7.1
Properties of
Stock Option Prices
Chapter 7
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Effect of Variables on Option
Pricing (Table 7.1, page 169)
Variable
S0
X
T

r
D
c
+
–
?
+
+
–
p
–
+?
+
–
+
C
+
–
+
+
+
–
P
–
+
+
+
–
+
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.2
7.3
Assumptions
• There are no transaction costs
• All trading profits are subject to the
same tax rate
• Borrowing and lending at the risk-free
interest rate is possible
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.4
Notation
• c : European call
•
•
•
•
•
option price
p : European put
option price
S0 : Stock price today
X : Strike price
T : Life of option
: Volatility 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 © 1999 by John C. Hull
7.5
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 © 1999 by John C. Hull
Upper Bounds
• Stock price is an upper bound to the
American or European call
c ≤ So and C ≤ So, if this would not
hold, an easy arbitrage profit would be available
by buying the stock and selling the call option.
• Strike price is an upper bound to the
American or European put
p ≤ X and P ≤ X
• For European put
p  Xe
 rT
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.6
7.7
Lower bound
• European calls on non-dividend-paying
stock
S o  Xe
 rT
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.8
Calls: An Arbitrage
Opportunity?
• Suppose that
c =3
T =1
X = 18
S0 = 20
r = 10%
D=0
• Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.9
LBEurCall  S 0  Xe rT  20  18e 0,1  3,71
What if the price of a European Call is
c  3,00$
and arbitrageur buys the call and
shorts the stock
$20  $3  $17
17e  17e
rT
and invests on r for a year
0,1
 18,79
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.10
If the stock price is greater than 18,
arbitrageur exercises the option,
closes short and makes a profit of $0,79.
$18,79  $18  $0,79
If the stock price is less than 18,
(for ex. 17) the stock is bought in
the market and a short position is
closed out with even greater profit
of 1,79.
$18,79  $17  $1,79
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.11
Portfolio A: 1 Eur Call + Xe-rT
Portfolio B: 1 share of stock
Cash in A is invested in r and will grow to X in time T
If ST>X, c is exercised and A =ST
If ST<X, c is worthless and A =X
At time T portfolio A is worth max (ST, X)
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.12
Portfolio B is worth ST at time T
Portfolio A is worth at least as portfolio B at time T
Hence, A must be worth at least as B today
c  Xe
 rT
 S0
c  S 0  Xe
 rT
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.13
Lower Bound for European Call
Option Prices; No Dividends
(Equation 7.1, page 172)
Since c must be non negative i.e. c ≥ 0

c  max So  Xe
 rT
,0

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.14
Lower bound
• European puts on non-dividend-paying
stock
Xe
 rT
 S0
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Puts: An Arbitrage
Opportunity?
• Suppose that
p =1
T = 0.5
X = 40
S0 = 37
r =5%
D =0
• Is there an arbitrage
opportunity?
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.15
7.16
LBEurPut  Xe rT  S 0  40e 0,050,5  37  2,01
What if the price of a European Put is
p  1,00$
and arbitrageur borrows $38 for six
months and buys a put and the stock
$37  1$  $38
38e
0, 0505
at the end of six months
arbitrageur will have to pay
 $38,96
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.17
If the stock price is bellow 40,
arbitrageur exercises the option,
Sells the stock and makes a profit of $1,04.
$40,00  $38,96  $1,04
If the stock price is greater than 40,
(for ex. 42) the option is discarded
and the stock is sold in the market
with even greater profit of 3,04.
$42,00  $38,96  $3,04
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.18
Portfolio C: 1 Eur Put +1 share
Portfolio D: cash of Xe-rT
If ST<X, p is exercised at T and C =X
If ST>X, p is worthless and C =ST
At time T portfolio C is worth max (ST, X)
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.19
Assuming cash is invested at r, D is worth X at T
Portfolio C is worth at list as portfolio D at time T
Hence, C must be worth at least as D today
p  S 0  Xe
p  Xe
 rT
 rT
 S0
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Lower Bound for European Put7.20
Prices; No Dividends
(Equation 7.2, page 173)
Since p must be non negative i.e. p ≥ 0
p  max (Xe
-rT -
S0, 0)
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
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
This relationship is known as put-call parity
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.22
Arbitrage Opportunities
• Suppose that
c =3
S0 = 31
T = 0.25
r = 10%
X =30
D =0
• What are the arbitrage
possibilities when
p = 2.25 ?
p =1?
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.23
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
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.24
An Extreme Situation
• For an American call option:
S0 = 50; T = 0.083; X = 40; D = 0
Should you exercise immediately?
• What should you do if
1 You want to hold the stock for 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 © 1999 by John C. Hull
Reasons For Not Exercising a
Call Early
(No Dividends )
• No income is sacrificed
• We delay paying the strike price
• Holding the call provides
insurance against stock price
falling below strike price
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.25
7.26
Should Puts Be Exercised
Early ?
Are there any advantages to
exercising an American put
when
S0 = 60; T = 0.25; r=10%
X = 100; D = 0
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.27
The Impact of Dividends on
Lower Bounds to Option Prices
(Equations 7.5 and 7.6, page 179)
c  S 0  D  Xe
p  D  Xe
 rT
 rT
 S0
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.28
Extensions of Put-Call Parity
• American options; D = 0
S0 - X < C - P < S0 - Xe -rT
Equation 7.4 p. 178
• European options; D > 0
c + D + Xe -rT = p + S0
Equation 7.7 p. 180
• American options; D > 0
S0 - D - X < C - P < S0 - Xe -rT
Equation 7.8 p. 180
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull