthly carrying costs for copper?

a. 1.2¢ per pound.
b. 2.0¢ per pound.
c. 2.7¢ per pound.
d. 3.2¢ per pound.

ii. What is fair value for 3-month copper futures?

a. $3.504 per pound.
b. $3.660 per pound.
c. $3.696 per pound.
d. $3.744 per pound.

Answer 22

i. d. 3.2¢ per pound.

(($3.60 × 0.04 ÷ 12) + $0.015 + $0.005) × 100
Multiply by 100 to convert the answer to cents.

ii. c. $3.696 per pound.

$3.60 + ($0.032 × 3)

The $0.032 is the monthly costs calculated in Question i) above.

Question 23

In a contango commodity futures market, would the basis narrow or widen as the futures expiration
approaches?

Answer 23

Regardless of the type of market (contango or inverted), the basis always narrows as futures expiration
approaches.

Question 24

As a general rule, are futures prices more easily underpriced or overpriced relative to fair value?
Explain your answer.

Answer 24

Futures are generally more easily underpriced, due to the fact that for some commodities it is often difficult if
not impossible to do a reverse cash and carry arbitrage.

Question 25

Which of the following situations would present an opportunity for risk-free profits from a cash and carry
arbitrage strategy?

a. The spot price is lower than the futures price.
b. The spot price is lower than its fair value.
c. The futures price is higher than its fair value.
d. The futures price is lower than its fair value.

Answer 25

c. The futures price is higher than its fair value.

Arbitrage opportunities arise when the futures price deviates from its fair value price. Cash and carry
arbitrage will produce risk-free profits when the futures price is higher than its fair value price.

Question 26

If 6-month gold futures are trading at $1,775 per ounce and the spot price is $1,750 per ounce, what is the
implied annualized cost-of-carry rate?

Answer 26

The implied cost of carry rate is simply the annualized basis divided by the cash price. In this case, it is 2,86%
(($1,775 − $1,750) × 2 ÷ $1,750).

Question 27

A farmer is currently holding 600 tonnes of canola. An elevator is offering to pay the farmer $400 per tonne
for the canola. The 3-month canola futures contract is trading at $420 per tonne. The cost of carrying canola
is $5 per tonne per month.

i. What is the better option, selling the canola to the elevator or holding the canola and selling the futures
contract (and then making delivery against the futures contract in three months)?

ii. At what futures price is the farmer indifferent about choosing between the two options?

a. $395
b. $405
c. $410
d. $415

Answer 27

i. The better option is the option that will bring the farmer the maximum price after taking into consideration
any carrying costs. The farmer in this case has two choices: sell the canola now at $400 per tonne, or sell a
futures contract at $420 per tonne and deliver the canola in three months’ time. For the second option, the
farmer will incur $15 worth of carrying costs, so his net price received is $405 ($420 − $15). This is greater
than $400, so the second option, selling the futures and delivering the canola in three months, is better.

ii. d. $415

The futures price at which the farmer is indifferent is equal to the cash price plus the costs of carrying
the underlying asset (in other words, the futures fair value price).

Question 28

In an inverted commodity futures market, convergence dictates that futures prices will _____________ relative
to cash prices.

Answer 28

Rise. In an inverted commodity futures market, the futures price is lower than the cash price. As expiration of the
futures approaches, the price must rise relative to the cash price to ensure that the two are equal at expiration.

Question 29

With which option is the concept of convenience yield most closely associated?

a. Contango markets.
b. Convergence.
c. Cash and carry arbitrage.
d. Inverted markets.

Answer 29

d. Inverted markets.

A convenience yield is the benefit from holding the cash commodity when there is excess demand for the
commodity relative to supply. When this happens, the cash price could rise above the futures price, thereby
creating an inverted market.

Question 30

List three reasons why a commodity futures price may trade lower than its cash price.

Answer 30

A commodity futures price may trade lower than its cash price for any one of the following reasons…
- Arbitrage may be very expensive or not possible. As a result reverse cash and carry arbitrage (buying the
futures and selling short the cash asset) may not be carried out, meaning that futures prices can remain
under cash prices.
- The market may put a premium on holding the cash asset due to supply tightness.
- Market participants may feel prices will decline in the future.

Question 31

What is the annualized cost of carry rate?

Answer 31

The annualized cost of carry rate = the annualized basis divided by the cash price

It is a measure of the total cost of holding a financial asset or physical commodity over a year; it includes ALL associated costs

Example…

In the case of a 6-month futures contract…

[(1775-1750)x2]/1750 = 2.86%

Notes about the math…

In this example we multiplied the difference by 2 to get the annualized figure because we were dealing with a 6-month contract. Had we been dealing with a 4-month contract we would have multiplied by 3, and so on. Finally, always remember that when attempting to get percentage figures (for the most part; and at least in this example) you will need to multiply the final answer by 100.

Other than this one question, the rest of the math in this chapter was pretty straight forward.

Question 32

How does arbitrage resolve price discrepancies?

Answer 32

Arbitrage involves buying an asset at a lower price in one market and simultaneously selling it at a higher price in another market. This activity increases demand in the cheaper market (raising the price) and increases supply in the expensive market (lowering the price). As more arbitrageurs exploit the price difference, the prices in both markets converge, eliminating the discrepancy. This process helps ensure that the same asset has the same price across different markets.

Question 33

Provide a short and simple example of a situation where the value of futures contracts are trading more on the value of future expectations rather than the value of cost of carry.

Answer 33

Imagine it’s November, and a futures contract for wheat, expiring in March, is trading at a much higher price than the current spot price. This is because market participants expect a poor wheat harvest in the coming months due to predicted bad weather, which would reduce supply and drive up prices. In this case, the high futures price reflects future expectations of scarcity and higher prices, rather than just the cost of carrying wheat over the winter.