Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.1
Options on
Stock Indices,Currencies,and
Futures
Chapter 13
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.2European Options on Stocks
Providing a Dividend Yield
We get the same probability
distribution for the stock price at time
T in each of the following cases:
1,The stock starts at price S0 and
provides a dividend yield = q
2,The stock starts at price S0e–q T
and provides no income
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.3European Options on Stocks
Providing Dividend Yield
continued
We can value European options by
reducing the stock price to S0e–q T and
then behaving as though there is no
dividend
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.4
Extension of Chapter 8 Results
(Equations 13.1 to 13.3)
rTqT KeeSc ?? ??
0
Lower Bound for calls:
Lower Bound for puts
qTrT eSKep ?? ??
0
Put Call Parity
qTrT eSpKec ?? ??? 0
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.5Extension of Chapter 12
Results (Equations 13.4 and 14.5)
T
TqrKS
d
T
TqrKS
d
dNeSdNKep
dNKedNeSc
qTrT
rTqT
?
????
?
?
????
?
????
??
??
??
)2/
2
()/l n (
)2/
2
()/l n (
)()(
)()(
0
2
0
1
102
210
w h e r e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.6
The Binomial Model
S0u
?u
S0d
?d
S0
?
f=e-rT[pfu+(1-p)fd ]
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.7The Binomial Model
continued
? In a risk-neutral world the stock price
grows at r-q rather than at r when there
is a dividend yield at rate q
? The probability,p,of an up movement
must therefore satisfy
pS0u+(1-p)S0d=S0e(r-q)T
so that p
e d
u d
r q T
? ??
?( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.8
Index Options
? Option contracts are on 100 times the
index
? The most popular underlying indices are
– the Dow Jones Industrial (European) DJX
– the S&P 100 (American) OEX
– the S&P 500 (European) SPX
? Contracts are settled in cash
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.9
Index Option Example
? Consider a call option on an
index with a strike price of 560
? Suppose 1 contract is exercised
when the index level is 580
? What is the payoff?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.10Using Index Options for
Portfolio Insurance
? Suppose the value of the index is S0 and the strike
price is K
? If a portfolio has a b of 1.0,the portfolio insurance
is obtained by buying 1 put option contract on the
index for each 100S0 dollars held
? If the b is not 1.0,the portfolio manager buys b put
options for each 100S0 dollars held
? In both cases,K is chosen to give the appropriate
insurance level
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.11
Example 1
? Portfolio has a beta of 1.0
? It is currently worth $500,000
? The index currently stands at 1000
? What trade is necessary to provide
insurance against the portfolio value
falling below $450,000?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.12
Example 2
? Portfolio has a beta of 2.0
? It is currently worth $500,000 and index
stands at 1000
? The risk-free rate is 12% per annum
? The dividend yield on both the portfolio
and the index is 4%
? How many put option contracts should
be purchased for portfolio insurance?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.13
? If index rises to 1040,it provides a
40/1000 or 4% return in 3 months
? Total return (incl dividends)=5%
? Excess return over risk-free rate=2%
? Excess return for portfolio=4%
? Increase in Portfolio Value=4+3-1=6%
? Portfolio value=$530,000
Calculating Relation Between Index
Level and Portfolio Value in 3 months
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.14Determining the Strike Price
(Table 13.2,page 274)
Value of Inde x in 3
months
Expecte d Portfo lio Value
in 3 months ($)
1,08 0 570,000
1,04 0 530,000
1,00 0 490,000
96 0 450,000
92 0 410,000
An option with a strike price of 960 will provide protection
against a 10% decline in the portfolio value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.15
Valuing European Index
Options
We can use the formula for an option
on a stock paying a dividend yield
Set S0 = current index level
Set q = average dividend yield
expected during the life of the option
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.16
Currency Options
? Currency options trade on the Philadelphia
Exchange (PHLX)
? There also exists an active over-the-counter
(OTC) market
? Currency options are used by corporations
to buy insurance when they have an FX
exposure
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.17
The Foreign Interest Rate
? We denote the foreign interest rate by rf
? When a U.S,company buys one unit of
the foreign currency it has an
investment of S0 dollars
? The return from investing at the foreign
rate is rf S0 dollars
? This shows that the foreign currency
provides a,dividend yield” at rate rf
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.18Valuing European Currency
Options
? A foreign currency is an asset that
provides a,dividend yield” equal to rf
? We can use the formula for an option
on a stock paying a dividend yield,
Set S0 = current exchange rate
Set q = r?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.19Formulas for European
Currency Options
(Equations 13.9 and 13.10,page 277)
T
T
f
rrKS
d
T
T
f
rrKS
d
dNeSdNKep
dNKedNeSc
TrrT
rTTr
f
f
?
