1
Finance School of Management
Chapter 14,Forward &
Futures Prices
Objective
?How to price forward and futures
?Storage of commodities
?Cost of carry
?Understanding financial
futures
2
Finance School of Management
Chapter 14,Contents
? Distinction Between Forward &
Futures Contracts
? The Economic Function of
Futures Markets
? The Role of Speculators
? Relationship Between
Commodity Spot & Futures
Prices
? Extracting Information from
Commodity Futures Prices
? Spot-Futures Price Parity for
Gold
? Financial Futures
? The,Implied” Risk-Free Rate
? The Forward Price is not a
Forecast of the Spot Price
? Forward-Spot Parity with Cash
Payouts
?,Implied” Dividends
? The Foreign Exchange Parity
Relation
? The Role of Expectations in
Determining Exchange Rates
3
Finance School of Management
? Two parties agree to exchange some item on a specified
future date at a delivery price specified now
? The forward price is defined as the delivery price which
makes the current market value of the contract zero
? No money is paid in the present by either party to the other
? The face value of the contract is the quantity of the item
specified in the contract times the forward price
? The party who agrees to buy the specified item is said to
take a long position,and the party who agrees to sell the
item is said to take a short position
Features of Forward Contracts
4
Finance School of Management
Features of Forward Contracts
?,Customization”,difficulty of,closing out”
positions,low liquidity
? The risk of contract default,credit risk
5
Finance School of Management
Characteristics of Futures
?Futures are,
– standard contracts
– immune from the credit worthiness of buyer
and seller because
? exchange stands between traders
? contracts marked to market daily
? margin requirements (enough collateral)
6
Finance School of Management
Terms
? Open,High,Low,Settle,Change,Lifetime high,
Lifetime low,Open interest
? Mark-to-market
? Margin requirement
? Margin call
7
Finance School of Management
An Illustration
? You place an order
to take a long
position in a
September wheat
futures contract on
August 4,1991
? The broker requires
you to deposit an
initial margin of
$1,500 in your
account
? On August 5,the futures price closes 71/4 cents per
bushel lower
? You have lost 71/4 cents*5,000 bushels=$362.50
that day
? The broker takes that amount out of your account
and transfers it to the future exchange,which
transfers it to one of the parties who was on the
short side of the contract (marking to market)
? If you do not have enough money in your account
to meet the margin requirement (variation /
maintenance margin),you’ll receive a margin call
from the broker asking you to add money
? If you do not respond immediately,then the
broker liquidates your position at the prevailing
market price
8
Finance School of Management
Spot-Futures Price Parity for Gold
? There are two ways to invest in gold
? buy an ounce of gold at S0,store it for a year at a storage
cost of $hS0,and sell it for S1
? invest S0 in a 1-year T-bill with return rf,and purchase a
1-ounce of gold forward,F,for delivery in 1-year
? ? 0
0
1
)(
0
01 1 ShrFr
S
FSrrh
S
SS
ffsy ng o l dg o l d ?????
??????
9
Finance School of Management
– The spot price of gold is $300,the storage costs is 2% per
year,and the risk-free rate is 8%
– The forward price is $340
Arbitrage Opportunity when Forward
Price of Gold Is Too High
A r b i t r a g e I m m e d i a t e C a s h C a s h F l o w 1 Y e a r
P o s i t i o n F l o w F r o m N o w
S e l l a f o r w a r d c o n t r a c t 0 $ 3 4 0 - S 1
B o r r o w $ 3 0 0 $300 ( $ 3 2 4 )
B u y a n o u n c e o f g o l d ( $ 3 0 0 ) S 1
P a y s t o r a g e c o s t s ( $ 6 )
N e t c a s h f l o w s 0 $ 3 4 0 - $ 3 3 0 = $ 1 0
10
Finance School of Management
Arbitrage Opportunity when Forward
Price of Gold Is Too Low A r b i t r a g e I m m e d i a t e C a s h C a s h F l o w 1 Y e a r
P o s i t i o n F l o w F r o m N o w
S e l l s h o r t a n o u n c e o f g o l d $300 0 - S
1
B u y a f o r w a r d c o n t r a c t 0 S
1 - $ 3 2 0
I n v e s t $ 3 0 0 i n 1 - y e a r p u r e ( $ 3 0 0 ) $324
d i s c o u n t b o n d s
R e c e i v e s t o r a g e c o s t s $6
N e t c a s h f l o w s 0 $ 3 3 0 - $ 3 2 0 = $ 1 0
– The forward price is $320
11
Finance School of Management
Spot-Futures Price Parity for Gold
?A contract with life T,
? ? 01 ShrF Tf ???
