CHAPTER 3 Futures Prices

1
CHAPTER 3Futures Prices

• In this chapter, we discuss how futures contracts
  are priced. This chapter is organized into the
  following sections
• Reading Futures Prices
• The Basis and Spreads
• Models of Futures Prices
• Futures Prices and Expectations
• Future Prices and Risk Aversion
• Characteristics of Futures Prices

2
Reading Futures Prices

• TERMINOLOGY
• To understand how to read the Wall Street Journal
  futures price quotations, we need to first
  understand some terminology.
• Spot Price
• Spot price is the price of a good for immediate
  delivery.
• Nearby Contract
• Nearby contracts are the next contract to mature.
• Distant Contract
• Distant contracts are contracts that mature
  sometime after the nearby contracts.

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Reading Futures Prices

• TERMINOLOGY
• Settlement Price
• Settlement price is the price that contracts are
  traded at the end of the trading day.
• Trading Session Settlement Price
• New term used to reflect round-the-clock trading.
• Open Interest
• Open interest is the number of futures contracts
  for which delivery is currently obligated.

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Reading Futures Prices

• Insert figure 3.1 Here

5
How Trading Affects Open Interest
The last column in Figure 3.1 shows the open
interest or total number of contracts outstanding
for each maturity month. Assume that today, Dec
1997, widget contract has just been listed for
trading, but that the contract has not traded
yet. Table 3.1 shows how trading affects open
interest at different times (t).
6
Open Interest Trading Volume Patterns

• Insert Figure 3.2 Here

• Insert Figure 3.3 Here

7
The Basis
The Basis The basis is the difference between the
current cash price of a commodity and the futures
price for the same commodity.

• S0 current spot price
• F0,t current futures price for delivery of the
  product at time t.
• The basis can be positive or negative at any
  given time.
• Normal MarketPrice for more distant futures are
  higher than for nearby futures.
• Inverted MarketDistant futures prices are lower
  than the price for contracts nearer to expiration.

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The Basis

• Example if the current price of gold in the
  cash market is 353.70 (July 11) and a
  futures contract with delivery in December
  is 364.20. How much is the
  basis?

9
The Basis

• Convergence
• As the time to delivery passes, the futures price
  will change to approach the spot price.
• When the futures contract matures, the futures
  price and the spot price must be the same. That
  is, the basis must be equal to zero, except for
  minor discrepancies due to transportation and
  other transactions costs.
• The relatively low variability of the basis is
  very important for hedging.

• Insert Figure 3.4 here

• Insert Figure 3.5 here

10
Spreads
Spread A spread is the difference in price
between two futures contracts on the same
commodity for two different maturity dates

• Where
• F0,t The current futures price for delivery
  of the product at time t.
• This might be the price of a futures contract
  on wheat for delivery in 3 months.
• F0,tk The current futures price for
  delivery of the product at time t k.
• This might be the price of a futures contract
  for wheat for delivery in 6 months.
• Spread relationships are important to speculators.

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Spreads

• Suppose that the price of a futures contract on
  wheat for delivery in 3 months is 3.25 per
  bushel.
• Suppose further that the price of a futures
  contract on wheat for delivery in 6 months is
  3.30/bushel.
• What is the spread?

• Insert Figure 3.7 Here

12
Repo Rate

• Repo Rate
• The repo rate is the finance charges faced by
  traders. The repo rate is the interest rate on
  repurchase agreements.
• A Repurchase Agreement
• An agreement where a person sells securities at
  one point in time with the understanding that
  he/she will repurchase the security at a certain
  price at a later time.
• Example Pawn Shop.