????
?
?
????
?
????
??
??
??
)2/
2
()/l n (
)2/
2
()/l n (
)()(
)()(
0
2
0
1
102
210
w h e r e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.20
Alternative Formulas
(Equations 13.11 and 13.12,page 278)
F S e r r Tf0 0? ?( )Using
Tdd
T
TKF
d
dNFdKNep
dKNdNFec
rT
rT
???
?
??
?
????
??
?
?
12
2
0
1
102
210
2/)/l n (
)]()([
)]()([
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.21Mechanics of Call Futures
Options
When a call futures option is exercised
the holder acquires
1,A long position in the futures
2,A cash amount equal to the excess of
the futures price over the strike price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.22
Mechanics of Put Futures
Option
When a put futures option is exercised
the holder acquires
1,A short position in the futures
2,A cash amount equal to the excess of
the strike price over the futures price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.23
The Payoffs
If the futures position is closed out
immediately:
Payoff from call = F0 – K
Payoff from put = K – F0
where F0 is futures price at time of
exercise
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.24
Put-Call Parity for Futures
Options (Equation 13.13,page 284)
Consider the following two portfolios:
1,European call plus Ke-rT of cash
2,European put plus long futures plus
cash equal to F0e-rT
They must be worth the same at time T
so that
c+Ke-rT=p+F0 e-rT
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.25
Futures Price = $33
Option Price = $4
Futures Price = $28
Option Price = $0
Futures price = $30
Option Price=?
Binomial Tree Example
A 1-month call option on futures has a strike price of
29,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.26
? Consider the Portfolio,long D futures
short 1 call option
? Portfolio is riskless when 3D – 4 = -2D or
D = 0.8
3D – 4
-2D
Setting Up a Riskless Portfolio
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.27
Valuing the Portfolio
( Risk-Free Rate is 6% )
? The riskless portfolio is,
long 0.8 futures
short 1 call option
? The value of the portfolio in 1 month is
-1.6
? The value of the portfolio today is
-1.6e – 0.06/12 = -1.592
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.28
Valuing the Option
? The portfolio that is
long 0.8 futures
short 1 option
is worth -1.592
? The value of the futures is zero
? The value of the option must
therefore be 1.592
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.29
Generalization of Binomial Tree
Example (Figure 13.3,page 285)
? A derivative lasts for time T and is
dependent on a futures price
F0u
?u
F0d
?d
F0
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.30
Generalization
(continued)
? Consider the portfolio that is long D futures and
short 1 derivative
? The portfolio is riskless when
D ? ?
?
? u df
F u F d0 0
F0u D ? F0 D – ?u
F0d D? F0D – ?d
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.31Generalization
(continued)
? Value of the portfolio at time T
is F0u D–F0D–?u
? Value of portfolio today is – ?
? Hence
? = – [F0u D –F0D – ?u]e-rT
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.32
Generalization
(continued)
? Substituting for D we obtain
? = [ p ?u + (1 – p )?d ]e–rT
where p d
u d
? ?
?
1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.33Valuing European Futures
Options
? We can use the formula for an option on a
stock paying a dividend yield
Set S0 = current futures price (F0)
Set q = domestic risk-free rate (r )
? Setting q = r ensures that the expected
growth of F in a risk-neutral world is zero
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.34
Growth Rates For Futures
Prices
? A futures contract requires no initial
investment
? In a risk-neutral world the expected return
should be zero
? The expected growth rate of the futures
price is therefore zero
? The futures price can therefore be treated
like a stock paying a dividend yield of r
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.35Black’s Formula
(Equations 13.17 and 13.18,page 287)
? The formulas for European options on
futures are known as Black’s formulas
? ?
? ?
Td
T
TKF
d
T
TKF
d
dNFdNKep
dNKdNFec
rT
rT
???
?
??
?
?
??
?
????
??
?
?
1
0
2
0
1
102
210
2/
2
)/l n (
2/
2
)/l n (
)()(
)()(
w h e r e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.36
Futures Option Prices vs Spot
Option Prices
? If futures prices are higher than spot prices
(normal market),an American call on futures
is worth more than a similar American call on
spot,An American put on futures is worth less
than a similar American put on spot
? When futures prices are lower than spot
prices (inverted market) the reverse is true
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.37
Summary of Key Results
? We can treat stock indices,currencies,
and futures like a stock paying a
dividend yield of q
–For stock indices,q = average
dividend yield on the index over the
option life
–For currencies,q = r?
