Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.1
Properties of
Stock Option Prices
Chapter 7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.2
Notation
c,European call
option price
p,European put
option price
S0,Stock price today
X,Strike price
T,Life of option
,Volatility(波动率 ) of
stock price
C,American Call option
price
P,American Put option
price
ST,Stock price at time T
D,Present value of
dividends during option’s
life
r,Risk-free rate for
maturity T with cont comp
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.3Effect of Variables on Option
Pricing (Table 7.1,page 169)
c p C PVariable
S0
X
T
r
D
+ + –+
+ ++ + + +
+ – + –
–– – +
– + – +
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.4
American vs European Options
An American option is worth
at least as much as the
corresponding European
option
C? c
P? p
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.5
Upper Bound for
Options Prices
It should be relatively easy to see that
c? S0 and C? S0
Otherwise,you could make a risk-less profit
by buying the stock and selling the option
Likewise,
p? Xe-rT and P? X
Otherwise,you could make a risk-less profit
by selling the option and investing the
proceeds at r
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.6
Lower Bound for
European Call Prices
(NO Dividends)
Consider the following positions
t = 0 ST <X ST >X
Portfolio A
Buy Call -c 0 ST - X
Lend Xe-rT at r -Xe-rT X X
Net Flows -c-Xe-rT X ST
Portfolio B
Buy one share -S0 ST ST
A is worth more than B,so it must cost more to
set it up initially,So c + Xe-rT > S0
c > max[S0 -Xe-rT,0]
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.7
Calls,An Arbitrage Possibility?
Suppose that
c = 3
S0= 52
T = 1
r= 5%
X= 50
D= 0
Is there an arbitrage possibility?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.8
Calls,An Arbitrage Possibility?
(continued)
Is c > S0 - Xe-rT?
S0 - Xe-rT =
= 52 - 50e-0.05(1.00)
= 52 - 47.56
= 4.44
Yes,an arbitrage is possible as 3 < 4.44
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.9
Calls,An Arbitrage Possibility?
(continued)
Yes,an arbitrage is possible as 3 < 4.44
t = 0 ST<X ST>X
Buy the Call -c 0 ST -X
Sell Stock S0 -ST -ST
Lend Xe-rT -Xe-rT X X
Net Flows -c+S-Xe-rT X-ST 0
Numerically,
t = 0 ST<50 ST>50
Buy the Call -3 0 ST-50
Sell Stock 52 -ST -ST
Lend Xe-rT -50e-0.05*1 50 50
Net Flows -3+52-50e-0.05*1=1.44 50-ST 0
Now
Possibly
More Later
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.10Lower Bound for
European Put Prices
(NO Dividends)
Consider the following positions
t = 0 ST <X ST >X
Portfolio C
Buy Put -p X - ST 0
Buy Stock -S0 ST ST
Net Flows -p-S0 X ST
Portfolio D
Lend Xe-rT at r -Xe-rT X X
C is worth more than D,so it must cost more
to set it up initially,So,p+S0 > Xe-rT
p > max[Xe-rT - S0,0]
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.11
Puts,An Arbitrage Possibility?
Suppose that
p= 3
S0= 48
T= 0.25
r= 5%
X= 50
D= 0
Is there an arbitrage possibility?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.12
Puts,An Arbitrage Possibility?
(continued)
Is p > Xe-rT - S0?
Xe-rT - S0 =
= 50e-0.05(0.25) - 48
= 49.37 - 48
= 1.37
No arbitrage possibility
3 > 1.37
An arbitrage is possible if p=1
The risk-less profit 0.37 can be obtained by borrowing
Xe-rT,buying Put and Stock (See slide 7.10)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.13
Call Early Exercise on
Non-Dividend Paying Stock
From earlier we know that
c > S0 - Xe-rT
Given that C? c
C > S0 - Xe-rT > S0 - X
Thus,C > S0 - X,but if exercised early
C = S0 - X
Hence,no reason to exercise early
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.14
Early Exercise
Usually,there is some chance that an
American option will be exercised early
An exception is an American call on a
non-dividend paying stock
– This should NEVER be exercised early
1,NO income is sacrificed
2,Strike price is paid later
3,Holding the call provides insurance against
stock price falling below strike price
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.15
An Extreme Situation
For an American call option with
S0 = 100,T = 0.25,X = 60,r = 5%,D = 0
– Should you exercise if …
1,You want to hold the stock over
the next 3 months?