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Arbitrage

• An Arbitrageur attempts to exploit any
  discrepancies in price between the futures and
  cash markets.
• An academic arbitrage is a risk-free transaction
  consisting of purchasing an asset at one price
  and simultaneously selling it that same asset at
  a higher price, generating a profit on the
  difference.
• Example riskless arbitrage scenario for IBM
  stock trading on the NYSE and Pacific Stock
  Exchange.
• Assumptions
• Perfect futures market
• No taxes
• No transactions costs
• Commodity can be sold short
• Price Exchange
• Arbitrageur Buys IBM (105) Pacific Stock
  E.Arbitrageur Sells IBM 110 NYSERiskless
  Profit 5

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Models of Futures Prices

• Cost-of-Carry Model
• The common way to value a futures contract is by
  using the Cost-of-Carry Model. The Cost-of-Carry
  Model says that the futures price should depend
  upon two things
• The current spot price.
• The cost of carrying or storing the underlying
  good from now until the futures contract matures.
• Assumptions
• There are no transaction costs or margin
  requirements.
• There are no restrictions on short selling.
• Investors can borrow and lend at the same rate of
  interest.
• In the next section, we will explore two
  arbitrage strategies that are associated with the
  Cost-and-Carry Model
• Cash-and-carry arbitrage
• Reserve cash-and-carry arbitrage

15
Cash-and-Carry Arbitrage

• A cash-and-carry arbitrage occurs when a trader
  borrows money, buys the goods today for cash and
  carries the goods to the expiration of the
  futures contract. Then, delivers the commodity
  against a futures contract and pays off the loan.
  Any profit from this strategy would be an
  arbitrage profit.

16
Reverse Cash-and-Carry Arbitrage

• A reverse cash-and-carry arbitrage occurs when a
  trader sells short a physical asset. The trader
  purchases a futures contract, which will be used
  to honor the short sale commitment. Then the
  trader lends the proceeds at an established rate
  of interest. In the future, the trader accepts
  delivery against the futures contract and uses
  the commodity received to cover the short
  position. Any profit from this strategy would be
  an arbitrage profit.

Table 3.5 summarizes the cash-and-carry and the
reverse cash-and-carry strategies.
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Arbitrage Strategies
18
Cost-of-Carry Model
The Cost-of-Carry Model can be expressed as

• Where
• S0 the current spot price
• F0,t the current futures price for delivery
  of the product at time t.
• C0,t the percentage cost required to store (or
  carry) the commodity from today until time t.
• The cost of carrying or storing includes
• Storage costs
• Insurance costs
• Transportation costs
• Financing costs
• In the following section, we will examine the
  cost-of-carryrules.

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Cost-of-Carry Rule 1

• The futures price must be less than or equal to
  the spot price of the commodity plus the carrying
  charges necessary to carry the spot commodity
  forward to delivery.

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Cost-of-Carry Rule 1
21
The Cost-of-Carry Rule 2

• The futures price must be equal to or greater
  than the spot price of the commodity plus the
  carrying charges necessary to carry the spot
  commodity forward to delivery.

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The Cost-of-Carry Rule 2
23
The Cost-of-Carry Rule 3

• Since the futures price must be either less than
  or equal to the spot price plus the cost of
  carrying the commodity forward by rule 1.
• And the futures price must be greater than or
  equal to the spot price plus the cost of carrying
  the commodity forward by rule 2.
• The only way that these two rules can reconciled
  so there is no arbitrage opportunity is by the
  cost of carry rule 3.
• Rule 3 the futures price must be equal to the
  spot price plus the cost of carrying the
  commodity forward to the delivery date of the
  futures contract.

If prices were not to conform to cost of carry
rule 3, a cash-and carry arbitrage profit could
be earned. Recall that we have assumed away
transaction costs, margin requirements, and
restrictions against short selling.
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Spreads and The Cost-of-Carry

• As we have just seen, there must be a
  relationship between the futures price and the
  spot price on the same commodity.
• Similarly, there must be a relationship between
  the futures prices on the same commodity with
  differing times to maturity.
• The following rules address these relationships
• Cost-of-Carry Rule 4
• Cost-of-Carry Rule 5
• Cost-of-Carry Rule 6

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The Cost-of-Carry Rule 4

• The distant futures price must be less than or
  equal to the nearby futures price plus the cost
  of carrying the commodity from the nearby
  delivery date to the distant delivery date.

where d gt n
F0,d the futures price at t0 for the distant
delivery contract maturing at
td. Fo,n the futures price at t0 for the
nearby delivery contract maturing at
tn. Cn,d the percentage cost of carrying the
good from tn to td. If prices were
not to conform to cost of carry rule 4, a
cash-and-carry arbitrage profit could be earned.
26
Spreads and the Cost-of-Carry

• Table 3.6 shows that the spread between two
  futures contracts can not exceed the cost of
  carrying the good from one delivery date forward
  to the next, as required by the cost-of-carry
  rule 4.