–For futures,q = r
13.1
Options on
Stock Indices,Currencies,and
Futures
Chapter 13
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.2European Options on Stocks
Providing a Dividend Yield
We get the same probability
distribution for the stock price at time
T in each of the following cases:
1,The stock starts at price S0 and
provides a dividend yield = q
2,The stock starts at price S0e–q T
and provides no income
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.3European Options on Stocks
Providing Dividend Yield
continued
We can value European options by
reducing the stock price to S0e–q T and
then behaving as though there is no
dividend
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.4
Extension of Chapter 8 Results
(Equations 13.1 to 13.3)
rTqT KeeSc ?? ??
0
Lower Bound for calls:
Lower Bound for puts
qTrT eSKep ?? ??
0
Put Call Parity
qTrT eSpKec ?? ??? 0
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.5Extension of Chapter 12
Results (Equations 13.4 and 14.5)
T
TqrKS
d
T
TqrKS
d
dNeSdNKep
dNKedNeSc
qTrT
rTqT
?
????
?
?
????
?
????
??
??
??
)2/
2
()/l n (
)2/
2
()/l n (
)()(
)()(
0
2
0
1
102
210
w h e r e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.6
The Binomial Model
S0u
?u
S0d
?d
S0
?
f=e-rT[pfu+(1-p)fd ]
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.7The Binomial Model
continued
? In a risk-neutral world the stock price
grows at r-q rather than at r when there
is a dividend yield at rate q
? The probability,p,of an up movement
must therefore satisfy
pS0u+(1-p)S0d=S0e(r-q)T
so that p
e d
u d
r q T
? ??
?( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.8
Index Options
? Option contracts are on 100 times the
index
? The most popular underlying indices are
– the Dow Jones Industrial (European) DJX
– the S&P 100 (American) OEX
– the S&P 500 (European) SPX
? Contracts are settled in cash
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.9
Index Option Example
? Consider a call option on an
index with a strike price of 560
? Suppose 1 contract is exercised
when the index level is 580
? What is the payoff?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.10Using Index Options for
Portfolio Insurance
? Suppose the value of the index is S0 and the strike
price is K
? If a portfolio has a b of 1.0,the portfolio insurance
is obtained by buying 1 put option contract on the
index for each 100S0 dollars held
? If the b is not 1.0,the portfolio manager buys b put
options for each 100S0 dollars held
? In both cases,K is chosen to give the appropriate
insurance level
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.11
Example 1
? Portfolio has a beta of 1.0
? It is currently worth $500,000
? The index currently stands at 1000
? What trade is necessary to provide
insurance against the portfolio value
falling below $450,000?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.12
Example 2
? Portfolio has a beta of 2.0
? It is currently worth $500,000 and index
stands at 1000
? The risk-free rate is 12% per annum
? The dividend yield on both the portfolio
and the index is 4%
? How many put option contracts should
be purchased for portfolio insurance?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.13
? If index rises to 1040,it provides a
40/1000 or 4% return in 3 months
? Total return (incl dividends)=5%
? Excess return over risk-free rate=2%
? Excess return for portfolio=4%
? Increase in Portfolio Value=4+3-1=6%
? Portfolio value=$530,000
Calculating Relation Between Index
Level and Portfolio Value in 3 months
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.14Determining the Strike Price
(Table 13.2,page 274)
Value of Inde x in 3
months
Expecte d Portfo lio Value
in 3 months ($)
1,08 0 570,000
1,04 0 530,000
1,00 0 490,000
96 0 450,000
92 0 410,000
An option with a strike price of 960 will provide protection
against a 10% decline in the portfolio value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.15
Valuing European Index
Options
We can use the formula for an option
on a stock paying a dividend yield
Set S0 = current index level
Set q = average dividend yield
expected during the life of the option
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.16
Currency Options
? Currency options trade on the Philadelphia
Exchange (PHLX)
? There also exists an active over-the-counter
(OTC) market
? Currency options are used by corporations
to buy insurance when they have an FX
exposure
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.17
The Foreign Interest Rate
? We denote the foreign interest rate by rf
? When a U.S,company buys one unit of
the foreign currency it has an
investment of S0 dollars
? The return from investing at the foreign
rate is rf S0 dollars
? This shows that the foreign currency
provides a,dividend yield” at rate rf
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.18Valuing European Currency
Options
? A foreign currency is an asset that
provides a,dividend yield” equal to rf
? We can use the formula for an option
on a stock paying a dividend yield,
Set S0 = current exchange rate
Set q = r?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.19Formulas for European
Currency Options
(Equations 13.9 and 13.10,page 277)
T
T
f
rrKS
d
T
T
f
rrKS
d
dNeSdNKep
dNKedNeSc
TrrT
rTTr
f
f
?
