1Charles Cao
4,Hedging Using Futures
Summary
Why Hedge
How to Hedge
What is the Optimal Hedge
2Charles Cao
Why Hedge?
Objective,Neutralize the risk
An example:
A firm is due to sell an asset at a particular
time in the future,The firm can hedge by
shorting a futures contract
3Charles Cao
Why Hedge?
If the price of the asset goes down,the
firm will lose money when it sells the
asset,but gain on the short futures
position
If the price of the asset goes up,the
firm will gain from the sale of the asset,
but will take a loss on the futures
position
4Charles Cao
Short Hedges
Involves a short position in the futures
contract
It is appropriate when the hedger owns
the asset and expects to sell it in the
future
It is also appropriate when the hedger
does not own the asset now,but will
own it in the future
5Charles Cao
Short Hedges,Example
Today,May 15
Exxon signed a contract to sell 1 million
barrels of oil,The selling price is the
spot price on August 15
Spot Price,$19/barrel
August oil futures price,$18.75/barrel
6Charles Cao
Short Hedges,Example
Hedging:
May 15,short 1000 August oil futures
(each contract is written on 1000 barrels)
August 15,close out the position
Outcome:
Exxon effectively locks in a selling price of
$18.75/barrel
7Charles Cao
Short Hedges,Example
If the oil price proves to be $17.50 on
August 15 (spot price)
Exxon receives $17.50 under the sale contract
Exxon receives $1.25 from the futures contract
If the oil price proves to be $19.50 on
August 15 (spot price)
Exxon receives $19.50 under the sale contract
Exxon loses $0.75 (19.50-18.75) from the
futures contract
8Charles Cao
Long Hedges
Involves a long position in the futures
contract
It is appropriate when the hedger has
to purchase assets in the future and
wants to lock in a price now
9Charles Cao
Hedging Using Futures Contracts
It is not perfect:
There may be a difference between the
hedged asset and the asset underlying the
futures contracts
You do not know the date when the asset will
be purchased or sold
Mismatch between the expiration date and the
date when the asset will be bought or sold
10Charles Cao
Hedging Using Futures Contracts
All these issues relate to the basis risk
Basis risk = spot price of asset to be
hedged - futures price of contract used
If the hedged asset is the same as the asset
underlying the futures contract,the basis risk is
0 on the expiration day
Before the expiration day,the basis risk could
be positive or negative (see the picture)
11Charles Cao
Hedging Using Futures Contracts
Notations:
S1,spot price at t1
S2,spot price at t2
F1,futures price at t1
F2,futures price at t2
b1,basis risk at t1
b2,basis risk at t2
t1,the first day of the hedge (open the position)
t2,the last day of the hedge (close out the position)
12Charles Cao
Hedging Using Futures Contracts
An example:
S1 = $2.5 F1 = $2.2
S2 = $2.0 F2 = $1.9
b1 = S1 - F1 = $0.3
b2 = S2 - F2 = $0.1
13Charles Cao
Hedging Using Futures Contracts
If a hedger knows that he will sell assets at
t2,and take a short position in t1,he will
receive S2 at t2
gain F1 - F2 on the futures position
thus,he will receive
S2 + F1 - F2 = F1 + b2
where b2 = S2 - F2
14Charles Cao
Hedging Using Futures Contracts
F1 is known at t1,but
b2 is unknown since S2 is unknown at t1
The hedging risk is the uncertainty
associated with the basis risk,b2
15Charles Cao
Hedging Using Futures Contracts
Sometimes the asset whose price is to be
hedged is different from the asset
underlying the futures contract
Example,heating oil vs,jet fuel
S1,Spot price of the asset being hedged at t1
S2,Spot price of the asset being hedged at t2
S2*,Spot price of the asset underlying the
futures contract at t2
16Charles Cao
Hedging Using Futures Contracts
Since the price that will be paid (or received)
for the asset is
where S2* - F2 is the basis that would exist if
the asset being hedged is the same as the
asset underlying the futures contract
)S - (S + )F - (S + F
S - S + F - F + S
22221
22212
212
FFS
17Charles Cao
What Factors Affect
The Basis Risk?