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The Cost-of-Carry Rule 4
28
The Cost-of-Carry Rule 5

• The nearby futures price plus the cost of
  carrying the commodity from the nearby delivery
  date to the distant delivery date cannot exceed
  the distant futures price.
• Or alternatively, the distant futures price must
  be greater than or equal to the nearby futures
  price plus the cost of carrying the commodity
  from the nearby futures date to the distant
  futures date.

If prices were not to conform to cost of carry
rule 5, a reverse cash-and-carry arbitrage
profit could be earned.
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The Cost-of-Carry Rule 5

• Table 3.7 illustrates what happens if the nearby
  futures price is too high relative to the distant
  futures price. When this is the case, a forward
  reverse cash-and-carry arbitrage is possible.

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The Cost-of-Carry Rule 5
31
Cost-of-Carry Rule 6

• Since the distant futures price must be either
  less than or equal to the nearby futures price
  plus the cost of carrying the commodity from the
  nearby delivery date to the distant delivery date
  by rule 4.
• And the nearby futures price plus the cost of
  carrying the commodity from the nearby delivery
  date to the distant delivery date can not exceed
  the distant futures price by rule 5.
• The only way that rules 4 and 5 can be reconciled
  so there is no arbitrage opportunity is by cost
  of carry rule 6.

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Cost-of-Carry Rule 6

• The distant futures price must equal the nearby
  futures price plus the cost of carrying the
  commodity from the nearby to the distant delivery
  date.

If prices were not to conform to cost of carry
rule 6, a cash-and-carry arbitrage profit or
reverse cash-and-carry arbitrage profit could be
earned. Recall that we have assumed away
transaction costs, margin requirements, and
restrictions against short selling.
33
Implied Repo Rates

• If we solve for C0,t in the above equation, and
  assume that financing costs are the only costs
  associated with holding an asset, the implied
  cost of carrying the asset from one time point to
  another can be estimated. This rate is called
  the implied repo rate.
• The Cost-of-Carry model gives us

Solving for
And
34
Implied Repo Rates

• Example cash price is 3.45 and the futures
  price is 3.75. The implied repo rate is?

That is, the cost of carrying the asset from
today until the expiration of the futures
contract is 8.6956.
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The Cost-of-Carry Model in Imperfect Markets

• In real markets, no less than four factors
  complicate the Cost-of-Carry Model
• Direct transactions costs
• Unequal borrowing and lending rates
• Margin and restrictions on short selling
• Limitations to storage

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Transaction Costs

• Transaction Costs
• Traders generally are faced with transaction
  costs when they trade. In this case, the profit
  on arbitrage transactions might be reduced or
  disappear altogether.
• Types of Transaction Costs
• Brokerage fees to have their orders executed
• A bid ask spread
• A market maker on the floor of the exchange needs
  to make a profit. He/She does so by paying one
  price (the bid price) for a product and selling
  it for a higher price (the ask price).

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Cost-of-Carry Rule 1 with Transaction Costs

• Recall that the futures price must be less than
  or equal to the spot price of the commodity plus
  the carrying charges necessary to carry the spot
  commodity forward to delivery.

We can modify this rule to account for
transaction costs

Where T is the percentage transaction cost.
38
Cost-of-Carry Rule 1 with Transaction Costs

• To show how transaction costs can frustrate an
  arbitrage consider Table 3.8.

39
Cost-of-Carry Rule 2with Transaction Costs

• Recall from Cost-of-Carry Rule 2 that the futures
  price must be equal to or greater than the spot
  price of the commodity plus the carrying charges
  necessary to carry the spot commodity forward to
  delivery.

We can modify this rule to allow for transaction
costs as follows
40
Cost-of-Carry Rule 2with Transaction Costs

• To show how transaction costs can frustrate an
  attempt to reserve cash-and-carry arbitrage.
  Consider Table 3.9.

