Chapter 3: Futures Prices

Report
CHAPTER 3
Futures Prices
In this chapter, we discuss how futures contracts are
priced. This chapter is organized into the following
sections:
1. Reading Futures Prices
2. The Basis and Spreads
3. Models of Futures Prices
4. Futures Prices and Expectations
5. Future Prices and Risk Aversion
6. Characteristics of Futures Prices
Chapter 3
1
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.
Chapter 3
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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.
Chapter 3
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Reading Futures Prices
Insert figure 3.1 Here
Chapter 3
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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).
Table 3.1
How Trading Affects Open Interest
Time
Action
Open Interest
t=0
Trading opens for the popular widget contract.
0
t=1
Trader A buys and Trader B sells 1 widget contract.
1
t=2
Trader C buys and Trader D sells 3 widget contracts.
4
t=3
Trader A sells and Trader D buys 1 widget contract.
(Trader A has offset 1 contract and is out of the market. Trader D has offset 1 contract and is now short
2 contracts.)
3
t=4
Trader C sells and Trader E buys 1 widget contract.
3
Ending
Positions
Trader
Long Position
B
C
D
E
All Traders
Short Position
1
2
2
1
3
Chapter 3
3
5
Open Interest &Trading Volume Patterns
Insert Figure 3.2 Here
Insert Figure 3.3 Here
Chapter 3
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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.
Basis  S 0  F 0, t
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 Market
Price for more distant futures are higher than for nearby
futures.
Inverted Market
Distant futures prices are lower than the price for contracts
nearer to expiration.
Chapter 3
7
The Basis
Table 3.2
Gold Prices and the Basis
(July 11)
Contract
Prices
CASH
JUL (this year)
AUG
OCT
DEC
FEB (next year)
APR
JUN
AUG
OCT
DEC
353.70
354.10
355.60
359.80
364.20
368.70
373.00
377.50
381.90
386.70
391.50
The Basis
-.40
-1.90
-6.10
-10.50
-15.00
-19.30
-23.80
-28.20
-33.00
-37.80
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?
Basis  S 0  F 0, t
Basis  $353 .7  364 .20  $10.50
Basis  $10.50
Chapter 3
8
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
Chapter 3
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Spreads
Spread
A spread is the difference in price between two futures
contracts on the same commodity for two different
maturity dates:
Spread  F 0, t  k  F 0, t
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,t+k =
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.
Chapter 3
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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?
Spread  F 0, t  k  F 0, t
Spread  $3.30  $3.25  $0.05
Insert Figure 3.7 Here
Chapter 3
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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.
Chapter 3
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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:
1. Perfect futures market
2. No taxes
3. No transactions costs
4. Commodity can be sold short
Arbitrageur Buys IBM
Arbitrageur Sells IBM
Riskless Profit
Price
Exchange
($105)
$110
$ 5
Pacific Stock E.
NYSE
Chapter 3
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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
Chapter 3
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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.
0
1. Borrow money
2. Sell futures contract
3. Buy commodity
1
4. Deliver the commodity
against the futures contract
5. Recover money & payoff
loan
Chapter 3
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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.
0
1. Sell short the commodity
2. Lend money received
from short sale
3. Buy futures contract
1
4. Accept delivery from futures
contract
5. Use commodity received
to cover the short sale
Table 3.5 summarizes the cash-and-carry and the
reverse cash-and-carry strategies.
Chapter 3
16
Arbitrage Strategies
Table 3.5
Transactions for Arbitrage Strategies
Market
Cash-and-Carry
Debt
Borrow funds
Lend short sale proceeds
Physical
Buy asset and store; deliver
against futures
Sell asset short; secure
proceeds from short sale
Futures
Sell futures
Buy futures; accept delivery;
return physical asset to honor
short sale commitment
Reverse Cash-and-Carry
Chapter 3
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Cost-of-Carry Model
The Cost-of-Carry Model can be expressed as:
F 0, t  S 0(1  C0, t )
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:
1. Storage costs
2. Insurance costs
3. Transportation costs
4. Financing costs
In the following section, we will examine the cost-of-carry
rules.
Chapter 3
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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.
F 0, t  S 0(1  C 0, t )
Table 3.3
Cash-and-Carry Gold Arbitrage Transactions
Prices for the Analysis:
Spot price of gold
Future price of gold (for delivery in one year)
Interest rate
Transaction
t=0
$400
$450
10%
Cash Flow
Borrow $400 for one year at 10%.
Buy 1 ounce of gold in the spot market for $400.
Sell a futures contract for $450 for delivery of
one ounce in one year.
