Chapter 15
Principles of
Options and Option Pricing
Portfolio Construction, Management, & Protection, 4e, Robert A. Strong
Copyright ©2006 by South-Western, a division of Thomson Business & Economics. All rights reserved.
1
We sent the first draft of our paper to the Journal of Political
Economy and promptly got back a rejection letter. We then
sent it to the Review of Economics and Statistics, where it
was also rejected.
Merton Miller and Eugene Fama…then took an interest in the
paper and gave us extensive comments on it. They
suggested to the JPE that perhaps the paper was worth
more serious consideration. The journal then accepted the
paper.
Fischer Black, on his journal
article with Myron Scholes that
gave birth to the Black-Scholes
2
Option Pricing Model
Outline
 Introduction
 Option
Principles
 Option Pricing
3
Introduction
 Innovations
in stock options have been
among the most important developments in
finance in the last 20 years
 The cornerstone of option pricing is the
Black-Scholes Option Pricing Model
(OPM)
• Delta is the most important OPM progeny to
the portfolio manager
4
Option Principles
 Why
Options Are a Good Idea
 What Options Are
 Standardized Option Characteristics
 Where Options Come From
 Where and How Options Trade
 The Option Premium
 Sources of Profits and Losses with Options
5
Why Options Are a Good Idea
 Options:
• Give the marketplace opportunities to adjust
risk or alter income streams that would
otherwise not be available
• Provide financial leverage
• Can be used to generate additional income from
investment portfolios
6
Why Options Are
a Good Idea (cont’d)
 The
investment process is dynamic:
• The portfolio manager needs to constantly
reassess and adjust portfolios with the arrival of
new information
 Options
are more convenient and less
expensive than wholesale purchases or sales
of stock
7
What Options Are
 Call
Options
 Put Options
8
Call Options
 A call
option gives you the right to buy
within a specified time period at a specified
price
 The
owner of the option pays a cash
premium to the option seller in exchange
for the right to buy
9
Practical Example
of a Call Option
10
Put Options
 A put
option gives you the right to sell
within a specified time period at a specified
price
 It
is not necessary to own the asset before
acquiring the right to sell it
11
Standardized
Option Characteristics
 All
exchange-traded options have
standardized expiration dates
• The Saturday following the third Friday of
designated months for most options
• Investors typically view the third Friday of the
month as the expiration date
12
Standardized
Option Characteristics (cont’d)
 The
striking price of an option is the
predetermined transaction price
• In multiples of $2.50 (for stocks priced $25.00
or below) or $5.00 (for stocks priced higher
than $25.00)
• There is usually at least one striking price
above and one below the current stock price
13
Standardized
Option Characteristics (cont’d)
 Puts
and calls are based on 100 shares of the
underlying security
• The underlying security is the security that the
option gives you the right to buy or sell
• It is not possible to buy or sell odd lots of
options
14
Where Options Come From
 Introduction
 Opening
and Closing Transactions
 Role of the Options Clearing Corporation
15
Introduction
 If
you buy an option, someone has to sell it
to you
 No
set number of put or call options exists
• The number of options in existence changes
every day
• Option can be created and destroyed
16
Opening and
Closing Transactions
 The
first trade someone makes in a
particular option is an opening transaction
• An opening transaction that is the sale of an
option is called writing an option
17
Opening and
Closing Transactions (cont’d)
 The
trade that terminates a position by
closing it out is a closing transaction
• Options have fungibility
– Market participants can reverse their positions by
making offsetting trades
– e.g., the writer of an option can close out the
position by buying a similar one
18
Opening and
Closing Transactions (cont’d)
 The
owner of an option will ultimately:
• Sell it to someone else
• Let it expire or
• Exercise it
19
Role of the
Options Clearing Corporation
 The
Options Clearing Corporation (OCC):
• Positions itself between every buyer and seller
• Acts as a guarantor of all option trades
• Regulates the trading activity of members of
the various options exchanges
• Sets minimum capital requirements
• Provides for the efficient transfer of funds
among members as gains or losses occur
20
OCC-Related
Information on the Web
21
Where and How Options Trade