41
No-Arbitrage Bounds

• Incorporating transaction costs and combining
  cost-of-carry rules 1 and 2, we have the
  following.

This equation defines the No Arbitrage Bounds.
That is, as long as the futures price trades
within this range, no cash-and-carry or reverse
cash-and-carry arbitrage transactions will be
profitable. Table 3.10 illustrates this
equation.
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No-Arbitrage Bounds

• In this case, as long as the futures price is
  between 426.80 and 453.20, arbitrage
  transactions will not be profitable.

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No-Arbitrage Bounds
44
Differential Transaction Costs

• Situations occur where all traders do not have
  equal transaction costs.
• For example, a floor trader, trading on his own
  behalf would have a lower transaction cost than
  others. So while he/she might be able to earn an
  arbitrage profit, others could not.
• Such a transaction is called a quasi-arbitrage.

45
Unequal Borrowing Lending Rates

• Thus far we have assumed that investors can
  borrow and lend at the same rate of interest.
  Anyone going to a bank knows that this
  possibility generally does not exist.
• Incorporating differential borrowing and lending
  rates into the Cost-of-Carry Model gives us

Where CL lending rate CB borrowing rate
46
Unequal Borrowing Lending Rates
47
Restrictions on Short Selling

• Thus far we have assumed that arbitrageurs can
  sell short commodities and have unlimited use of
  the proceeds.
• There are two limitations to this in the real
  world
• It is difficult to sell some commodities short.
• Investors are generally not allowed to use all
  proceeds from the short sale.
• How do limitations on the use of funds from a
  short sale affect the Cost-of-Carry Model?
• We can examine this by editing our transaction
  cost and differential borrowing Cost-of-Carry
  Model as follows

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Restrictions on Short Selling

• The transaction cost and differential cost of
  borrowing model is as follows

We modify this by recognizing that you will not
get all of the proceeds from the short sale. You
will get some portion of the proceeds.
Where ƒ the proportion of funds received
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Restrictions on Short Selling

• Table 3.12 illustrates the effect of limitations
  on the use of short sale proceeds.

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Restrictions on Short Selling

• The effect of the proceed use limitation is to
  widen the no-arbitrage trading bands.

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Limitations on Storage

• The ability to undertake certain arbitrage
  transactions requires storing the product. Some
  items are easier to store than others.
• Gold is very easy to store. You simply rent a
  safe deposit box at the bank and place your gold
  there for safekeeping.
• Wheat is moderately easy to store.
• How about milk or eggs?
• They can be stored, but not for long periods of
  time.
• To the extent that a commodity can not be stored,
  or has limited storage life, the Cost-of-Carry
  Model may not hold.

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How Traders Deal with Market Imperfections

• The costs associated with carrying commodities
  forward vary widely among traders.
• If you are a floor trader, your transaction costs
  will be very low. If you are a farmer with
  unused grain storage on your farm, your cost of
  storage will be very low.
• Individuals with lowest trading costs (storage
  costs, and cost of borrowing) will have the most
  profitable arbitrage opportunities.
• The ability to sell short varies between traders.

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The Concept of Full Carry Market
To the extent that markets adhere to the
following equations markets are said to be at
full carry

• If the futures price is higher than that
  specified by above equations, the market is said
  to be above full carry.
• If the futures price is below that specified by
  the above equations, the market is said to be
  below full carry.
• To determine if a market is at full carry,
  consider the following example
• Suppose that
• September Gold 410.20December
  Gold 417.90Bankers Acceptance Rate 7.8

54
The Concept of Full Carry Market

• Step 1 compute the annualized percentage
  difference between two futures contracts.

Where AD Annualized percentage difference M
Number of months between the maturity of the
futures contracts.
Step 2 compare the annualized difference to the
interest rate in the market. The
gold market is almost always at full carry.
Other markets can diverge substantially from full
carry.
55
The Concept of Full Carry Market

• Insert Figure 3.1 here
• Futures Price Quotations

56
Market Features That Promote Full Carry

• Ease of Short Selling
• To the extent that it is easy to short sell a
  commodity, the market will become closer to full
  carry.
• Difficulties in short selling will move a market
  away from full carry.
• Selling short of physical goods like wheat is
  more difficult, while selling short of financial
  assets like Eurodollars is much easier. For this
  reason, markets for financial assets tend to be
  closer to full carry than markets for physical
  assets.
• Large Supply
• If the supply of an asset is large relative to
  its consumption, the market will tend to be
  closer to full carry. If the supply of an asset
  is low relative to its consumption, the market
  will tend to be further away from full carry.