+$400
- 400
0
Total Cash Flow
t=1
Remove the gold from storage.
Deliver the ounce of gold against the futures
contract.
Repay loan, including interest.
Chapter 3
$0
$0
+450
-440
Total Cash Flow
+$10
19
Cost-of-Carry Rule 1
0
1
1. Borrow $400
2. Buy 1 oz gold
3. Sell futures contract
4. Deliver gold against
futures contract
5. Repay loan
Chapter 3
20
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.
F 0, t  S 0(1  C 0, t )
Table 3.4
Reverse Cash-and-Carry Gold Arbitrage Transactions
Prices for the Analysis
Spot price of gold
Future price of gold (for delivery in one year)
Interest rate
Transaction
t=0
$420
$450
10%
Cash Flow
Sell 1 ounce of gold short.
Lend the $420 for one year at 10%.
Buy 1 ounce of gold futures for delivery in 1
year.
+$420
- 420
0
Total Cash Flow
t=1
Collect proceeds from the loan ($420 x 1.1).
Accept delivery on the futures contract.
Use gold from futures delivery to repay short
sale.
$0
+$462
-450
0
Total Cash Flow +$12
Chapter 3
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The Cost-of-Carry Rule 2
0
1. Sell short 1 oz. gold
2. Lend $420 at 10%
interest
3. Buy a futures contract
1
4. Collect proceeds
from loan
5. Accept delivery on
futures contract
6. Use gold from futures
contract to repay the
short sale
Chapter 3
22
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.
F 0, t  S 0(1  C 0, t )
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.
Chapter 3
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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
Chapter 3
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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.
F 0, d  F 0, n(1  Cn, d )
where d > n
F0,d = the futures price at t=0 for the distant delivery
contract maturing at t=d.
Fo,n= the futures price at t=0 for the nearby delivery contract
maturing at t=n.
Cn,d= the percentage cost of carrying the good from t=n
to t=d.
If prices were not to conform to cost of carry rule # 4, a
cash-and-carry arbitrage profit could be earned.
Chapter 3
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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.
Table 3.6
Gold Forward Cash-and-Carry Arbitrage
Prices for the Analysis
Futures price for gold expiring in 1 year
Futures price for gold expiring in 2 years
Interest rate (to cover from year 1 to year 2)
Transaction
t=0
$400
$450
10%
Cash Flow
Buy the futures expiring in 1 year.
Sell the futures expiring in 2 years.
Contract to borrow $400 at 10% for year 1 to
year 2.
+$0
0
0
Total Cash Flow
t=1
t=2
Borrow $400 for 1 year at 10% as contracted at
t = 0.
Take delivery on the futures contract.
Begin to store gold for one year.
Deliver gold to honor futures contract.
Repay loan ($400 x 1.1)
$0
+$400
- 400
0
Total Cash Flow
$0
+$450
- 440
Total Cash Flow + $10
Chapter 3
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The Cost-of-Carry Rule 4
0
1
1. Buy futures
contract w/exp
in 1 yrs.
2. Sell futures
contract w/exp
in 2 years
3. Contract to
borrow $400
from yr 1-2
4. Borrow $400
5. Take delivery on 1
yr to exp futures
contract.
6. Place the gold in
storage for one yr.
Chapter 3
2
7. Remove gold
from storage
8. Deliver gold
against 2 yr.
futures contract
9. Pay back loan
27
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.
F0,d  F0,n 1 Cn,d 
If prices were not to conform to cost of carry rule # 5, a
reverse cash-and-carry arbitrage profit could be earned.
Chapter 3
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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.
Table 3.7
Gold Forward Reverse Cash-and-Carry Arbitrage
Prices for the Analysis:
Futures price for gold expiring in 1 year
Futures price for gold expiring in 2 years
Interest rate (to cover from year 1 to year 2)
Transaction
t=0
$440
$450
10%
Cash Flow
Sell the futures expiring in one year.
Buy the futures expiring in two years.
Contract to lend $440 at 10% from year 1 to
year 2.
+$0
0
0
Total Cash Flow
t=1
Borrow 1 ounce of gold for one year.
Deliver gold against the expiring futures.
Invest proceeds from delivery for one year.
$0
+ 440
- 440
Total Cash Flow
t=2
Accept delivery on expiring futures.
Repay 1 ounce of borrowed gold.
Collect on loan of $440 made at t = 1.
$0
$0
- $450
0
+ 484
Total Cash Flow + $34
Chapter 3
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The Cost-of-Carry Rule 5
0
1
1. Sell futures
contract w/exp
in 1 yrs.