????
?
?
????
?
????
??
??
??
)2/
2
()/l n (
)2/
2
()/l n (
)()(
)()(
0
2
0
1
102
210
w h e r e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.20
Alternative Formulas
(Equations 13.11 and 13.12,page 278)
F S e r r Tf0 0? ?( )Using
Tdd
T
TKF
d
dNFdKNep
dKNdNFec
rT
rT
???
?
??
?
????
??
?
?
12
2
0
1
102
210
2/)/l n (
)]()([
)]()([
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.21Mechanics of Call Futures
Options
When a call futures option is exercised
the holder acquires
1,A long position in the futures
2,A cash amount equal to the excess of
the futures price over the strike price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.22
Mechanics of Put Futures
Option
When a put futures option is exercised
the holder acquires
1,A short position in the futures
2,A cash amount equal to the excess of
the strike price over the futures price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.23
The Payoffs
If the futures position is closed out
immediately:
Payoff from call = F0 – K
Payoff from put = K – F0
where F0 is futures price at time of
exercise
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.24
Put-Call Parity for Futures
Options (Equation 13.13,page 284)
Consider the following two portfolios:
1,European call plus Ke-rT of cash
2,European put plus long futures plus
cash equal to F0e-rT
They must be worth the same at time T
so that
c+Ke-rT=p+F0 e-rT
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.25
Futures Price = $33
Option Price = $4
Futures Price = $28
Option Price = $0
Futures price = $30
Option Price=?
Binomial Tree Example
A 1-month call option on futures has a strike price of
29,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.26
? Consider the Portfolio,long D futures
short 1 call option
? Portfolio is riskless when 3D – 4 = -2D or
D = 0.8
3D – 4
-2D
Setting Up a Riskless Portfolio
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.27
Valuing the Portfolio
( Risk-Free Rate is 6% )
? The riskless portfolio is,
long 0.8 futures
short 1 call option
? The value of the portfolio in 1 month is
-1.6
? The value of the portfolio today is
-1.6e – 0.06/12 = -1.592
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.28
Valuing the Option
? The portfolio that is
long 0.8 futures
short 1 option
is worth -1.592
? The value of the futures is zero
? The value of the option must
therefore be 1.592
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.29
Generalization of Binomial Tree
Example (Figure 13.3,page 285)
? A derivative lasts for time T and is
dependent on a futures price
F0u
?u
F0d
?d
F0
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.30
Generalization
(continued)
? Consider the portfolio that is long D futures and
short 1 derivative
? The portfolio is riskless when
D ? ?
?
? u df
F u F d0 0
F0u D ? F0 D – ?u
F0d D? F0D – ?d
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.31Generalization
(continued)
? Value of the portfolio at time T
is F0u D–F0D–?u
? Value of portfolio today is – ?
? Hence
? = – [F0u D –F0D – ?u]e-rT
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.32
Generalization
(continued)
? Substituting for D we obtain
? = [ p ?u + (1 – p )?d ]e–rT
where p d
u d
? ?
?
1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.33Valuing European Futures
Options
? We can use the formula for an option on a
stock paying a dividend yield
Set S0 = current futures price (F0)
Set q = domestic risk-free rate (r )
? Setting q = r ensures that the expected
growth of F in a risk-neutral world is zero
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.34
Growth Rates For Futures
Prices
? A futures contract requires no initial
investment
? In a risk-neutral world the expected return
should be zero
? The expected growth rate of the futures
price is therefore zero
? The futures price can therefore be treated
like a stock paying a dividend yield of r
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.35Black’s Formula
(Equations 13.17 and 13.18,page 287)
? The formulas for European options on
futures are known as Black’s formulas
? ?
? ?
Td
T
TKF
d
T
TKF
d
dNFdNKep
dNKdNFec
rT
rT
???
?
??
?
?
??
?
????
??
?
?
1
0
2
0
1
102
210
2/
2
)/l n (
2/
2
)/l n (
)()(
)()(
w h e r e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.36
Futures Option Prices vs Spot
Option Prices
? If futures prices are higher than spot prices
(normal market),an American call on futures
is worth more than a similar American call on
spot,An American put on futures is worth less
than a similar American put on spot
? When futures prices are lower than spot
prices (inverted market) the reverse is true
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
13.37
Summary of Key Results
? We can treat stock indices,currencies,
and futures like a stock paying a
dividend yield of q
–For stock indices,q = average
dividend yield on the index over the
option life
–For currencies,q = r?
–For futures,q = r
