3.1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward and
Futures Prices
Chapter 3
3.2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Two Kinds of Underlying Assets
Investment assets,held for investment
purposes by a significant numbers of
investors,Examples,stocks,bonds,gold,
Three different situations:
1,The asset provides no income
2,The asset provides a known dollar income
3,The asset provides a known dividend yield
Consumption assets,held primarily for
consumption,Examples,commodities such
as copper,oil and live hogs,
3.3
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Arbitrage Arguments
is workable for the determination of the
forward and futures prices of investment
assets from spot and other observable
variables.
is not possible to determine the forward and
futures prices of consumption.
The forward price and futures price are very
close to each other when the maturities of the
two contracts are the same.
3.4
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Compounding Frequency
The compounding frequency
used for an interest rate is the
unit of measurement
The difference between
quarterly and annual
compounding is analogous to
the difference between miles and
kilometers
3.5
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Continuous Compounding(Page 51)
In the limit as we compound more and more
frequently we obtain continuously
compounded interest rates
$100 grows to $100eRT when invested at a
continuously compounded rate R for time T
$100 received at time T discounts to $100e-RT
at time zero when the continuously
compounded discount rate is R
Rnmn
m Aem
RA
)1(lim
3.6
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Conversion Formulas
(Pages 52,53)
Define
Rc,continuously compounded rate
Rm,same rate with compounding m times per
year
Examples,3.1,3.2(page 53)
3.7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Short Selling (Page 53)
Short selling involves selling
securities you do not own
Your broker borrows the
securities from another client
and sells them in the market in
the usual way
3.8
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Short Selling
(continued)
At some stage you must
buy the securities back so
they can be replaced in the
account of the client
You must pay dividends &
other benefits the owner of
the securities receives
3.9
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Assumptions and Notations
The market participants
are subject to no transaction costs when they trade
are subject to the same tax rate on all net trading profits
can borrow money at the same risk-free rate of interest as they
lend money
take advantage of arbitrage opportunities as they occor
Notations:
T,time when the forward contract matures (years)
S0,price of asset underlying the forward contract today
F0,forward price today
r,risk-free rate of interest per annual,with continuous
compounding,for an investment maturing at T
3.10
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Gold Example
For the gold example in chapter 1,
F0 = S0(1 + r )T
(assuming no storage costs)
If r is compounded continuously instead
of annually
F0 = S0erT
3.11
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Extension of the Gold Example
(Page 55)
For any investment asset that provides
no income and has no storage costs
F0 = S0erT
3.12
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
When an Investment Asset
Provides a Known Dollar
Income (page 58)
F0 = (S0 – I )erT
where I is the present value of the
income
3.13
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
When an Investment Asset
Provides a Known Dividend
Yield (Page 59)
F0 = S0 e(r–q )T
where q is the average dividend yield
during the life of the contract
3.14
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Valuing a Forward Contract
(Pages 59,60)
Suppose that
K is delivery price in a forward contract &
F0 is forward price that would apply to the
contract today
The value of a long forward contract,?,is
= (F0 – K )e–rT
Similarly,the value of a short forward contract
is
(K – F0 )e–rT
3.15
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward vs Futures Prices
Forward and futures prices are usually assumed
to be the same,When interest rates are
uncertain they are,in theory,slightly different:
A strong positive correlation between interest
rates and the asset price implies the futures
price is slightly higher than the forward price
A strong negative correlation implies the
reverse
3.16
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Stock Index (Page 64)
Porfolio of stocks with weights,stock proportion,
market prices or market capitalization
Examples,DJIA (10 times),S&P500 (250 times),
Nikkei 225 (5 times),NASDAQ100,CAC-40,FT-
SE100
Can be viewed as an investment asset paying a
continuous dividend yield
The futures price & spot price relationship is
therefore
F0 = S0 e(r–q )T
where q is the dividend yield on the portfolio
represented by the index
3.17
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Stock Index
(continued)
For the formula to be true it is
important that the index represent
an investment asset
In other words,changes in the index
must correspond to changes in the
value of a tradable portfolio
The Nikkei index viewed as a dollar
number does not represent an
investment asset (see P67)
3.18
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Index Arbitrage
When F0>S0e(r-q)T an arbitrageur buys
the stocks underlying the index and
sells futures
When F0<S0e(r-q)T an arbitrageur buys
futures and shorts or sells the stocks
underlying the index
3.19
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Index Arbitrage
(continued)
Index arbitrage involves simultaneous
trades in futures & many different stocks
Very often a computer is used to
generate the trades
Occasionally (e.g.,on Black Monday)
simultaneous trades are not possible
and the theoretical no-arbitrage
relationship between F0 and S0 may not
hold
3.20
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Hedging Using Index Futures
(Page 65)
To hedge the risk in a portfolio(投资组合 )
the number of contracts that should be
shorted is
where P is the value of the portfolio,b is
its beta,and A is the value of the assets
underlying one futures contract
Example 3.8,page 66
b PA
3.21
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Changing Beta
What position in index futures is
appropriate to change the beta of a
portfolio from b to b*
When,a short position in
contracts is required
When,a long position in
contracts is required
A
P*)( bb?*bb?