2. Buy futures
contract w/exp
in 2 years
3. Contract to
lend $400
from yr 1-2
4. Borrow 1 oz. gold
5. Deliver gold on 1
yr to exp futures
contract.
6. Invest proceeds
from delivery for
one yr.
Chapter 3
2
7. Accept delivery
on exp 2 yr
futures contract
8. Repay 1 oz.
borrowed gold.
9. Collect $400
loan
30
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.
Chapter 3
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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.
F 0, d  F 0, n(1  Cn, d )
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.
Chapter 3
32
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:
F 0, t  S 0(1  C 0, t )
Solving for
C 0, t
F 0, t
 (1  C 0, t )
S0
And
F 0, t
 1  C 0, t
S0
Chapter 3
33
Implied Repo Rates
Example: cash price is $3.45 and the futures price is
$3.75. The implied repo rate is?
F 0, t
 1  C 0, t
S0
$3.75
 1  0.086956
$3.45
That is, the cost of carrying the asset from today until the
expiration of the futures contract is 8.6956%.
Chapter 3
34
The Cost-of-Carry Model in Imperfect
Markets
In real markets, no less than four factors complicate the
Cost-of-Carry Model:
1. Direct transactions costs
2. Unequal borrowing and lending rates
3. Margin and restrictions on short selling
4. Limitations to storage
Chapter 3
35
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).
Chapter 3
36
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.
F 0, t  S 0(1  C 0, t )
We can modify this rule to account for transaction
costs:
F 0, t  S 0(1  T )(1  C 0, t )
Where T is the percentage transaction cost.
Chapter 3
37
Cost-of-Carry Rule 1
with Transaction Costs
To show how transaction costs can frustrate an arbitrage
consider Table 3.8.
Table 3.8
Attempted Cash-and-Carry Gold Arbitrage Transactions
Prices for the Analysis:
Spot price of gold
Future price of gold (for delivery in one year)
Interest rate
Transaction cost (T )
Transaction
t=0
t=1
$400
$450
10%
3%
Cash Flow
Borrow $412 for one year at 10%.
Buy 1 ounce of gold in the spot market for
$400 and pay 3% transaction costs, to total
$412.
Sell a futures contract for $450 for delivery of
one ounce in one year.
Remove the gold from storage.
Deliver the ounce of gold to close futures
contract.
Repay loan, including interest.
Chapter 3
+$412
- 412
0
Total Cash Flow
$0
$0
+450.00
-453.20
Total Cash Flow -$3.20
38
Cost-of-Carry Rule 2
with 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.
F 0, t  S 0(1  C 0, t )
We can modify this rule to allow for transaction costs as
follows:
F 0, t  S 0(1  T )(1  C 0, t )
Chapter 3
39
Cost-of-Carry Rule 2
with Transaction Costs
To show how transaction costs can frustrate an attempt to
reserve cash-and-carry arbitrage. Consider Table 3.9.
Table 3.9
Attempted Reverse Cash-and-Carry Gold Arbitrage
Prices for the Analysis:
Spot price of gold
Future price of gold (for delivery in one year)
Interest rate
Transaction costs (T )
Transaction
t=0
t=1
$420
$450
10%
3%
Cash Flow
Sell 1 ounce of gold short, paying 3%
transaction costs. Receive $420(.97) =
$407.40.
Lend the $407.40 for one year at 10%.
Buy 1 ounce of gold futures for delivery in 1
year.
Collect loan proceeds ($407.40 x 1.1).
Accept gold delivery on the futures contract.
Use gold from futures delivery to repay short
sale.
+$407.40
- 407.40
0
Total Cash Flow
$0
+$448.14
-450.00
0
Total Cash Flow
-$1.86
Chapter 3
40
No-Arbitrage Bounds
Incorporating transaction costs and combining cost-ofcarry rules 1 and 2, we have the following.
S 0(1  T )(1  C 0, t )  F 0, t  S 0(1  T )(1  C 0, t )
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.
Chapter 3
41
No-Arbitrage Bounds
Table 3.10
Illustration of No-Arbitrage Bounds
Prices for the Analysis:
Spot price of gold
$400
Interest rate
Transaction costs (T )
10%
3%
No-Arbitrage Futures Price in Perfect Markets
F0,t = S0(1 + C ) = $400(1.1) = $440
Upper No-Arbitrage Bound with Transaction Costs
F0,t < S0(1 + T )(1 + C ) = $400(1.03)(1.1) = $453.20
Lower No-Arbitrage Bound with Transaction Costs
F0,t > S0(1 B T )(1 + C ) = $400(.97)(1.1) = $426.80
In this case, as long as the futures price is between
$426.80 and $453.20, arbitrage transactions will not be
profitable.
