Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.1
Determination of
Forward and
Futures Prices
Chapter 3
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.2
Consumption vs Investment
Assets
? Investment assets are assets held by
significant numbers of people purely for
investment purposes (Examples,gold,
silver)
? Consumption assets are assets held
primarily for consumption (Examples,
copper,oil)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.3
Short Selling (Page 41-42)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.4
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
and other benefits the
owner of the securities
receives
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.5
Measuring Interest Rates
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.6
Continuous Compounding
(Page 43)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.7
Conversion Formulas
(Page 44)
Define
Rc, continuously compounded rate
Rm,same rate with compounding m times
per year
? ?
R m
R
m
R m e
c
m
m
R mc
? ?
?
?
?
?
?
?
? ?
ln
/
1
1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.8
Notation
S0,Spot price today
F0,Futures or forward price today
T,Time until delivery date
r,Risk-free interest rate for
maturity T
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.9Gold Example
(From Chapter 1)
? For gold
F0 = S0(1 + r )T
(assuming no storage costs)
? If r is compounded continuously instead
of annually
F0 = S0erT
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.10
Extension of the Gold Example
(Page 46,equation 3.5)
? For any investment asset that provides
no income and has no storage costs
F0 = S0erT
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.11When an Investment Asset
Provides a Known Dollar
Income (page 48,equation 3.6)
F0 = (S0 – I )erT
where I is the present value of the
income
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.12When an Investment Asset
Provides a Known Yield
(Page 49,equation 3.7)
F0 = S0 e(r–q )T
where q is the average yield during the
life of the contract (expressed with
continuous compounding)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.13Valuing a Forward Contract
Page 50
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.14
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.15
Stock Index (Page 52)
? Can be viewed as an investment asset
paying a dividend yield
? The futures price and spot price
relationship is therefore
F0 = S0 e(r–q )T
where q is the dividend yield on the
portfolio represented by the index
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.16
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.17
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.18
Index Arbitrage
(continued)
? Index arbitrage involves simultaneous
trades in futures and 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 does not
hold
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.19
? A foreign currency is analogous to a security
providing a 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 55-58)
F S e r r Tf0 0? ?( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.20Futures on Consumption
Assets
(Page 59)
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.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.21
The Cost of Carry (Page 60)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.22
Futures Prices & Expected
Future Spot Prices (Page 61)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.23Futures 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 )
3.1
Determination of
Forward and
Futures Prices
Chapter 3
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.2
Consumption vs Investment
Assets
? Investment assets are assets held by
significant numbers of people purely for
investment purposes (Examples,gold,
silver)
? Consumption assets are assets held
primarily for consumption (Examples,
copper,oil)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.3
Short Selling (Page 41-42)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.4
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
and other benefits the
owner of the securities
receives
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.5
Measuring Interest Rates
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.6
Continuous Compounding
(Page 43)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.7
Conversion Formulas
(Page 44)
Define
Rc, continuously compounded rate
Rm,same rate with compounding m times
per year
? ?
R m
R
m
R m e
c
m
m
R mc
? ?
?
?
?
?
?
?
? ?
ln
/
1
1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.8
Notation
S0,Spot price today
F0,Futures or forward price today
T,Time until delivery date
r,Risk-free interest rate for
maturity T
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.9Gold Example
(From Chapter 1)
? For gold
F0 = S0(1 + r )T
(assuming no storage costs)
? If r is compounded continuously instead
of annually
F0 = S0erT
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.10
Extension of the Gold Example
(Page 46,equation 3.5)
? For any investment asset that provides
no income and has no storage costs
F0 = S0erT
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.11When an Investment Asset
Provides a Known Dollar
Income (page 48,equation 3.6)
F0 = (S0 – I )erT
where I is the present value of the
income
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.12When an Investment Asset
Provides a Known Yield
(Page 49,equation 3.7)
F0 = S0 e(r–q )T
where q is the average yield during the
life of the contract (expressed with
continuous compounding)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.13Valuing a Forward Contract
Page 50
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.14
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.15
Stock Index (Page 52)
? Can be viewed as an investment asset
paying a dividend yield
? The futures price and spot price
relationship is therefore
F0 = S0 e(r–q )T
where q is the dividend yield on the
portfolio represented by the index
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.16
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.17
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.18
Index Arbitrage
(continued)
? Index arbitrage involves simultaneous
trades in futures and 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 does not
hold
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.19
? A foreign currency is analogous to a security
providing a 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 55-58)
F S e r r Tf0 0? ?( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.20Futures on Consumption
Assets
(Page 59)
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.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.21
The Cost of Carry (Page 60)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.22
Futures Prices & Expected
Future Spot Prices (Page 61)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
3.23Futures 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 )