*bb? AP)*( bb?
3.22
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
A foreign currency is analogous to a security
providing a continuous dividend yield
The continuous dividend yield is the foreign
risk-free interest rate
It follows that if rf is the foreign risk-free
interest rate
Futures and Forwards on
Currencies (Page 68)
F S e r r Tf0 0( )
3.23
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Futures on Consumption
Assets
(Page 71)
F0? S0 e(r+u )T
where u is the storage cost per unit
time as a percent of the asset value.
Alternatively,
F0? (S0+U )erT
where U is the present value of the
storage costs.
3.24
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Cost of Carry (Page 73)
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
3.25
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Futures Prices & Expected
Future Spot Prices (Page 75)
Suppose k is the expected return
required by investors on an asset
We can invest F0e–r T now to get ST
back at maturity of the futures
contract
This shows that
F0 = E (ST )e(r–k )T
3.26
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Futures Prices & Future Spot
Prices (continued)
If the asset has
–no systematic risk,then
k = r and F0 is an unbiased
estimate of ST
–positive systematic risk,then
k > r and F0 < E (ST )
–negative systematic risk,then
k < r and F0 > E (ST )
3.27
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Assignments
3.3,3.4,3.11,3.13,3.14,3.16,3.17,
3.18,3.24,3.25,3.29-3.33
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward and
Futures Prices
Chapter 3
3.2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Two Kinds of Underlying Assets
Investment assets,held for investment
purposes by a significant numbers of
investors,Examples,stocks,bonds,gold,
Three different situations:
1,The asset provides no income
2,The asset provides a known dollar income
3,The asset provides a known dividend yield
Consumption assets,held primarily for
consumption,Examples,commodities such
as copper,oil and live hogs,
3.3
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Arbitrage Arguments
is workable for the determination of the
forward and futures prices of investment
assets from spot and other observable
variables.
is not possible to determine the forward and
futures prices of consumption.
The forward price and futures price are very
close to each other when the maturities of the
two contracts are the same.
3.4
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Compounding Frequency
The compounding frequency
used for an interest rate is the
unit of measurement
The difference between
quarterly and annual
compounding is analogous to
the difference between miles and
kilometers
3.5
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Continuous Compounding(Page 51)
In the limit as we compound more and more
frequently we obtain continuously
compounded interest rates
$100 grows to $100eRT when invested at a
continuously compounded rate R for time T
$100 received at time T discounts to $100e-RT
at time zero when the continuously
compounded discount rate is R
Rnmn
m Aem
RA
)1(lim
3.6
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Conversion Formulas
(Pages 52,53)
Define
Rc,continuously compounded rate
Rm,same rate with compounding m times per
year
Examples,3.1,3.2(page 53)
3.7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Short Selling (Page 53)
Short selling involves selling
securities you do not own
Your broker borrows the
securities from another client
and sells them in the market in
the usual way
3.8
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Short Selling
(continued)
At some stage you must
buy the securities back so
they can be replaced in the
account of the client
You must pay dividends &
other benefits the owner of
the securities receives
3.9
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Assumptions and Notations
The market participants
are subject to no transaction costs when they trade
are subject to the same tax rate on all net trading profits
can borrow money at the same risk-free rate of interest as they
lend money
take advantage of arbitrage opportunities as they occor
Notations:
T,time when the forward contract matures (years)
S0,price of asset underlying the forward contract today
F0,forward price today
r,risk-free rate of interest per annual,with continuous
compounding,for an investment maturing at T
3.10
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Gold Example
For the gold example in chapter 1,
F0 = S0(1 + r )T
(assuming no storage costs)
If r is compounded continuously instead
of annually
F0 = S0erT
3.11
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Extension of the Gold Example
(Page 55)
For any investment asset that provides