57
Market Features that Promote Full Carry

• Non-Seasonal Production
• To the extent that production of a crop is
  seasonal, temporary imbalances between supply and
  demand can occur. In this case, prices can vary
  widely.
• Example in North America, wheat harvest occurs
  between May and September.
• Non-Seasonal Consumption
• To the extent that consumption of commodity is
  seasonal, temporary imbalances between supply and
  demand can occur.
• Example propane gas during winter
  Turkeys during thanksgiving
• High Storability
• A market moves closer to full carry if its
  underline commodity can be stored easily.
• The Cost-of-Carry Model is not likely to apply to
  commodities that have poor storage
  characteristics.
• Example eggs

58
Convenience Yield

• When there is a return for holding a physical
  asset, we say there is a convenience yield. A
  convenience yield can cause futures prices to be
  below full carry. In extreme cases, the cash
  price can exceed the futures price. When the
  cash price exceeds the futures price, the market
  is said to be in backwardation.

59
Futures Prices and Expectations

• If futures contracts are priced appropriately,
  the current futures price should tell us
  something about what the spot price will be at
  some point in the future.
• There are four theories about futures prices and
  future spot prices
• Expectations or Risk Neutral Theory
• Normal Backwardation
• Contango
• Capital Asset Pricing Model (CAPM)
• Speculators play an important role in the futures
  market, they ensure that futures prices
  approximately equal the expected future spot
  price.

60
Expectations or Risk Neutral Theory

• The Expectations Theory says that the futures
  price equals the expected future spot price.

Where
the expected future spot price
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Normal Backwardation

• The Normal Backwardation Theory says that futures
  markets are primarily driven by hedgers who hold
  short positions. For example, farmers who have
  sold futures contracts to reduce their price
  risk.
• The hedgers must pay speculators a premium in
  order to assume the price risk that the farmer
  wishes to get rid of.
• So speculators take long positions to assume this
  price risk. They are rewarded for assuming this
  price risk when the futures price increases to
  match the spot price at maturity.
• So this theory implies that the futures price is
  less than the expected future spot price.

62
Normal Backwardation

• Figure 3.9 depicts a situation that might prevail
  in the futures market for a commodity.

• Insert figure 3.9 here

63
Contango

• The Contango Theory says that futures markets are
  primarily driven by hedgers who hold long
  positions. For example, grain millers who have
  purchased futures contracts to reduce their price
  risk.
• The hedgers must pay speculators a premium in
  order to assume the price risk that the grain
  miller wishes to get rid of.
• So speculators take short positions to assume
  this price risk. They are rewarded for assuming
  this price risk when the futures price declines
  to match the spot price at maturity.
• So this theory implies that the futures price is
  greater than the expected future spot price.

64
Contango

• Figure 3.10 illustrates the price patterns for
  futures under different scenarios.

• Insert Figure 3.10 here

65
Capital Asset Pricing Model (CAPM)

• The CAPM Theory is consistent with the Normal
  Backwardation Theory, the Contango Theory, and
  the Expectations Theories.
• However, the CAPM Theory suggests that
  speculators will be rewarded only for the
  systematic portion of the risk that they are
  assuming.

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Statistical Characteristics of Futures Prices

• Futures prices exhibit statistically significant
  first-order autocorrelation. However, not strong
  enough to allow profitable trading strategies.
• Autocorrelation
• A time series is correlated when one observation
  in the series is statistically related to
  another.
• first-order autocorrelation occurs when one
  observation is related to the immediately
  preceding observations.
• The Volatility of Futures Prices
• Evidence suggests that futures trading does not
  increase the volatility of the cash market.
• Time to Expiration and Futures Price Volatility
• Price changes are large when more information is
  known about the future spot price.