Choice of Contracts
The choice of the asset underlying the
futures contract
The correlation between prices of the asset
under the hedge and the asset underlying
the futures contract
The choice of the delivery month
18Charles Cao
What Factors Affect
The Basis Risk? Example
A U.S,firm expects to receive 50 million
Japanese Yen (JY) on July 31
JY futures,March,June,Sept.,and Dec
One contract is for the delivery of 12.5
million JY
Sept,contract is chosen
On March 1,the September futures are
traded at $0.78/JY
19Charles Cao
What Factors Affect
The Basis Risk? Example
Positions:
Short 4 Sept,Yen futures on March 1,and
close out the position on July 31
20Charles Cao
What Factors Affect
The Basis Risk? Example
F1 = 0.78,F2 = 0.725 (on July 31),S2 = 0.72
Basis = S2 - F2 = 0.72 - 0.725 = -$0.005
Gain on futures = F1 - F2 = 0.78 - 0.725 = $0.055
The effective price received by the hedger is:
the end-of-July spot price + gains on futures
0.72 + 0.055 = $0.775
21Charles Cao
Minimum Variance Hedge Ratio
Hedge ratio,the ratio of the size of the
position taken in futures contracts to
the size of exposure
The objective is to minimize the risk
The optimal hedge ratio is not 1
Criterion of the optimal hedge,
minimize the variance of the hedger’s
position
22Charles Cao
Minimum Variance Hedge Ratio
Notations:
S,changes in spot price S during the
life of the hedge
F,changes in futures price F during the
life of the hedge
S,standard deviation of?S
F,standard deviation of?F
23Charles Cao
Minimum Variance Hedge Ratio
Notations,(cont.)
,correlation between?S and?F
h,hedge ratio
h*,hedge ration that minimizes the
variance of the hedger’s position
F
Sh
24Charles Cao
Minimum Variance Hedge Ratio
If the hedger is short the futures and long
the asset,the change in the value of the
position is
The variance of the changes in value of the
hedged position is
FhS
FSFS hhva r 2222
25Charles Cao
Minimum Variance Hedge Ratio
To minimize the variance of hedger’s
position,choose h such that
022 2 FSFhhv a r
F
Sh
Thus,
26Charles Cao
Minimum Variance Hedge
Ratio,Example
A company will buy 1 million gallons of
jet fuel in 3 months
S = 0.032
F = 0.04
= 0.8
27Charles Cao
Minimum Variance Hedge
Ratio,Example
The optimal hedge ratio is
Since each futures contract is written on
42,000 gallons,the company needs to buy
64.0
0 4 0.0
0 3 2.08.0
F
Sh
c o n t r a c t s 15~2.150 0 0,42 0 0 00 0 0164.0,,
28Charles Cao
Rolling The Hedge Forward
The expiration date of the hedge is later
than the delivery dates of all the futures
contracts available
The hedger should roll the hedge forward
Close out one contract
Take the same position in the same
contract with a later delivery date
You can roll the hedge over many times
29Charles Cao
Example
The price of oil is $19 per barrel,A
company will sell 100,000 barrels in
June 2003,Although contracts are
available for every delivery month up to
one year,only the first six delivery
months provide sufficient liquidity,The
contract size is 1,000 barrels
30Charles Cao
Example
Strategy,Use the 6-month contract and roll
over the contract
April,2002,short 100 October 2002 contracts
September,2002,close out the 100 October
contracts,Then short 100 March 2003 contracts
February,2003,close out the 100 March
contracts
June,2003,close out the 100 July contracts,
Sell 100,000 barrels of oil
31Charles Cao
Example
The Outcome:
October 2002 futures contract,shorted in
April 2002 at $18.20 and closed out in
September 2002 at $17.40
March 2003 futures contract,shorted in
September 2002 at $17.00 and closed out
in February 2003 at $16.50
July 2003 futures contract,shorted in
February 2003 at $16.30 and closed out in
June 2003 at $15.90
32Charles Cao
Example
Spot oil price in June 2003,$16 per barrel
The gain from the futures contracts is:
(18.20-17.40)+(17.00-16.50)+(16.30-
15.90)=1.70
The decline in oil price from April 2002 to
June 2003 is $3,Thus,the gain from the
futures contracts partly offsets the $3 decline
in oil prices
4,Hedging Using Futures
Summary
Why Hedge
How to Hedge
What is the Optimal Hedge
2Charles Cao
Why Hedge?