Chapter 3
42
No-Arbitrage Bounds
$453.20
Futures
Price
$426.80
Time
Chapter 3
43
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.
Chapter 3
44
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:
S0 (1  T )(1  CL )  F 0, t  S 0(1  T )(1  CB )
Where:
CL = lending rate
CB = borrowing rate
Chapter 3
45
Unequal Borrowing & Lending Rates
Table 3.11
Illustration of No-Arbitrage Bounds
with Differential Borrowing and Lending Rates
Prices for the Analysis:
Spot price of gold
$400
Interest rate (borrowing)
12%
Interest rate (lending)
8%
Transaction costs (T )
3%
Upper No-Arbitrage Bound with Transaction Costs and a Borrowing Rate
F0,t < S0(1 + T )(1 + CB ) = $400(1.03)(1.12) = $461.44
Lower No-Arbitrage Bound with Transaction Costs and a Lending Rate
F0,t > S0(1 B T )(1 + CL ) = $400(.97)(1.08) = $419.04
Chapter 3
46
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:
Chapter 3
47
Restrictions on Short Selling
The transaction cost and differential cost of borrowing
model is as follows:
S0 (1  T )(1  CL )  F 0, t  S 0(1  T )(1  CB )
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.
S 0(1  T )(1  fCL )  F 0, t  S 0(1  T )(1  CB )
Where:
ƒ = the proportion of funds received
Chapter 3
48
Restrictions on Short Selling
Table 3.12 illustrates the effect of limitations on the use of
short sale proceeds.
Table 3.12
Illustration of No-Arbitrage Bounds
with Various Short Selling Restrictions
Prices for the Analysis:
Spot price of gold
$400
Interest rate (borrowing)
12%
Interest rate (lending)
8%
Transaction costs (T )
3%
Upper No-Arbitrage Bound with Transaction Costs and a Borrowing Rate
F0,t < S0(1 + T )(1 + CB ) = $400(1.03)(1.12) = $461.44
Lower No-Arbitrage Bound with Transaction Costs and a Lending Rate, f = 1.0
F0,t > S0(1 B T )(1 + fCL ) = $400(.97)[1 + (1.0)(.08)] = $419.04
Lower No-Arbitrage Bound with Transaction Costs and a Lending Rate, f = 0.75
F0,t > S0(1 B T)(1 + fCL ) = $400(.97)[1 + (.75)(.08)] = $411.28
Lower No-Arbitrage Bound with Transaction Costs and a Lending Rate, f = 0.5
F0,t > S0(1 B T )(1 + f CL ) = $400(.97)[1 + (0.5)(.08)] = $403.52
Chapter 3
49
Restrictions on Short Selling
The effect of the proceed use limitation is to widen the noarbitrage trading bands.
$461.44
Futures
Price
$403.52
Time
Chapter 3
50
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.
Chapter 3
51
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.
Chapter 3
52
The Concept of Full Carry Market
To the extent that markets adhere to the following equations
markets are said to be at “full carry”:
F 0, t  S 0(1  C 0, t )
F 0, d  F 0, n(1  Cn, d )
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
December Gold
Bankers Acceptance Rate
Chapter 3
$410.20
$417.90
7.8%
53
The Concept of Full Carry Market
Step 1: compute the annualized percentage difference
between two futures contracts.
AD 
( FF )
0, d
M
12  1
0. N
Where
AD = Annualized percentage difference
M = Number of months between the maturity of the
futures contracts.
AD 
(
3
$417.90 12
1
$410.20
)
AD  1.0772  1
AD  0.0772
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.
Chapter 3
54
The Concept of Full Carry Market
Insert Figure 3.1 here
Futures Price Quotations
Chapter 3
55
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.
Chapter 3
56
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
Chapter 3
57
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.”
Chapter 3
58
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.
Chapter 3
59
Expectations or Risk Neutral Theory
The Expectations Theory says that the futures price equals
the expected future spot price.
F 0, t  E(S 0)
Where
E(S 0)
= the expected future spot price
Chapter 3
60
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.
F 0, t  E(S 0)
Chapter 3
61
Normal Backwardation
Figure 3.9 depicts a situation that might prevail in the
futures market for a commodity.
Insert figure 3.9 here
Chapter 3
62
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.
F 0, t  E(S 0)
Chapter 3
63
Contango
Figure 3.10 illustrates the price patterns for futures under
different scenarios.
Insert Figure 3.10 here
Chapter 3
64
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.
Chapter 3
65
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.
Chapter 3
66

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