no income and has no storage costs
F0 = S0erT
3.12
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
When an Investment Asset
Provides a Known Dollar
Income (page 58)
F0 = (S0 – I )erT
where I is the present value of the
income
3.13
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
When an Investment Asset
Provides a Known Dividend
Yield (Page 59)
F0 = S0 e(r–q )T
where q is the average dividend yield
during the life of the contract
3.14
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Valuing a Forward Contract
(Pages 59,60)
Suppose that
K is delivery price in a forward contract &
F0 is forward price that would apply to the
contract today
The value of a long forward contract,?,is
= (F0 – K )e–rT
Similarly,the value of a short forward contract
is
(K – F0 )e–rT
3.15
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward vs Futures Prices
Forward and futures prices are usually assumed
to be the same,When interest rates are
uncertain they are,in theory,slightly different:
A strong positive correlation between interest
rates and the asset price implies the futures
price is slightly higher than the forward price
A strong negative correlation implies the
reverse
3.16
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Stock Index (Page 64)
Porfolio of stocks with weights,stock proportion,
market prices or market capitalization
Examples,DJIA (10 times),S&P500 (250 times),
Nikkei 225 (5 times),NASDAQ100,CAC-40,FT-
SE100
Can be viewed as an investment asset paying a
continuous dividend yield
The futures price & spot price relationship is
therefore
F0 = S0 e(r–q )T
where q is the dividend yield on the portfolio
represented by the index
3.17
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Stock Index
(continued)
For the formula to be true it is
important that the index represent
an investment asset
In other words,changes in the index
must correspond to changes in the
value of a tradable portfolio
The Nikkei index viewed as a dollar
number does not represent an
investment asset (see P67)
3.18
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Index Arbitrage
When F0>S0e(r-q)T an arbitrageur buys
the stocks underlying the index and
sells futures
When F0<S0e(r-q)T an arbitrageur buys
futures and shorts or sells the stocks
underlying the index
3.19
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Index Arbitrage
(continued)
Index arbitrage involves simultaneous
trades in futures & many different stocks
Very often a computer is used to
generate the trades
Occasionally (e.g.,on Black Monday)
simultaneous trades are not possible
and the theoretical no-arbitrage
relationship between F0 and S0 may not
hold
3.20
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Hedging Using Index Futures
(Page 65)
To hedge the risk in a portfolio(投资组合 )
the number of contracts that should be
shorted is
where P is the value of the portfolio,b is
its beta,and A is the value of the assets
underlying one futures contract
Example 3.8,page 66
b PA
3.21
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Changing Beta
What position in index futures is
appropriate to change the beta of a
portfolio from b to b*
When,a short position in
contracts is required
When,a long position in
contracts is required
A
P*)( bb?*bb?
*bb? AP)*( bb?
3.22
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
A foreign currency is analogous to a security
providing a continuous dividend yield
The continuous dividend yield is the foreign
risk-free interest rate
It follows that if rf is the foreign risk-free
interest rate
Futures and Forwards on
Currencies (Page 68)
F S e r r Tf0 0( )
3.23
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Futures on Consumption
Assets
(Page 71)
F0? S0 e(r+u )T
where u is the storage cost per unit
time as a percent of the asset value.
Alternatively,
F0? (S0+U )erT
where U is the present value of the
storage costs.
3.24
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Cost of Carry (Page 73)
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
3.25
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Futures Prices & Expected
Future Spot Prices (Page 75)
Suppose k is the expected return
required by investors on an asset
We can invest F0e–r T now to get ST
back at maturity of the futures
contract
This shows that
F0 = E (ST )e(r–k )T
3.26
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Futures Prices & Future Spot
Prices (continued)
If the asset has
–no systematic risk,then
k = r and F0 is an unbiased
estimate of ST
–positive systematic risk,then
k > r and F0 < E (ST )
–negative systematic risk,then
k < r and F0 > E (ST )
3.27
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Assignments
3.3,3.4,3.11,3.13,3.14,3.16,3.17,
3.18,3.24,3.25,3.29-3.33