? This is not a causal relationship,but the forward
and current spot are jointly determined in the
market
? If we know one,then the law of one price
determines that we know the other
12
Finance School of Management
Implied Cost of Carry
? Implied cost of carry = F - S0 = (rf + h) S0
? The implied cost of storing the gold (per $spot) is
h = (F - S0)/S0 - rf
? Suppose F = $330,S0 = $300,and rf = 8%,then
– Implied cost of carry = $330-300=$30
– Implied storage cost = (330-300)/300 – 8% = 2%
13
Finance School of Management
Financial Futures
? Financial futures contracts are usually settled in
cash
? With no storage cost,the relationship between the
forward and the spot is
? ?TfrSF ?? 1
? Any deviation from this will result in an arbitrage
opportunity
14
Finance School of Management
Replication of Non-Dividend-Paying Stock
Using a pure Discount Bond and a Stock
Forward Contract
I m m e di a t e C a s h C a s h F l ow 1 Y e a r
P os i t i on F l ow F r om N ow
B uy a s ha r e of s t oc k - S S
1
R e p l i c a t i n g P o r t f o l i o ( S y n t h e t i c S t o c k )
G o l on g a f or w a r d c on t r a c t 0 S
1 – F
on s t oc k
B uy a pu r e di s c ou nt bo nd - F / ( 1+ r
f )
F
w i t h f a c e va l ue of F
T ot a l r e pl i c a t i ng po r t f ol i o - F / ( 1+ r
f ) S 1
15
Finance School of Management
Arbitrage in Stock Futures
– The spot price of a stock is $100,and the risk-free rate is
8%
– The forward price is $109
A r b i t r a g e I m m e d i a t e C a s h C a s h F l o w 1 Y e a r
P o s i t i o n F l o w F r o m N o w
S e l l a f o r w a r d c o n t r a c t 0 $ 1 0 9 - S 1
B o r r o w $ 1 0 0 $100 ( $ 1 0 8 )
B u y a s h a r e o f s t o c k ( $ 1 0 0 ) S 1
N e t c a s h f l o w s 0 $1
16
Finance School of Management
The,Implied” Risk-Free Rate
? Rearranging the formula,the implied interest rate
on a forward given the spot is
0
0
1
0
1,T if;1
S
SF
r
S
F
r
T ?
???
?
?
?
?
?
?
?
?
?
? This is reminiscent of the formula for the interest
rate on a discount bond
17
Finance School of Management
Replication of a Pure Discount Bond Using a
Stock and a Forward Contract
I m m e d i a t e C a s h C a s h F l o w 1 Y e a r
P o s i t i o n F l o w F r o m N o w
B u y a T - b i l l w i t h f a c e
v a l u e o f F
B u y a s h a r e o f s t o c k - S S
1
G o s h o r t a f o r w a r d c o n t r a c t 0 F – S
1
T o t a l r e p l i c a t i n g p o r t f o l i o - S F
- F / ( 1 + r f ) F
R e p l i c a t i n g P o r t f o l i o ( S y n t h e t i c T - B i l l )
18
Finance School of Management
The Forward Price is not a Forecast of
the Spot Price
? Following the diagrams in Chapter 13 we might
suppose that the expected price of a stock is
FrSp r e m i u mr i s krSS ff ?????? )1()1( 001
? If this is indeed correct,then the forward price is
not an indicator of the expected spot price at the
maturity of the forward
19
Finance School of Management
Forward-Spot Parity with Cash Payouts
? The S - F relationship becomes
? Note,(forward price > the spot price) if (D < rf S)
? Because D is not known with certainty,this is a
quasi-arbitrage situation
DSrSF
r
FD
S f
f
????
?
?
?
1
20
Finance School of Management
Replication of Dividend-Paying Stock Using a
Pure Discount Bond and a Stock Forward
Contract
I m m e d i a t e C a s h C a s h F l o w 1 Y e a r
P o s i t i o n F l o w F r o m N o w
B u y a s h a r e o f s t o c k - S D + S
1
G o l o n g a f o r w a r d c o n t r a c t 0 S
1 – F
o n s t o c k
B u y a p u r e d i s c o u n t b o n d - ( D + F ) / ( 1 + r
f )
D + F
w i t h f a c e v a l u e o f D + F
R e p l i c a t i n g P o r t f o l i o ( S y n t h e t i c S t o c k )
21
Finance School of Management
,Implied” Dividends
?From the last slide,we may obtain the
implied dividend
? ? FSrD f ??? 1
22
Finance School of Management
Exchange Rate Example
15000 ￥
(Borrowed)
15450 ￥
15450 ￥
(Repaid)
￡100
(Invested)
￡109
(Matures)
Time
3% ￥/￥ (direct)
3% ￥/￡/￡/￥
150 ￥/￡
9%￡/￡
Forward ￥/￡
Japan U.K,
23
Finance School of Management
The Foreign Exchange Parity
Relation
? We used the diagram to show that
? ? ? ? tYt$ r1
Y e n f o r S p o tdD e n o m i n a t e $
r1
Y e n on F o r w a r d dD e n o m i n a t e $
?
?
?
Finance School of Management
Chapter 14,Forward &
Futures Prices
Objective
?How to price forward and futures
?Storage of commodities
?Cost of carry
?Understanding financial
futures
2
Finance School of Management
Chapter 14,Contents
? Distinction Between Forward &
Futures Contracts
? The Economic Function of
Futures Markets
? The Role of Speculators
? Relationship Between
Commodity Spot & Futures
Prices
? Extracting Information from
Commodity Futures Prices
? Spot-Futures Price Parity for
Gold
? Financial Futures
? The,Implied” Risk-Free Rate
? The Forward Price is not a
Forecast of the Spot Price
? Forward-Spot Parity with Cash
Payouts
?,Implied” Dividends
? The Foreign Exchange Parity
Relation
? The Role of Expectations in
Determining Exchange Rates
3
Finance School of Management
? Two parties agree to exchange some item on a specified
future date at a delivery price specified now
? The forward price is defined as the delivery price which
makes the current market value of the contract zero
? No money is paid in the present by either party to the other
? The face value of the contract is the quantity of the item
specified in the contract times the forward price
? The party who agrees to buy the specified item is said to
take a long position,and the party who agrees to sell the
item is said to take a short position
Features of Forward Contracts
4
Finance School of Management
Features of Forward Contracts
?,Customization”,difficulty of,closing out”
positions,low liquidity
? The risk of contract default,credit risk
5
Finance School of Management
Characteristics of Futures
?Futures are,
– standard contracts
– immune from the credit worthiness of buyer
and seller because
? exchange stands between traders
? contracts marked to market daily
? margin requirements (enough collateral)
6
Finance School of Management
Terms
? Open,High,Low,Settle,Change,Lifetime high,
Lifetime low,Open interest
? Mark-to-market
? Margin requirement
? Margin call
7
Finance School of Management
An Illustration
? You place an order
to take a long
position in a
September wheat
futures contract on
August 4,1991
? The broker requires
you to deposit an
initial margin of
$1,500 in your
account
? On August 5,the futures price closes 71/4 cents per
bushel lower
? You have lost 71/4 cents*5,000 bushels=$362.50
that day
? The broker takes that amount out of your account
and transfers it to the future exchange,which
transfers it to one of the parties who was on the
short side of the contract (marking to market)
? If you do not have enough money in your account
to meet the margin requirement (variation /
maintenance margin),you’ll receive a margin call
from the broker asking you to add money
? If you do not respond immediately,then the
broker liquidates your position at the prevailing
market price
8
Finance School of Management
Spot-Futures Price Parity for Gold
? There are two ways to invest in gold
? buy an ounce of gold at S0,store it for a year at a storage
cost of $hS0,and sell it for S1
? invest S0 in a 1-year T-bill with return rf,and purchase a
1-ounce of gold forward,F,for delivery in 1-year
? ? 0
0
1
)(
0
01 1 ShrFr
S
FSrrh
S
SS
ffsy ng o l dg o l d ?????
??????
9
Finance School of Management
– The spot price of gold is $300,the storage costs is 2% per
year,and the risk-free rate is 8%
– The forward price is $340
Arbitrage Opportunity when Forward
Price of Gold Is Too High
A r b i t r a g e I m m e d i a t e C a s h C a s h F l o w 1 Y e a r
P o s i t i o n F l o w F r o m N o w
S e l l a f o r w a r d c o n t r a c t 0 $ 3 4 0 - S 1
B o r r o w $ 3 0 0 $300 ( $ 3 2 4 )
B u y a n o u n c e o f g o l d ( $ 3 0 0 ) S 1
P a y s t o r a g e c o s t s ( $ 6 )
N e t c a s h f l o w s 0 $ 3 4 0 - $ 3 3 0 = $ 1 0
10
Finance School of Management
Arbitrage Opportunity when Forward
Price of Gold Is Too Low A r b i t r a g e I m m e d i a t e C a s h C a s h F l o w 1 Y e a r
P o s i t i o n F l o w F r o m N o w
S e l l s h o r t a n o u n c e o f g o l d $300 0 - S
1
B u y a f o r w a r d c o n t r a c t 0 S
1 - $ 3 2 0
I n v e s t $ 3 0 0 i n 1 - y e a r p u r e ( $ 3 0 0 ) $324
d i s c o u n t b o n d s
R e c e i v e s t o r a g e c o s t s $6
N e t c a s h f l o w s 0 $ 3 3 0 - $ 3 2 0 = $ 1 0
– The forward price is $320
11
Finance School of Management
Spot-Futures Price Parity for Gold
?A contract with life T,
? ? 01 ShrF Tf ???
? This is not a causal relationship,but the forward
and current spot are jointly determined in the
market
? If we know one,then the law of one price
determines that we know the other
12
Finance School of Management
Implied Cost of Carry
? Implied cost of carry = F - S0 = (rf + h) S0
? The implied cost of storing the gold (per $spot) is
h = (F - S0)/S0 - rf
? Suppose F = $330,S0 = $300,and rf = 8%,then
– Implied cost of carry = $330-300=$30
– Implied storage cost = (330-300)/300 – 8% = 2%
13
Finance School of Management
Financial Futures
? Financial futures contracts are usually settled in
cash
? With no storage cost,the relationship between the
forward and the spot is
? ?TfrSF ?? 1
? Any deviation from this will result in an arbitrage
opportunity
14
Finance School of Management
Replication of Non-Dividend-Paying Stock
Using a pure Discount Bond and a Stock
Forward Contract
I m m e di a t e C a s h C a s h F l ow 1 Y e a r
P os i t i on F l ow F r om N ow
B uy a s ha r e of s t oc k - S S
1
R e p l i c a t i n g P o r t f o l i o ( S y n t h e t i c S t o c k )
G o l on g a f or w a r d c on t r a c t 0 S
1 – F
on s t oc k
B uy a pu r e di s c ou nt bo nd - F / ( 1+ r
f )
F
w i t h f a c e va l ue of F
T ot a l r e pl i c a t i ng po r t f ol i o - F / ( 1+ r
f ) S 1
15
Finance School of Management
Arbitrage in Stock Futures
– The spot price of a stock is $100,and the risk-free rate is
8%
– The forward price is $109
A r b i t r a g e I m m e d i a t e C a s h C a s h F l o w 1 Y e a r
P o s i t i o n F l o w F r o m N o w
S e l l a f o r w a r d c o n t r a c t 0 $ 1 0 9 - S 1
B o r r o w $ 1 0 0 $100 ( $ 1 0 8 )
B u y a s h a r e o f s t o c k ( $ 1 0 0 ) S 1
N e t c a s h f l o w s 0 $1
16
Finance School of Management
The,Implied” Risk-Free Rate
? Rearranging the formula,the implied interest rate
on a forward given the spot is
0
0
1
0
1,T if;1
S
SF
r
S
F
r
T ?
???
?
?
?
?
?
?
?
?
?
? This is reminiscent of the formula for the interest
rate on a discount bond
17
Finance School of Management
Replication of a Pure Discount Bond Using a
Stock and a Forward Contract
I m m e d i a t e C a s h C a s h F l o w 1 Y e a r
P o s i t i o n F l o w F r o m N o w
B u y a T - b i l l w i t h f a c e
v a l u e o f F
B u y a s h a r e o f s t o c k - S S
1
G o s h o r t a f o r w a r d c o n t r a c t 0 F – S
1
T o t a l r e p l i c a t i n g p o r t f o l i o - S F
- F / ( 1 + r f ) F
R e p l i c a t i n g P o r t f o l i o ( S y n t h e t i c T - B i l l )
18
Finance School of Management
The Forward Price is not a Forecast of
the Spot Price
? Following the diagrams in Chapter 13 we might
suppose that the expected price of a stock is
FrSp r e m i u mr i s krSS ff ?????? )1()1( 001
? If this is indeed correct,then the forward price is
not an indicator of the expected spot price at the
maturity of the forward
19
Finance School of Management
Forward-Spot Parity with Cash Payouts
? The S - F relationship becomes
? Note,(forward price > the spot price) if (D < rf S)
? Because D is not known with certainty,this is a
quasi-arbitrage situation
DSrSF
r
FD
S f
f
????
?
?
?
1
20
Finance School of Management
Replication of Dividend-Paying Stock Using a
Pure Discount Bond and a Stock Forward
Contract
I m m e d i a t e C a s h C a s h F l o w 1 Y e a r
P o s i t i o n F l o w F r o m N o w
B u y a s h a r e o f s t o c k - S D + S
1
G o l o n g a f o r w a r d c o n t r a c t 0 S
1 – F
o n s t o c k
B u y a p u r e d i s c o u n t b o n d - ( D + F ) / ( 1 + r
f )
D + F
w i t h f a c e v a l u e o f D + F
R e p l i c a t i n g P o r t f o l i o ( S y n t h e t i c S t o c k )
21
Finance School of Management
,Implied” Dividends
?From the last slide,we may obtain the
implied dividend
? ? FSrD f ??? 1
22
Finance School of Management
Exchange Rate Example
15000 ￥
(Borrowed)
15450 ￥
15450 ￥
(Repaid)
￡100
(Invested)
￡109
(Matures)
Time
3% ￥/￥ (direct)
3% ￥/￡/￡/￥
150 ￥/￡
9%￡/￡
Forward ￥/￡
Japan U.K,
23
Finance School of Management
The Foreign Exchange Parity
Relation
? We used the diagram to show that
? ? ? ? tYt$ r1
Y e n f o r S p o tdD e n o m i n a t e $
r1
Y e n on F o r w a r d dD e n o m i n a t e $
?
?
?