2,You do NOT feel that the stock
is worth holding for the next 3 months?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.16Call Valuation Prior to
Maturity Date
Figure 7.1 (p,176)
Call Option Price
Time Value
Intrinsic value
X Stock price,S
As r or? increases,the call price moves in the direction of the arrows
XSXSCc rT 00 e
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.17
Put Early Exercise on
Non-Dividend Paying Stock
Early exercise of a put is a trade-off between
– Receiving the strike price sooner
– Risk that stock price will rise above the strike price
If the interest on the strike price outweighs the
insurance benefit,early exercise is beneficial
Example is deep in the money puts
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.18
Should Puts Be Exercised Early
Are there any advantages to
exercising a put when
S0= 60
T= 0.25
X= 100
D = 0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.19American Put Valuation Prior to
Maturity Date
Figure 7.2 (p,177)
AmericanPut Option Price
A X Stock price,S
As? or T increases,the put price moves in the direction of the arrows
0SXP
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.20European Put Valuation Prior to
Maturity Date
Figure 7.3 (p,178)
Put Option Price
Time Value (negative)
Intrinsic Value
A B X Stock price,S
As? increases,the put price moves in the direction of the arrows
0e SXp rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.21
Put-Call Parity; No Dividends
(Equation 7.3,page 174)
Consider the following 2 portfolios:
– Portfolio A,European call on a stock + PV of the
strike price in cash
– Portfolio C,European put on the stock + the stock
Both are worth MAX(ST,X ) at the maturity of the
options
They must therefore be worth the same today
– This means that
c + Xe -rT = p + S0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.22Put-Call Parity Cash Flows
(NO Dividends)
Consider the following 2 portfolios
– Portfolio A,European call on a stock +
the present value of the strike price in cash
– Portfolio C,European put on the stock + the
stock
ST <X ST >X
Buy Call 0 ST -X
Lend Xe-rT at r X X Portfolio A = MAX(ST,X)
Net Flows X ST
Buy Put X-ST 0
Buy Stock ST ST Portfolio C= MAX(ST,X)
Net Flows X ST
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.23
Put-Call Parity Graphically
(NO Dividends)
Consider the following 2 portfolios
– Portfolio X,Buy a European call on a stock +
Sell a European put
– Portfolio Y,Buy the stock + borrow the present
value of the strike price
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.24
Put-Call Parity Graphs
(NO Dividends)? Portfolio X
Portfolio Y
X ST
Payoff
X S
T
Payoff
Short PutLong Call
X S
T
Payoff
X ST
Payoff
X S
T
Payoff
Borrow Xe-rTLong Stock
X S
T
Payoff
Net Position
-X
Net Position
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.25
Put-Call Parity Graphs
(NO Dividends)
(continued)
Because these two portfolios have identical
payoffs,arbitrage theory states that they
must cost the same to establish.
Therefore,
)3.7(e
e
0
0
SpXc
XSpc
rT
rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.26
Puts,An Arbitrage Possibility?
Suppose that
c = 3
S0= 48
T= 1
r = 5%
X = 50
D = 0
What are the arbitrage possibilities
when?
p= 3?
p= 2?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.27
Puts,An Arbitrage Possibility?
(continued)
If p = 3
Therefore,because 0 < 0.44,need to
buy the left-hand side and sell the right
buy low,sell high
44.00
56.47480
e504833
e
e
1*05.0
0
0
rT
rT
XSpc
SpXc
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.28
Puts,An Arbitrage Possibility?
