1Charles Cao
3,Forward and Futures Prices
Summary
Short Selling
The Repo Rate
Forward Contracts
Forward Price vs,Futures Price
Stock Index Futures
Forward and Futures Contracts on Currencies
2Charles Cao
Short Selling
Selling securities that you do not own
and buying them back later
Yield a profit if the price of a security
goes down
Yield a loss if the price of a security
goes up
Real world application
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Short Selling,Example
You call your broker and instruct him to
short 100 shares of ATT
He will borrow 100 shares of ATT from
another client
Sell them in the open market at $50 per
share
Deposit the cash in your account
4Charles Cao
Short Selling,Example
You instruct your broker to close out
the position 10 days later,He will buy
100 shares of ATT,and return them to
the client
If the price is $48,the profit is $200
If the price is $51,the loss is $100
5Charles Cao
Short Selling
Shares can only be sold on an uptick
(price increases)
Margin requirement
Some brokers pay interest on margin
accounts
Dividends and interest received by the
investor will be transferred to the client
from whom the shares were borrowed
6Charles Cao
The Repo Rate
The risk-free rate available to investors is
the repo rate,repurchase agreement
You sell securities to another investor
Buy them back at a higher price
The difference between the two prices is
the interest earned by the counterparty
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The Repo Rate
Repo is slightly higher than the T-bill
rate
Overnight repo,term repo
8Charles Cao
Forward Contracts
Notations:
T,expiration day
t,current time
T-t,the life of the contract (in years)
St,price of asset underlying forward
contract at t
T
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Forward Contracts
Notations,(cont.)
ST,price of asset underlying forward
contract at T
F,forward price at t
r,risk-free rate per year
K,delivery price in forward contract
f,value of a long forward contract
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Forward Contracts
Forward contracts on a security that
provides no income
Determine today’s forward price using no-
arbitrage argument
Forward Contracts
Action
Buy 1 Unit
of Asset
Short 1 Forward
Contract
Borrow $
at rate r
Cash Flow at t Cash Flow at T
tS? TS
TSF?
tS )( tTrteS
0)( tTrtTT eSSFS0
tS
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12Charles Cao
Forward Contracts
Therefore,the price of the forward contract
at t is:
)( tTrt eSF
Intuition
You walk in a local Mercedes-Benz
dealer’s office
You want to buy a white color CL500
model
Today,the price is $90,000
However,only black and silver colors
are available
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Intuition
The salesman agrees to get a white
color CL500 from Florida in 30 days
You agree to pay when the car is ready
to be delivered
How much should the salesman charge
you 30 days later?
Interest rate is 5%
14Charles Cao
Intuition
or
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16Charles Cao
Forward Contracts
If F > St e r(T-t),you can earn an
arbitrage profit by taking the following
positions at t (cash is not required):
Borrow St dollars at rate r
Buy 1 unit of asset
Take a short position in the forward contract
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Forward Contracts
At time T:
Sold the asset for ST
Pay back the loan,-St e r(T-t)
Realize a profit,F – St e r(T-t) > 0
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Forward Contracts,Example 1
A forward contract is written on a stock,
The maturity of the contract is 6
months,The stock price is $50 today
and the risk-free rate is 10% per year
Recall )( tTr
t eSF
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Forward Contracts,Example 1
T-t = 0.5 year
r = 0.1
St = $50
Thus,the forward price is:
56.52$50 5.01.0eF
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Forward Contracts,Example 2
A forward contract is written on a
discount bond,The maturity of the
contract is 4 months,The bond price is
$950 today and the risk-free rate is 8%
per year
Recall
)( tTrt eSF
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Forward Contracts,Example 2
T-t = 0.33 year (4/12)
r = 0.08
St = $950
Thus,the forward price is:
41.9 7 5$9 5 0 33.008.0eF
22Charles Cao
Forward Contracts
Forward contracts on a security that
provides a known cash income
Dividends-paying stock with known
(discrete) dividends
Coupon bonds
Determine today’s forward price using no-
arbitrage argument
I,present value of all known cash income
Action
Buy Underlying
Asset
Short 1 Forward
Contract
Borrow
at rate r
Cash Flow at t Cash Flow at T
TS
TSF?