Objective,Neutralize the risk
An example:
A firm is due to sell an asset at a particular
time in the future,The firm can hedge by
shorting a futures contract
3Charles Cao
Why Hedge?
If the price of the asset goes down,the
firm will lose money when it sells the
asset,but gain on the short futures
position
If the price of the asset goes up,the
firm will gain from the sale of the asset,
but will take a loss on the futures
position
4Charles Cao
Short Hedges
Involves a short position in the futures
contract
It is appropriate when the hedger owns
the asset and expects to sell it in the
future
It is also appropriate when the hedger
does not own the asset now,but will
own it in the future
5Charles Cao
Short Hedges,Example
Today,May 15
Exxon signed a contract to sell 1 million
barrels of oil,The selling price is the
spot price on August 15
Spot Price,$19/barrel
August oil futures price,$18.75/barrel
6Charles Cao
Short Hedges,Example
Hedging:
May 15,short 1000 August oil futures
(each contract is written on 1000 barrels)
August 15,close out the position
Outcome:
Exxon effectively locks in a selling price of
$18.75/barrel
7Charles Cao
Short Hedges,Example
If the oil price proves to be $17.50 on
August 15 (spot price)
Exxon receives $17.50 under the sale contract
Exxon receives $1.25 from the futures contract
If the oil price proves to be $19.50 on
August 15 (spot price)
Exxon receives $19.50 under the sale contract
Exxon loses $0.75 (19.50-18.75) from the
futures contract
8Charles Cao
Long Hedges
Involves a long position in the futures
contract
It is appropriate when the hedger has
to purchase assets in the future and
wants to lock in a price now
9Charles Cao
Hedging Using Futures Contracts
It is not perfect:
There may be a difference between the
hedged asset and the asset underlying the
futures contracts
You do not know the date when the asset will
be purchased or sold
Mismatch between the expiration date and the
date when the asset will be bought or sold
10Charles Cao
Hedging Using Futures Contracts
All these issues relate to the basis risk
Basis risk = spot price of asset to be
hedged - futures price of contract used
If the hedged asset is the same as the asset
underlying the futures contract,the basis risk is
0 on the expiration day
Before the expiration day,the basis risk could
be positive or negative (see the picture)
11Charles Cao
Hedging Using Futures Contracts
Notations:
S1,spot price at t1
S2,spot price at t2
F1,futures price at t1
F2,futures price at t2
b1,basis risk at t1
b2,basis risk at t2
t1,the first day of the hedge (open the position)
t2,the last day of the hedge (close out the position)
12Charles Cao
Hedging Using Futures Contracts
An example:
S1 = $2.5 F1 = $2.2
S2 = $2.0 F2 = $1.9
b1 = S1 - F1 = $0.3
b2 = S2 - F2 = $0.1
13Charles Cao
Hedging Using Futures Contracts
If a hedger knows that he will sell assets at
t2,and take a short position in t1,he will
receive S2 at t2
gain F1 - F2 on the futures position
thus,he will receive
S2 + F1 - F2 = F1 + b2
where b2 = S2 - F2
14Charles Cao
Hedging Using Futures Contracts
F1 is known at t1,but
b2 is unknown since S2 is unknown at t1
The hedging risk is the uncertainty
associated with the basis risk,b2
15Charles Cao
Hedging Using Futures Contracts
Sometimes the asset whose price is to be
hedged is different from the asset
underlying the futures contract
Example,heating oil vs,jet fuel
S1,Spot price of the asset being hedged at t1
S2,Spot price of the asset being hedged at t2
S2*,Spot price of the asset underlying the
futures contract at t2
16Charles Cao
Hedging Using Futures Contracts
Since the price that will be paid (or received)
for the asset is
where S2* - F2 is the basis that would exist if
the asset being hedged is the same as the
asset underlying the futures contract
)S - (S + )F - (S + F
S - S + F - F + S
22221
22212
212
FFS
17Charles Cao
What Factors Affect
The Basis Risk?