(continued)
Set up the tables p = 3
Algebraically,
t = 0 ST <X ST >X
Buy Call -c 0 ST-X
Sell Put p -(X-ST) 0
Sell Stock S0 -ST -ST
Lend Xe-rT at r -Xe-rT X X
Net Flows -c+p+S0- Xe-rT 0 0
Numerically,
t = 0 ST <X ST >X
Buy Call -3.00 0 ST-50
Sell Put 3.00 -(50-ST) 0
Sell Stock 48.00 -ST -ST
Lend Xe-rT at r 47.56 50 50
Net Flows 0.44 0 0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.29
Puts,An Arbitrage Possibility?
(continued)
If p = 2
Therefore,because 1.00 > 0.44,need to
buy the right-hand side and sell the left
buy low,sell high
44.01
56.47481
e504823
e
e
1*05.0
0
0
rT
rT
XSpc
SpXc
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.30
Puts,An Arbitrage Possibility?
(continued)
Set up the tables p = 2
Algebraically,
t = 0 ST <X ST >X
Buy Stock -S0 ST ST
Borrow Xe-rT at r Xe-rT -X -X
Sell Call c 0 -(ST-X)
Buy Put -p X-ST 0
Net Flows c-p-S+Xe-rT 0 0
Numerically,
t = 0 ST <X ST >X
Buy Stock -48.00 ST ST
Borrow Xe-rT at r 47.56 -50 -50
Sell Call 3.00 0 -(ST-50)
Buy Put -2.00 50-ST 0
Net Flows 0.56 0 0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.31Lower Bound for
European Options
(with Dividends)
If the stock pays dividends,whose net present value
today is denoted as D,then all that we need to do is to
replace S0 with S0-D and pretend like there are no
dividends
c > (S0-D) -Xe-rT = S0-D -Xe-rT
p > Xe-rT - (S0-D) = Xe-rT + D - S0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.32
Extensions of Put-Call Parity
American options; D = 0 (Equation 7.4 )
S0 - X < C - P < S0 - Xe -rT
European options; D > 0 Equation 7.7)
c + D + Xe -rT = p + S0
American options; D > 0 (Equation 7.8)
S0 - D - X < C - P < S0 - Xe -rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.33
Assignments
7.3,7.7,7.10,7.11,7.12,7.13,7.15,
7.17,719,7.20
Assignment Questions
Tang Yincai,Shanghai Normal University
7.1
Properties of
Stock Option Prices
Chapter 7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.2
Notation
c,European call
option price
p,European put
option price
S0,Stock price today
X,Strike price
T,Life of option
,Volatility(波动率 ) of
stock price
C,American Call option
price
P,American Put option
price
ST,Stock price at time T
D,Present value of
dividends during option’s
life
r,Risk-free rate for
maturity T with cont comp
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.3Effect of Variables on Option
Pricing (Table 7.1,page 169)
c p C PVariable
S0
X
T
r
D
+ + –+
+ ++ + + +
+ – + –
–– – +
– + – +
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.4
American vs European Options
An American option is worth
at least as much as the
corresponding European
option
C? c
P? p
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.5
Upper Bound for
Options Prices
It should be relatively easy to see that
c? S0 and C? S0
Otherwise,you could make a risk-less profit
by buying the stock and selling the option
Likewise,
p? Xe-rT and P? X
Otherwise,you could make a risk-less profit
by selling the option and investing the
proceeds at r
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.6
Lower Bound for
European Call Prices
(NO Dividends)
Consider the following positions
t = 0 ST <X ST >X
Portfolio A
Buy Call -c 0 ST - X
Lend Xe-rT at r -Xe-rT X X
Net Flows -c-Xe-rT X ST
Portfolio B
Buy one share -S0 ST ST
A is worth more than B,so it must cost more to
set it up initially,So c + Xe-rT > S0
c > max[S0 -Xe-rT,0]
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.7
Calls,An Arbitrage Possibility?
Suppose that
c = 3
S0= 52
T = 1
r= 5%
X= 50
D= 0
Is there an arbitrage possibility?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.8
Calls,An Arbitrage Possibility?
(continued)
Is c > S0 - Xe-rT?
S0 - Xe-rT =
= 52 - 50e-0.05(1.00)
= 52 - 47.56
= 4.44
Yes,an arbitrage is possible as 3 < 4.44
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.9
Calls,An Arbitrage Possibility?