0
ISt
)( tTrt eIS
)()( tTrtTT eISSFS
ISt?
Forward Contracts
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Forward Contracts
Therefore,the price of the forward
contract at t is
)()( tTrt eISF
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Forward Contracts,Example
A 10-month contract is written on a
dividend-paying stock,The stock price
is $50 today,The risk free rate is 8%
per year,The dividend is $2.00 per
share and will be paid after 6 months
and 9 months
Recall )()( tTr
t eISF
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Forward Contracts,Example
T-t = 0.83 year (10/12)
r = 0.08
St = $50
36.49$)81.350( 83.008.0eF
81.322 )12/9(08.0)12/6(08.0 eeI
Thus,the forward price is:
27Charles Cao
Forward Contracts
Forward contracts on a security that
provides a known (continuous) dividend
yield
Dividend-paying stocks,Dividend yield is
paid continuously at an annual rate of q
Determine today’s forward price using
no-arbitrage argument
Forward Contracts
Action
Buy
Unit of Asset
Short 1 Forward
Contract
Borrow
at rate r
Cash Flow at t Cash Flow at T
TS
TSF?
0
)( tTqt eS
)()( tTrtTqt eeS
))(( tTqrtTT eSSFS
)( tTqteS
)( tTqe
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29Charles Cao
Forward Contracts
Therefore,the price of the forward
contract at t is:
))(( tTqrt eSF
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Forward Contracts,Example
A 6-month forward contract is written
on a dividend-paying stock,The
dividend yield is 4% per year,The
stock price is $25 today,The risk free
rate is 10% per year
Recall ))(( tTqr
t eSF
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Forward Contracts,Example
T-t = 0.5 year (6/12)
r = 0.10
St = $25
q = 0.04
76.25$25 5.0)04.010.0(eF
Thus,the forward price is:
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Forward Price vs,Futures Price
The forward price and futures price are
the same when the interest rate is
constant
When the interest rates change over
time (unpredictable),the forward and
futures prices are different
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Forward Price vs,Futures Price
If the maturity is only a few months,the
difference between the two prices is small
and can be ignored
If asset price is positively correlated with
interest rates,futures prices tend to be
higher then forward prices
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Forward Price vs,Futures Price
If asset price is negatively correlated with
interest rates,futures price tend to be
lower than forward prices
In general,it is reasonable to assume that
forward and futures prices are equal
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Stock Index Futures
Stock Indices
Track the changes in the underlying index
The weight of the portfolio (value-weighted
or equal-weighted)
The weight changes when the price of the
stock changes
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Stock Index Futures
A stock index is not usually adjusted for a
cash dividend
If a stock’s price increases sharply,more
weight will be given to the stock
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Stock Index,Example
S&P 500 Index
400 industrials,40 utilities,20 transportations,
and 40 financial institutions
Autos and Transportation 3%
Consumer Services 14%
Financial Services 22%
Health Care 13%
Oils 4%
Technology 16%
Utilities 6%
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Stock Index,Example
S&P 500 Index
Value-weighted portfolio
Accounts for 80% of the market
capitalization of all stocks listed on the
NYSE
Each futures contract is based on 250
times the index
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Stock Index,Example
NYSE Composite Index
Based on all stocks listed on the NYSE.
Value-weighted
Each futures contract is based on 250 times the
index
Major Market Index (MMI)
Based on 20 blue-chip stocks listed on the NYSE
Price-weighted
Very close to the Dow Jones Index
S&P 500 index (CME) - $250 x index
40
Open High Low Settle
Dec 105,680 105,960 105,210 105,810
Mr04 105,300 105,800 105,100 105,670
Est Vol 46,123; Vol Tues 49,444;
Open Int 583,222,+2,378
Index Price,Hi 1,059.62; Lo 1,052.96;
Close 1,058.41,-.15
(Source,Wall Street Journal,11/12/2003)
Charles Cao
41Charles Cao
Stock Index Futures
Stock Index Futures
Treat stock index as a dividend-paying
stock
Assume the dividend is paid continuously
with a dividend yield of rate q
The price of the futures contract at t is:
))(( tTqrt eSF
42Charles Cao
Stock Index Futures
Index Arbitrage
Implementation,Program Trading
If F > Ste(r-q)(T-t),you can make a profit by
taking the following positions:
Buy stocks underlying the index
Short futures contracts
Done by investors who have money market investments
or cash
43Charles Cao
Stock Index Futures
Implementation,Program Trading (cont.)
