Options
Dr. Lynn Phillips Kugele
FIN 338
Options Review
• Mechanics of Option Markets
• Properties of Stock Options
• Valuing Stock Options:
– The Black-Scholes Model
OPT-2
Mechanics of Options
Markets
OPT-3
Option Basics
• Option = derivative security
– Value “derived” from the value of the
underlying asset
• Stock Option Contracts
– Exchange-traded
– Standardized
• Facilitates trading and price reporting.
– Contract = 100 shares of stock
OPT-4
Put and Call Options
• Call option
– Gives holder the right but not the
obligation to buy the underlying asset at
a specified price at a specified time
• Put option
– Gives the holder the right but not the
obligation to sell the underlying asset at
a specified price at a specified time
OPT-5
Options on Common Stock
1.
2.
3.
4.
5.
•
Identity of the underlying stock
Strike or Exercise price
Contract size
Expiration date or maturity
Exercise cycle
American or European
6. Delivery or settlement procedure
OPT-6
Option Exercise
• American-style
– Exercisable at any time up to and
including the option expiration date
– Stock options are typically American
• European-style
– Exercisable only at the option expiration
date
OPT-7
Option Positions
• Call positions:
– Long call = call “holder”
• Hopes/expects asset price will increase
– Short call = call “writer”
• Hopes asset price will stay or decline
• Put Positions:
– Long put = put “holder”
• Expects asset price to decline
– Short put = put “writer”
• Hopes asset price will stay or increase
OPT-8
Option Writing
• The act of selling an option
• Option writer = seller of an option
contract
– Call option writer obligated to sell the
underlying asset to the call option holder
– Put option writer obligated to buy the
underlying asset from the put option holder
– Option writer receives the option premium
when contract entered
OPT-9
Option Payoffs & Profits
Notation:
•
•
•
•
S0 = current stock price per share
ST = stock price at expiration
X = option exercise or strike price
C = American call option premium per share
• c = European call option premium
• P = American put option premium per share
• p = European put option premium
• r = risk free rate
• T = time to maturity in years
OPT-10
Option Payoffs & Profits
Call Holder
Payoff to Call Holder
(S - X)
0
if S >X
if S < X
= Max (S-X,0)
Profit to Call Holder
Payoff - Option Premium
Profit =Max (S-X, 0) - C
OPT-11
Option Payoffs & Profits
Call Writer
Payoff to Call Writer
- (S - X)
0
if S > X = -Max (S-X, 0)
if S < X = Min (X-S, 0)
Profit to Call Writer
Payoff + Option Premium
Profit = Min (X-S, 0) + C
OPT-12
Payoff & Profit Profiles for Calls
Call Payoff and Profit
X = $20 c = $5.00
Stock
Call Holder
Price
Payoff
Profit
0
$0
-$5
$10
$0
-$5
$20
$0
-$5
$30
$10
$5
$40
$20
$15
Payoff:
Profit:
Max(S-X,0)
Max (S-X,0) – c
Call Writer
Payoff
Profit
$0
$5
$0
$5
$0
$5
-$10
-$5
-$20
-$15
-Max(S-X,0)
-[Max (S-X, 0)-p]
OPT-13
Payoff & Profit Profiles for Calls
Payoff
Profit
Call Holder
Profit
0
Call Writer
Profit
Stock Price
OPT-14
Option Payoffs and Profits
Put Holder
Payoffs to Put Holder
0
(X - S)
if S > X
if S < X
= Max (X-S, 0)
Profit to Put Holder
Payoff - Option Premium
Profit = Max (X-S, 0) - P
OPT-15
Option Payoffs and Profits
Put Writer
Payoffs to Put Writer
0
if S > X
-(X - S) if S < X
= -Max (X-S, 0)
= Min (S-X, 0)
Profits to Put Writer
Payoff + Option Premium
Profit = Min (S-X, 0) + P
OPT-16
Payoff & Profit Profiles for Puts
Put Payoff and Profit
X = $20 p = $5.00
Stock
Put Holder
Price
Payoff
Profit
0
$20
$15
$10
$10
$5
$20
$0
-$5
$30
$0
-$5
$40
$0
-$5
Payoff:
Profit:
Max(X-S,0)
Max (X-S,0) – p
Put Writer
Payoff
Profit
-$20
-$15
-$10
-$5
$0
$5
$0
$5
$0
$5
-Max(X-S,0)
-[Max (X-S, 0)-p]
OPT-17
Payoff & Profit Profiles for Puts
Profits
Put Writer
Profit
0
Put Holder
Profit
Stock Price
OPT-18
Option Payoffs and Profits
CALL
PUT
Holder: Payoff
(Long) Profit
Max (S-X,0)
Max (S-X,0) - C
“Bullish”
Max (X-S,0)
Max (X-S,0) - P
“Bearish”
Writer: Payoff
(Short) Profit
Min (X-S,0)
Min (X-S,0) + C
“Bearish”
Min (S-X,0)
Min (S-X,0) + P
“Bullish”
S = P = Value of firm at expiration
X = Face Value of Debt
OPT-19
Long Call
Long Call Profit = Max(S-X,0) - C
Call option premium (C) = $5, Strike price (X) = $100.
