7.1
Properties of
Stock Option Prices
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
FACTORS AFFECTING OPTION
PRICES
•The current stock price
•The strike price
•The time to expiration
•The volatility of the stock price
•The risk free rate
•The dividend
expected during the life of the option
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.2
7.3
The owner of a EUROPEAN option can only exercise
at the maturity of the option.
The owner of an AMERICAN option can exercise any
time before the maturity of the option
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
WHICH ONE IS MORE EXPENSIVE ?
MARCH CALL
or
JULY CALL
JULY 30 CALL IF IT IS AMERICAN
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.4
7.5
TIME TO EXPIRATION
American Put and call options become MORE valuable as the time
to expiration increase because the holder of the option has all the
exercise opportunity and time.
European call and put options do not necessarily increase in value
as the time to expiration increases because the owner of the option
can only exercise at expiration.
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.6
VOLATILITY
It is a measure of how uncertain we are about
future stock price movements.
Volatility (s) = Standard deviation of the return on a stock
in a length of time t
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
DIVIDENDS
7.7
WHAT DOES "EX DIVIDEND " MEAN ?
A stock goes ex-dividend the day the company pays the dividend
WHAT HAPPENS TO THE VALUE OF THE STOCK ?
Stock price is reduced by the amount of the dividend at the opening
HOW DOES IT AFFECT CALL & PUT ?
•The value of a put option is positively related to the size of any
anticipated dividend
•The value of a call option is inversely related to the size of any
anticipated dividend.
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.8
Notation
• c : European call
•
•
•
•
•
option price
p : European put
option price
S0 : Stock price today
X : Strike price
T : Life of option
s: 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
7.9
UPPER AND LOWER BOUNDS FOR OPTION
PRICES
Can the price of a call option be worth more than the stock ?
NO
C  S0
The stock price is an upper bound to the option price.
What if that did not hold ?
ARBITRAGE
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.10
No matter how low the stock price becomes, the option (p) can never
be worth more than the price of the stock (X)
pX
For EUROPEAN put options, we know, that at expiration (T), its
value will not be worth more than X. Its value today cannot be more
than the present value of X :
p  Xe-rT
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.11
LOWER BOUNDS FOR EUROPEAN CALLS
or
THE THEORETICAL MINIMUM
A lower bound for the price of a EUROPEAN call option is :
S0 -
-rT
Xe
- D
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
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
7.12
EXAMPLE
7.13
Suppose a stock is trading at $20 (S0). The strike price (X) of the
call option is $18, the risk free rate r is 10% and T = 1 year.
S - Xe-rT = $3.71
The lower bound is :
0
The call option is trading at $3.
An arbitrageur can buy the call and short the stock
In cash flow analysis
20 - 3 = $17 invested at 10% per
annum
17e0.1 = $18.79
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.14
$19
At expiration
$17
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.15
STOCK IS AT $19
The arbitrageur exercises the option :
$18.79 - $18.00 = $0.79
RIGHT TO BUY AT 18
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.16
STOCK = $17
The arbitrageur buys the stock back in the market
and the short position is closed out.
$18.79 - $17.00 = $1.79
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Effect of Variables on Option
Pricing
Variable
S0
X
T
s
r
D
c
+
–
?
+
+
–
p
–
+?
+
–
+
C
+
–
+
+
+
–
P
–
+
+
+
–
+
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.17
7.18
Lower Bound for European Call
Option
c  S0 -Xe
-rT
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
LOWER BOUNDS FOR EUROPEAN PUT
OPTIONS
or
THE THEORETICAL MINIMUM
7.19
A lower bound for the price of a EUROPEAN put option is :
-rT
Xe
- S0 + D
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
7.20
EXAMPLE
7.21
Suppose a stock is trading at $37(S0). The strike price (X) of the
put option is $40, the risk free rate r is 5% and T = 0.5 year.
-rT
The lower bound is :
Th put option is trading at $1.00
Xe
- S0 = $2.01
An arbitrageur can buy the put and the stock by
borrowing $38
In cash flow analysis
$38 borrowed at 5% per
annum
38e0.5 X 0.05 = $38.96 will have to be paid back in 6
months.
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.22
$45
At expiration
$39
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.23
STOCK IS AT $45
The arbitrageur discards the put, sells the stock in the market and
repays the loan :
$45 - $38.96 = $6.04
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.24
STOCK IS AT $39
The arbitrageur exercises the option to sell the stock for $40,
repays the loan and makes a profit of :
$40 - $38.96 = $1.04
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.25
Lower Bound for European Put
options
p  Xe
-rT -
S0
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.26
PUT - CALL PARITY
Put-call parity is a fundamental relationship that must exist
between the prices of a put option and call option if both have
the same underlier, strike price and expiration date.
The relationship is derived using arbitrage arguments.
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.27
Put-Call Parity; No Dividends
• Consider the following 2 portfolios:
– Portfolio A: European call on a stock + PV of
the strike price in cash
– Portfolio B: 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 © 1999 by John C. Hull
7.28
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.29
BOTH PORTFOLIOS HAVE IDENTICAL PAYOFF PATTERNS
THEY MUST HAVE THE SAME VALUE TODAY
ARBITRAGE OPPORTUNITY
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.30
PUT - CALL PARITY
C+
-rt
Xe
= P + S0
Put-call parity is often used as a simple test of option
pricing models. Any option pricing model which
produces put and call prices that do not satisfy put-call
parity must be rejected as unsound. Such a model will
suggest trading opportunities where none exist.
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.31
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
C+
A
-rt
Xe
= P + S0
B
VALUE OF PORTFOLIO "A" ?
$32.26
VALUE OF PORTFOLIO "B" ?
$33.25
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.32
7.33
BUY "A" AND SELL "B"
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
For p= 2.25
7.34
Portfolio B is overpriced relative to portfolio A
Portf A :c + Xe-rT = $32.26 Portf B : p + S0 = $33.25
Arbitrage : buy securities in A and sell securities in B
Buy the call and short the put and the stock...
CASH FLOW ANALYSIS…
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.35
Cash flow : -3 + 2.25 +31 = 30.25
(invested at 10% for 3months = 31.02)
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Suppose the stock at expiration
is greater than $30.
Call is exercised
7.36
Suppose the stock at expiration
is less than $30
Put is exercised
LONG THE SHARE AT $30
Cash Flow: $31.02 - $30 = $1.02 (close out the short position
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
For p= 1
7.37
Portfolio A is overpriced relative to portfolio B
Portf A :c + Xe-rT = $32.26 Portf B: p + S0 = $32.00
Arbitrage : buy securities in B and sell securities in A
Buy the put and the stock and short the call
Investment of : - $31 - $1 + $3 = -$29
(29e0.10 x 0.25 = $29.73)
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
Suppose the stock at expiration
is greater than $30.
Call is exercised
7.38
Suppose the stock at expiration
is less than $30
Put is exercised
SHORT THE SHARE AT $30
Cash Flow: $30.00 - $29.73 = $0.27
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.39
DOES PUT/CALL PARITY EXIST
FOR AMERICAN TYPE
OPTIONS ?
NO
WHY ?
Because American options can be exerciced before
Expiration and would not the same value today
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.40
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
7.41
An Extreme Situation
• For an American call option:
S0 = 100; T = 0.25; X = 60; 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
7.42
7.43
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
7.44
The Impact of Dividends on
Lower Bounds to Option Prices
c  S 0  D  Xe
p  D  Xe
 rT
 rT
 S0
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.45
RELATIONSHIP BETWEEN
AMERICAN PUT AND CALL PRICES
Put-Call parity apply for European options but can be derived to
American option :
S0 - X  C - P  S0 - Xe-rT
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.46
EXAMPLE
An American Call option with strike price (X) = $20.00 and
maturity in 5 months (T) is worth $1.50.
Current stock price (S0) = $19.00 and r = 10%
S0 - X  C - P  S0 - Xe-rT
19 - 20  1.50 -P  19 - 20e-0.10 x 5/12
$1.68  P  $2.50
Upper and lower bounds for price of an American put with the same strike
price and expiration date as the American call are $2.50 and $1.68
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull