Lecture 10
Purchase of shares
April: Purchase 500 shares for $120
-$60,000
May: Receive dividend
+500
July: Sell 500 shares for $100 per share
+50,000
Net profit = -$9,500
Short Sale of shares
April: Borrow 500 shares and sell for $120
May: Pay dividend
July: Buy 500 shares for $100 per share
Replace borrowed shares to close short position
+60,000
-$500
-$50,000
.
Net profit = + 9,500
S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for maturity
T
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
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The price of a non interest bearing asset futures
contract.
The price is merely the future value of the spot
price of the asset.
F0  S0e
rT
Example
 IBM stock is selling for $68 per share. The zero
coupon interest rate is 4.5%. What is the likely price
of the 6 month futures contract?
F0  S0e
rT
.045.50
F0  68e
F0  $69.55
Example - continued
If the actual price of the IBM futures contract is selling for
$70, what is the arbitrage transactions?
NOW
 Borrow $68 at 4.5% for 6 months
 Buy one share of stock
 Short a futures contract at $70
Month 6
Sell stock for $70
Repay loan at $69.55
Profit
+70.00
-69.55
$0.45
Example - continued
If the actual price of the IBM futures contract is selling for
$65, what is the arbitrage transactions?
NOW
 Short 1 share at $68
 Invest $68 for 6 months at 4.5%
 Long a futures contract at $65
Month 6
Buy stock for $65
Receive 68 x e.5x.045
Profit
-65.00
69.55
$4.55
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The price of a non interest bearing asset futures
contract.
The price is merely the future value of the spot
price of the asset, less dividends paid.
I = present value of dividends
F0  ( S0  I )e
rT
Example
 IBM stock is selling for $68 per share. The zero
coupon interest rate is 4.5%. It pays $.75 in
dividends in 3 and 6 months. What is the likely price
of the 6 month futures contract?
I
.75
e.045.25
I  $1.47

.75
e.045.50
F0  ( S 0  I )e
rT
F0  (68  1.47)e.045.50
F0  $68.04
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If an asset provides a known % yield, instead of a
specific cash yield, the formula can be modified to
remove the yield.
q = the known continuous compounded yield
F0  S0e
( r  q )T
Example
 A stock index is selling for $500. The zero coupon
interest rate is 4.5% and the index is known to
produce a continuously compounded dividend yield
of 2.0%. What is the likely price of the 6 month
futures contract?
F0  S0e( r q )T
F0  500e(.045.02).50
F0  $506.29

The profit (or value) from a properly priced futures
contract can be calculated from the current spot
price and the original price as follows, where K is
the delivery price in the contract (this should have
been the original futures price.
Long Contract Value
( F 0 K )
Value 
rT
e
Short Contract Value
( K  F 0)
Value 
rT
e
Example
 IBM stock is selling for $71 per share. The zero
coupon interest rate is 4.5%. What is the likely value
of the 6 month futures contract, if it only has 3
months remaining? Recall the original futures price
was 69.55.
F0  S0e rT
F0  71e
.045.25
F0  $71.80
(71.80  69.55)
Value 
.045.25
e
 $2.22
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Commodities require storage
Storage costs money. Storage can be charged as either a constant yield or a
set amount.
The futures price of a commodity can be modified to incorporate both, as in
a dividend yield.
Futures price given set
price storage cost
F 0 S 0U e
U
t Storage Cost
e rT
rT
Futures price given
constant yield storage
cost
F 0 S 0 e
( r  u )T
u =continuously compounded
cost of storage, listed as a
percentage of the asset price
Example
 The spot price of copper is $3.60 per pound. The 6 month cost to store
copper is $0.10 per pound. What is the price of a 6 month futures contract
on copper given a risk free interest rate of 3.5%?
U
.10
e.035.50
 .098
F 0 S 0U e
rT
 (3.60  .098)e
 $3.76
.035.50
Example
 The spot price of copper is $3.60 per pound. The annual cost to store copper
is quoted as a continuously compounded yield of 0.5%. What is the price of a
6 month futures contract on copper given a risk free interest rate of 3.5%?
F 0 S 0 e
( r  u )T
(.035.005).50
 3.60e
 $3.67
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
Shortages in an asset may cause a lower than
expected futures price.
This lower price is the result of a reduction in
the interest rate in the futures equation.
The reduction is called the “convenience
yield” or y.
F 0 S 0e
( r u  y )T
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The cost of carry, c, is the storage cost plus the
interest costs less the income earned
For an investment asset F0 = S0ecT
For a consumption asset F0  S0ecT
The convenience yield on the consumption asset, y,
is defined so that
F0 = S0 e(c–y )T
c can be thought of as the difference between the
borrowing rate and the income earned on the asset.
C=r-q
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
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