30 Profit ($)
20
10
70
0
-5
80
90
100
Terminal
stock price (S)
110 120 130
OPT-20
Properties of Stock Options
OPT-21
Notation
c
p
S0
ST
X
T

r
= European call option price (C = American)
= European put option price (P = American)
= Stock price today
=Stock price at option maturity
= Strike price
= Option maturity in years
= Volatility of stock price
= Risk-free rate for maturity T with
continuous compounding
OPT-22
American vs. European Options
An American option is worth at least
as much as the corresponding
European option
Cc
Pp
OPT-23
Factors Influencing Option
Values
Effect on Option
Value
American
Call
Put
Input Factor
Underlying stock price
S
Strike price of option contract
X
Time remaining to expiration
T
Volatility of the underlying stock price
σ
Risk-free interest rate
r
+
+
+
+
+
+
+
-
OPT-24
Effect on Option Values
Underlying Stock Price (S) & Strike Price (K)
• Payoff to call holder: Max (S-X,0)
– As S , Payoff increases; Value increases
– As X , Payoff decreases; Value decreases
• Payoff to Put holder: Max (X-S, 0)
– As S , Payoff decreases; Value decreases
– As X , Payoff increases; Value increases
OPT-25
Option Price Quotes
Calls
MSFT (MICROSOFT CORP)
$ 25.98
July 2008 CALLS
Strike
Last Sale
Bid
Ask
Vol
Open Int
15.00
10.85
10.95
11.10
10
85
17.50
10.54
8.45
8.55
0
33
20.00
6.00
6.00
6.05
4
729
22.50
3.60
3.55
3.65
195
3891
24.00
2.30
2.24
2.27
422
2464
25.00
1.50
1.45
1.48
3190
10472
26.00
0.83
0.83
0.85
2531
15764
27.50
0.31
0.29
0.31
2554
61529
OPT-26
Option Price Quotes
Puts
MSFT (MICROSOFT CORP)
$ 25.98
July 2008 PUTS
Strike
Last Sale
Bid
Ask
Vol
Open Int
15.00
0.01
0.00
0.01
0
2751
17.50
0.01
0.00
0.02
0
2751
20.00
0.01
0.01
0.02
0
5013
22.50
0.03
0.03
0.04
13
4788
24.00
0.11
0.11
0.12
50
25041
25.00
0.25
0.24
0.25
399
7354
26.00
0.45
0.45
0.47
10212
51464
27.50
0.80
0.82
0.84
2299
39324
OPT-27
Effect on Option Values
Time to Expiration = T
• For an American Call or Put:
– The longer the time left to maturity, the greater
the potential for the option to end in the
money, the grater the value of the option
• For a European Call or Put:
– Not always true due to restriction on exercise
timing
OPT-28
Option Price Quotes
MSFT (MICROSOFT CORP)
STRIKE = $25.00
CALLS
Last Sale
July 2008
1.42
August 2008
1.80
October 2008
2.36
January 2009
3.10
Bid
1.45
1.85
2.43
3.15
Ask
1.48
1.87
2.46
3.20
Vol
355
257
41
454
Open Int
10472
927
3309
59244
PUTS
July 2008
August 2008
October 2008
January 2009
Bid
0.45
0.80
1.39
2.06
Ask
0.47
0.82
1.41
2.08
Vol
419
401
215
2524
Open Int
51464
1591
25323
155877
Last Sale
0.47
0.81
1.43
2.09
25.98
OPT-29
Effect on Option Values
Volatility = σ
• Volatility = a measure of uncertainty about
future stock price movements
– Increased volatility increased upside
potential and downside risk