If F < Ste(r-q)(T-t),you can make a profit by
taking the following positions:
Short stocks underlying the index
Long futures contracts
Done by pension fund managers,They own index
portfolios
44Charles Cao
Forward and Futures Contracts
on Currencies
St = price in dollars of one unit of the
foreign currency
r = domestic (U.S.) interest rate with
continuous compounding
rf = foreign interest rate with continuous
compounding
45Charles Cao
Forward and Futures Contracts
on Currencies
Recall,
Replace q by rf,we obtain the futures price
on a foreign currency
))(( tTqrt eSF
))(( tTrrt feSF
46Charles Cao
Forward and Futures Contracts
on Currencies
A foreign currency is analogous to a stock
paying a known dividend yield
The,dividend yield” is the risk-free rate in the
foreign currency
Interest earned on a foreign currency holding is
denominated in the foreign currency
The,present value of the dividend” is the
present value of the interest earned on a foreign
currency
47Charles Cao
Forward and Futures Contracts
on Currencies
When r - rf < 0,
futures price is less than the spot price St
When r - rf > 0,
futures price is greater than the spot
price St
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Futures on Commodities
Commodities can be held for investment
Commodities can be held for consumption
Summary
Gold futures
The cost of carry
Delivery options
Futures written on dividend-paying stocks
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Gold Futures
If no storage costs,gold can be considered as
a security paying no income,Thus the
futures price is:
)( tTrt eSF
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Gold Futures
If there are storage costs,storage costs can
be regarded as negative income,Let U be
the present value of the storage costs that
will be incurred during the life of the contract,
replace I by -U in the formula
)()( tTrt eISF
)()( tTrt eUSF
we obtain the futures price
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Gold Futures
If the storage costs are incurred continuously,
regard the storage costs as negative dividend
yield,Let storage costs be u,and replace q
by -u in the formula ))(( tTqr
t eSF
))(( tTurt eSF
we obtain the futures price
where u is the storage costs per annum as a
proportion of the spot price.
52Charles Cao
Gold Futures,Example
A futures contract is written on gold,
The maturity of the contract is 1 year,
The storage cost is $2 per ounce per
year,The payment will be made at the
end of the year,The spot price is $450,
and the risk-free rate is 7% per year
Recall )()( tTr
t eUSF
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Gold Futures,Example
T-t = 1 year
r = 0.07
St = $450
865.122 107.0)( eeU tTr
6.484$)865.1450( 107.0eF
Thus the futures price is:
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The Cost of Carry
The storage cost
(+) the interest paid to finance the
asset
(-) the income earned on the asset
For a non-dividend paying stock,c = r
For a dividend-paying stock,c = r - q
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The Cost of Carry
For a currency,c = r - rf
For a commodity,c = r + u
For an investment asset,the futures
price is,)( tTc
t eSF
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Delivery Options
If you short a futures contract,you can
deliver the underlying asset at any time
up until the date of contract maturity
How should you determine the maturity?
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Futures Contract Written on
Dividend-paying Stocks
Consider a futures contract written on a
dividend-paying stock,Recall
When r > q
You earn the dividend yield at rate q and lose
the interest at rate r if you keep the stock for
one more day (i.e.,deliver it later)
The benefits of holding the stock are less than
the interest that has occurred
))(( tTqrt eSF
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Futures Contract Written on
Dividend-paying Stocks
When r > q (cont.)
It is optimal for the party with the short
position to deliver as early as possible
Thus,the futures price will be calculated on the
basis that delivery will take place at the
beginning of the delivery period
When r < q
It is optimal for the party with the short
position to deliver as late as possible