Choice of Contracts
The choice of the asset underlying the
futures contract
The correlation between prices of the asset
under the hedge and the asset underlying
the futures contract
The choice of the delivery month
18Charles Cao
What Factors Affect
The Basis Risk? Example
A U.S,firm expects to receive 50 million
Japanese Yen (JY) on July 31
JY futures,March,June,Sept.,and Dec
One contract is for the delivery of 12.5
million JY
Sept,contract is chosen
On March 1,the September futures are
traded at $0.78/JY
19Charles Cao
What Factors Affect
The Basis Risk? Example
Positions:
Short 4 Sept,Yen futures on March 1,and
close out the position on July 31
20Charles Cao
What Factors Affect
The Basis Risk? Example
F1 = 0.78,F2 = 0.725 (on July 31),S2 = 0.72
Basis = S2 - F2 = 0.72 - 0.725 = -$0.005
Gain on futures = F1 - F2 = 0.78 - 0.725 = $0.055
The effective price received by the hedger is:
the end-of-July spot price + gains on futures
0.72 + 0.055 = $0.775
21Charles Cao
Minimum Variance Hedge Ratio
Hedge ratio,the ratio of the size of the
position taken in futures contracts to
the size of exposure
The objective is to minimize the risk
The optimal hedge ratio is not 1
Criterion of the optimal hedge,
minimize the variance of the hedger’s
position
22Charles Cao
Minimum Variance Hedge Ratio
Notations:
S,changes in spot price S during the
life of the hedge
F,changes in futures price F during the
life of the hedge
S,standard deviation of?S
F,standard deviation of?F
23Charles Cao
Minimum Variance Hedge Ratio
Notations,(cont.)
,correlation between?S and?F
h,hedge ratio
h*,hedge ration that minimizes the
variance of the hedger’s position
F
Sh
24Charles Cao
Minimum Variance Hedge Ratio
If the hedger is short the futures and long
the asset,the change in the value of the
position is
The variance of the changes in value of the
hedged position is
FhS
FSFS hhva r 2222
25Charles Cao
Minimum Variance Hedge Ratio
To minimize the variance of hedger’s
position,choose h such that
022 2 FSFhhv a r
F
Sh
Thus,
26Charles Cao
Minimum Variance Hedge
Ratio,Example
A company will buy 1 million gallons of
jet fuel in 3 months
S = 0.032
F = 0.04
= 0.8
27Charles Cao
Minimum Variance Hedge
Ratio,Example
The optimal hedge ratio is
Since each futures contract is written on
42,000 gallons,the company needs to buy
64.0
0 4 0.0
0 3 2.08.0
F
Sh
c o n t r a c t s 15~2.150 0 0,42 0 0 00 0 0164.0,,
28Charles Cao
Rolling The Hedge Forward
The expiration date of the hedge is later
than the delivery dates of all the futures
contracts available
The hedger should roll the hedge forward
Close out one contract
Take the same position in the same
contract with a later delivery date
You can roll the hedge over many times
29Charles Cao
Example
The price of oil is $19 per barrel,A
company will sell 100,000 barrels in
June 2003,Although contracts are
available for every delivery month up to
one year,only the first six delivery
months provide sufficient liquidity,The
contract size is 1,000 barrels
30Charles Cao
Example
Strategy,Use the 6-month contract and roll
over the contract
April,2002,short 100 October 2002 contracts
September,2002,close out the 100 October
contracts,Then short 100 March 2003 contracts
February,2003,close out the 100 March
contracts
June,2003,close out the 100 July contracts,
Sell 100,000 barrels of oil
31Charles Cao
Example
The Outcome:
October 2002 futures contract,shorted in
April 2002 at $18.20 and closed out in
September 2002 at $17.40
March 2003 futures contract,shorted in
September 2002 at $17.00 and closed out
in February 2003 at $16.50
July 2003 futures contract,shorted in
February 2003 at $16.30 and closed out in
June 2003 at $15.90
32Charles Cao
Example
Spot oil price in June 2003,$16 per barrel
The gain from the futures contracts is:
(18.20-17.40)+(17.00-16.50)+(16.30-
15.90)=1.70
The decline in oil price from April 2002 to
June 2003 is $3,Thus,the gain from the
futures contracts partly offsets the $3 decline
in oil prices