• Increased volatility is NOT good for the
holder of a share of stock
• Increased volatility is good for an option
holder
– Option holder has no downside risk
– Greater potential for higher upside payoff
OPT-30
Effect on Option Values
Risk-free Rate = r
• As r :
–Investor’s required return increases
–The present value of future cash
flows decreases
= Increases value of calls
= Decreases value of puts
OPT-31
Valuing Stock Options:
The Black-Scholes Model
OPT-32
BSOPM
Black-Scholes (-Merton) Option Pricing Model
• “BS” = Fischer Black and Myron Scholes
– With important contributions by Robert Merton
• BSOPM published in 1973
• Nobel Prize in Economics in 1997
• Values European options on non-dividend
paying stock
OPT-33
Concepts Underlying Black-Scholes
• Option price and stock price depend on
same underlying source of uncertainty
• A portfolio consisting of the stock and the
option can be formed which eliminates
this source of uncertainty (riskless).
– The portfolio is instantaneously riskless
– Must instantaneously earn the risk-free
rate
OPT-34
Assumptions Underlying BSOPM
1. Stock price behavior corresponds to the
lognormal model with μ and σ constant
2. No transactions costs or taxes. All securities
are perfectly divisible
3. No dividends on stocks during the life of the
option
4. No riskless arbitrage opportunities
5. Security trading is continuous
6. Investors can borrow & lend at the risk-free
rate
7. The short-term rate of interest, r, is constant
OPT-35
Notation
• c and p = European option prices
(premiums)
• S0 = stock price
• X = strike or exercise price
• r = risk-free rate
• σ = volatility of the stock price
• T = time to maturity in years
OPT-36
Formula Functions
• ln(S/X) = natural log of the "moneyness" term
• N(d) = the probability that a normally
distributed variable with a mean of zero
and a standard deviation of 1 is less than x
• N(d1) and N(d2) denote the standard normal
probability for the values of d1 and d2.
• Formula makes use of the fact that:
N(-d1) = 1 - N(d1)
OPT-37
The Black-Scholes Formulas
c  S0 N ( d 1 )  X e
 rT
N( d 2 )
p  Xe  rT N ( d 2 )  S0 N ( d 1 )
where :
d1 
2
ln( S0 / X )  ( r   / 2 )T
 T
d 2  d1   T
OPT-38
BSOPM
Example
Given:
S0 = $42
X = $40
d1 
r = 10%
T = 0.5
σ = 20%
ln( S0 / X )  ( r   2 / 2 )T
 T
ln( 42 40 )  ( 0.10  0.20 2 2 )  0.5
d1 
 0.7693
0.20 0.50
d 2  d1   T
d 2  0.7693  0.20 0.50  0.6278
OPT-39
BSOPM
Call Price Example
d1 = 0.7693
N(0.7693) = 0.7791
d2 = 0.6278
N(0.6278) = 0.7349
c  S0 N ( d 1 )  X e  rT N ( d 2 )
c  40(0.7791) - 42e -.10.5 (0.7349)
c  $4.76
OPT-40
BSOPM in Excel
• N(d1):
=NORMSDIST(d1)
Note the “S” in the function
“S” denotes “standard normal”
~ Φ(0,1)
=NORMDIST() → Normal distribution
Mean and variance must be specified
~N(μ,σ2 )
OPT-41