In the United States, options trade on five
principal exchanges:
• Chicago Board Options Exchange (CBOE)
• American Stock Exchange (AMEX)
• Philadelphia Stock Exchange
• Pacific Stock Exchange
• International Securities Exchange
22
Where and How
Options Trade (cont’d)
 AMEX
and Philadelphia Stock Exchange
options trade via the specialist system
• All orders to buy or sell a particular security
pass through a single individual (the specialist)
• The specialist:
– Keeps an order book with standing orders from
investors and maintains the market in a fair and
orderly fashion
– Executes trades close to the current market price if
no buyer or seller is available
23
Where and How
Options Trade (cont’d)
 CBOE
and Pacific Stock Exchange options
trade via the marketmaker system
• Competing marketmakers trade in a specific
location on the exchange floor near the order
book official
• Marketmakers compete against one another for
the public’s business
24
Where and How
Options Trade (cont’d)
 Any
given option has two prices at any
given time:
• The bid price is the highest price anyone is
willing to pay for a particular option
• The asked price is the lowest price at which
anyone is willing to sell a particular option
25
The Option Premium
 Intrinsic Value
and Time Value
 The Financial Page Listing
26
Intrinsic Value and Time Value
 The
price of an option has two components:
• Intrinsic value:
– For a call option equals the stock price minus the
striking price
– For a put option equals the striking price minus the
stock price
• Time value equals the option premium minus
the intrinsic value
27
Intrinsic Value and
Time Value (cont’d)
 An
option with no intrinsic value is out of
the money
 An
option with intrinsic value is in the
money
 If
an option’s striking price equals the stock
price, the option is at the money
28
The Financial Page Listing
 The
following slide shows an example from
the online edition of The Wall Street
Journal:
• The current price for a share of Disney stock is
$21.95
• Striking prices from $20 to $25 are available
• The expiration month is in the second column
• The option premiums are provided in the “Last”
column
29
The Financial Page Listing
30
The Financial
Page Listing (cont’d)
 Investors
identify an option by company,
expiration, striking price, and type of
option:
Disney JUN 22.50 Call
Company
Expiration
Striking
Price
Type
31
The Financial
Page Listing (cont’d)

The Disney JUN 22.50 Call is out of the money
• The striking price is greater than the stock price
• The time value is $0.25

The Disney JUN 22.50 Put is in the money
• The striking price is greater than the stock price
• The intrinsic value is $22.50 - $21.95 = $0.55
• The time value is $1.05 - $0.55 = $0.50
32
The Financial
Page Listing (cont’d)
 As
an option moves closer to expiration, its
time value decreases
• Time value decay
 An
option is a wasting asset
• Everything else being equal, the value of an
option declines over time
33
Sources of Profits and
Losses with Options
 Option
Exercise
 Exercise Procedures
34
Option Exercise
 An
American option can be exercised at
any time prior to option expiration
• It is typically not advantageous to exercise
prior to expiration since this amount to
foregoing time value
 European
options can be exercised only at
expiration
35
Exercise Procedures
 The
owner of an option who decides to
exercise the option:
• Calls her broker
• Must put up the full contract amount for the
option
– The premium is not a down payment on the option
terms
36
Exercise Procedures (cont’d)
 The
option writer:
• Must be prepared to sell the necessary shares to
the call option owner
• Must be prepared to buy shares of stock from
the put option owner
37
Exercise Procedures (cont’d)
 In
general, you should not buy an option
with the intent of exercising it:
• Requires two commissions
• Selling the option captures the full value
contained in an option
38
Profit and Loss Diagrams
 For
the Disney JUN 22.50 Call buyer:
Breakeven Point = $22.75
Maximum profit
is unlimited
$0
-$0.25
Maximum loss
$22.50
39
Profit and Loss Diagrams
(cont’d)
 For
the Disney JUN 22.50 Call writer:
Maximum profit
Breakeven Point = $22.75
$0.25
$0
Maximum loss
is unlimited
$22.50
40
Profit and Loss Diagrams
(cont’d)
 For
the Disney JUN 22.50 Put buyer:
Maximum profit = $21.45
Breakeven Point = $21.45
$0
-$1.05
Maximum loss
$22.50
41
Profit and Loss Diagrams
(cont’d)
 For
the Disney JUN 22.50 Put writer:
Maximum profit
Breakeven Point = $21.45
$1.05
$0
Maximum loss = $21.45
$22.50
42
Option Pricing
Determinants of the Option Premium
 Black-Scholes Option Pricing Model
 Development and Assumptions of the Model
 Insights into the Black-Scholes Model
 Delta
 Theory of Put/Call Parity
 Stock Index Options

43
Determinants of the
Option Premium
 Market
Factors
 Accounting Factors
44
Market Factors
 Striking
price
• For a call option, the lower the striking price,
the higher the option premium
 Time
to expiration
• For both calls and puts, the longer the time to
expiration, the higher the option premium
45
Market Factors (cont’d)
 Current
stock price
• The higher the stock price, the higher the call
option premium and the lower the put option
premium
 Volatility
of the underlying stock
• The greater the volatility, the higher the call and
put option premium
46
Market Factors (cont’d)
 Dividend
yield on the underlying stock
• Companies with high dividend yields have a
smaller call option premium than companies
with low dividend yields
 Risk-free
interest rate
• The higher the risk-free rate, the higher the call
option premium
47
Accounting Factors
 Stock
splits:
• The OCC will make the following adjustments:
– The striking price is reduced by the split ratio
– The number of options is increased by the split ratio
• For odd-lot generating splits:
– The striking price is reduced by the split ratio
– The number of shares covered by your options is
increased by the split ratio
48
Black-Scholes
Option Pricing Model