(continued)
Yes,an arbitrage is possible as 3 < 4.44
t = 0 ST<X ST>X
Buy the Call -c 0 ST -X
Sell Stock S0 -ST -ST
Lend Xe-rT -Xe-rT X X
Net Flows -c+S-Xe-rT X-ST 0
Numerically,
t = 0 ST<50 ST>50
Buy the Call -3 0 ST-50
Sell Stock 52 -ST -ST
Lend Xe-rT -50e-0.05*1 50 50
Net Flows -3+52-50e-0.05*1=1.44 50-ST 0
Now
Possibly
More Later
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.10Lower Bound for
European Put Prices
(NO Dividends)
Consider the following positions
t = 0 ST <X ST >X
Portfolio C
Buy Put -p X - ST 0
Buy Stock -S0 ST ST
Net Flows -p-S0 X ST
Portfolio D
Lend Xe-rT at r -Xe-rT X X
C is worth more than D,so it must cost more
to set it up initially,So,p+S0 > Xe-rT
p > max[Xe-rT - S0,0]
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.11
Puts,An Arbitrage Possibility?
Suppose that
p= 3
S0= 48
T= 0.25
r= 5%
X= 50
D= 0
Is there an arbitrage possibility?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.12
Puts,An Arbitrage Possibility?
(continued)
Is p > Xe-rT - S0?
Xe-rT - S0 =
= 50e-0.05(0.25) - 48
= 49.37 - 48
= 1.37
No arbitrage possibility
3 > 1.37
An arbitrage is possible if p=1
The risk-less profit 0.37 can be obtained by borrowing
Xe-rT,buying Put and Stock (See slide 7.10)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.13
Call Early Exercise on
Non-Dividend Paying Stock
From earlier we know that
c > S0 - Xe-rT
Given that C? c
C > S0 - Xe-rT > S0 - X
Thus,C > S0 - X,but if exercised early
C = S0 - X
Hence,no reason to exercise early
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.14
Early Exercise
Usually,there is some chance that an
American option will be exercised early
An exception is an American call on a
non-dividend paying stock
– This should NEVER be exercised early
1,NO income is sacrificed
2,Strike price is paid later
3,Holding the call provides insurance against
stock price falling below strike price
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.15
An Extreme Situation
For an American call option with
S0 = 100,T = 0.25,X = 60,r = 5%,D = 0
– Should you exercise if …
1,You want to hold the stock over
the next 3 months?
2,You do NOT feel that the stock
is worth holding for the next 3 months?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.16Call Valuation Prior to
Maturity Date
Figure 7.1 (p,176)
Call Option Price
Time Value
Intrinsic value
X Stock price,S
As r or? increases,the call price moves in the direction of the arrows
XSXSCc rT 00 e
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.17
Put Early Exercise on
Non-Dividend Paying Stock
Early exercise of a put is a trade-off between
– Receiving the strike price sooner
– Risk that stock price will rise above the strike price
If the interest on the strike price outweighs the
insurance benefit,early exercise is beneficial
Example is deep in the money puts
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.18
Should Puts Be Exercised Early
Are there any advantages to
exercising a put when
S0= 60
T= 0.25
X= 100
D = 0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.19American Put Valuation Prior to
Maturity Date
Figure 7.2 (p,177)
AmericanPut Option Price
A X Stock price,S
As? or T increases,the put price moves in the direction of the arrows
0SXP
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.20European Put Valuation Prior to
Maturity Date
Figure 7.3 (p,178)
Put Option Price
Time Value (negative)
Intrinsic Value
A B X Stock price,S
As? increases,the put price moves in the direction of the arrows
0e SXp rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.21
Put-Call Parity; No Dividends
(Equation 7.3,page 174)
Consider the following 2 portfolios:
– Portfolio A,European call on a stock + PV of the
strike price in cash
– Portfolio C,European put on the stock + the stock
Both are worth MAX(ST,X ) at the maturity of the
options
They must therefore be worth the same today
– This means that
c + Xe -rT = p + S0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.22Put-Call Parity Cash Flows
(NO Dividends)
Consider the following 2 portfolios
– Portfolio A,European call on a stock +
the present value of the strike price in cash
– Portfolio C,European put on the stock + the
stock
ST <X ST >X
Buy Call 0 ST -X
Lend Xe-rT at r X X Portfolio A = MAX(ST,X)
Net Flows X ST
Buy Put X-ST 0
Buy Stock ST ST Portfolio C= MAX(ST,X)
Net Flows X ST
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.23
Put-Call Parity Graphically
(NO Dividends)
Consider the following 2 portfolios
– Portfolio X,Buy a European call on a stock +
Sell a European put
– Portfolio Y,Buy the stock + borrow the present
value of the strike price
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.24
Put-Call Parity Graphs
(NO Dividends)? Portfolio X
Portfolio Y
X ST
Payoff
X S
T
Payoff
Short PutLong Call
X S
T
Payoff
X ST
Payoff
X S
T
Payoff
Borrow Xe-rTLong Stock
X S
T
Payoff
Net Position
-X
Net Position
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.25
Put-Call Parity Graphs
(NO Dividends)
(continued)
Because these two portfolios have identical
payoffs,arbitrage theory states that they
must cost the same to establish.
Therefore,
)3.7(e
e
0
0
SpXc
XSpc
rT
rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.26
Puts,An Arbitrage Possibility?
Suppose that
c = 3
S0= 48
T= 1
r = 5%
X = 50
D = 0
What are the arbitrage possibilities
when?
p= 3?
p= 2?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.27
Puts,An Arbitrage Possibility?
(continued)
If p = 3
Therefore,because 0 < 0.44,need to
buy the left-hand side and sell the right
buy low,sell high
44.00
56.47480
e504833
e
e
1*05.0
0
0
rT
rT
XSpc
SpXc
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.28
Puts,An Arbitrage Possibility?
(continued)
Set up the tables p = 3
Algebraically,
t = 0 ST <X ST >X
Buy Call -c 0 ST-X
Sell Put p -(X-ST) 0
Sell Stock S0 -ST -ST
Lend Xe-rT at r -Xe-rT X X
Net Flows -c+p+S0- Xe-rT 0 0
Numerically,
t = 0 ST <X ST >X
Buy Call -3.00 0 ST-50
Sell Put 3.00 -(50-ST) 0
Sell Stock 48.00 -ST -ST
Lend Xe-rT at r 47.56 50 50
Net Flows 0.44 0 0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.29
Puts,An Arbitrage Possibility?
(continued)
If p = 2
Therefore,because 1.00 > 0.44,need to
buy the right-hand side and sell the left
buy low,sell high
44.01
56.47481
e504823
e
e
1*05.0
0
0
rT
rT
XSpc
SpXc
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.30
Puts,An Arbitrage Possibility?
(continued)
Set up the tables p = 2
Algebraically,
t = 0 ST <X ST >X
Buy Stock -S0 ST ST
Borrow Xe-rT at r Xe-rT -X -X
Sell Call c 0 -(ST-X)
Buy Put -p X-ST 0
Net Flows c-p-S+Xe-rT 0 0
Numerically,
t = 0 ST <X ST >X
Buy Stock -48.00 ST ST
Borrow Xe-rT at r 47.56 -50 -50
Sell Call 3.00 0 -(ST-50)
Buy Put -2.00 50-ST 0
Net Flows 0.56 0 0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.31Lower Bound for
European Options
(with Dividends)
If the stock pays dividends,whose net present value
today is denoted as D,then all that we need to do is to
replace S0 with S0-D and pretend like there are no
dividends
c > (S0-D) -Xe-rT = S0-D -Xe-rT
p > Xe-rT - (S0-D) = Xe-rT + D - S0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.32
Extensions of Put-Call Parity
American options; D = 0 (Equation 7.4 )
S0 - X < C - P < S0 - Xe -rT
European options; D > 0 Equation 7.7)
c + D + Xe -rT = p + S0
American options; D > 0 (Equation 7.8)
S0 - D - X < C - P < S0 - Xe -rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
7.33
Assignments
7.3,7.7,7.10,7.11,7.12,7.13,7.15,
7.17,719,7.20
Assignment Questions