The Black-Scholes OPM:
C  S N (d1 )  Ke

 Rt
N (d 2 )

ln( S / K )  R  ( / 2) t
d1 
 t
2
d 2  d1   t
49
Black-Scholes
Option Pricing Model (cont’d)
 Variable
•
•
•
•
•
definitions:
C = theoretical call premium
S = current stock price
t = time in years until option expiration
K = option striking price
R = risk-free interest rate
50
Black-Scholes
Option Pricing Model (cont’d)
 Variable
definitions (cont’d):
•  = standard deviation of stock returns
• N(x) = probability that a value less than “x” will
occur in a standard normal distribution
• ln = natural logarithm
• e = base of natural logarithm (2.7183)
51
Black-Scholes
Option Pricing Model (cont’d)
Example
Stock ABC currently trades for $30. A call option on ABC
stock has a striking price of $25 and expires in three
months. The current risk-free rate is 5%, and ABC stock
has a standard deviation of 0.45.
According to the Black-Scholes OPM, what should be the
premium for this option?
52
Black-Scholes
Option Pricing Model (cont’d)
Example (cont’d)
Solution: We must first determine d1 and d2:
d1 

ln( S / K )   R  ( 2 / 2)  t
 t
ln(30 / 25)  0.05  (0.452 / 2)  0.25
0.45 0.25
0.1823  0.0378

 0.978
0.225
53
Black-Scholes
Option Pricing Model (cont’d)
Example (cont’d)
Solution (cont’d):
d 2  d1   t
 0.978  (0.45) 0.25
 0.978  0.225
 0.753
54
Black-Scholes
Option Pricing Model (cont’d)
Example (cont’d)
Solution (cont’d): The next step is to find the normal
probability values for d1 and d2. Using Microsoft Excel’s
NORMSDIST function yields:
N (d1 )  0.836
N (d 2 )  0.774
55
Black-Scholes
Option Pricing Model (cont’d)
Example (cont’d)
Solution (cont’d): The final step is to calculate the option premium:
C  S N (d1 )  Ke  Rt N (d 2 )
 $300.836  $25( 0.05)( 0.25) 0.774
 $25.08  $19.11
 $5.97
56
Using Microsoft Excel’s
NORMSDIST Function
 The
Excel portion below shows the input
and the result of the function:
57
Development and
Assumptions of the Model
Introduction
 The Stock Pays No Dividends during the Option’s
Life
 European Exercise Terms
 Markets are Efficient
 No Commissions
 Constant Interest Rates
 Lognormal Returns

58
Introduction
 Many
of the steps used in building the
Black-Scholes OPM come from:
• Physics
• Mathematical shortcuts
• Arbitrage arguments
 The
actual development of the OPM is
complicated
59
The Stock Pays No Dividends
during the Option’s Life
 The
OPM assumes that the underlying
security pays no dividends
 Valuing
securities with different dividend
yields using the OPM will result in the same
price
60
The Stock Pays No Dividends
during the Option’s Life (cont’d)
 The
OPM can be adjusted for dividends:
• Discount the future dividend assuming
continuous compounding
• Subtract the present value of the dividend from
the stock price in the OPM
• Compute the premium using the OPM with the
adjusted stock price
61
European Exercise Terms
 The
OPM assumes that the option is
European
 Not
a major consideration since very few
calls are ever exercised prior to expiration
62
Markets are Efficient
 The
OPM assumes markets are
informationally efficient
• People cannot predict the direction of the
market or of an individual stock
63
No Commissions
 The
OPM assumes market participants do
not have to pay any commissions to buy or
sell
 Commissions paid by individual investors
can significantly affect the true cost of an
option
• Trading fee differentials cause slightly different
effective option prices for different market
participants
64
Constant Interest Rates
 The
OPM assumes that the interest rate R in
the model is known and constant
 It
is common use to use the discount rate on
a U.S. Treasury bill that has a maturity
approximately equal to the remaining life of
the option
• This interest rate can change
65
Lognormal Returns
 The
OPM assumes that the logarithms of
returns of the underlying security are
normally distributed
 A reasonable
assumption for most assets on
which options are available
66
Insights into the
Black-Scholes Model

Divide the OPM into two parts:
C  S N (d1 )  Ke
Part A
 Rt
N (d2 )
Part B
67
Insights into the
Black-Scholes Model (cont’d)
 Part A is
the expected benefit from
acquiring the stock:
• S is the current stock price and the discounted
value of the expected stock price at any future
point
• N(d1) is a pseudo-probability
– It is the probability of the option being in the money
at expiration, adjusted for the depth the option is in
the money
68
Insights into the
Black-Scholes Model (cont’d)
 Part
B is the present value of the exercise
price on the expiration day:
• N(d2) is the actual probability the option will be
in the money on expiration day
69
Insights into the
Black-Scholes Model (cont’d)
 The
value of a call option is the difference
between the expected benefit from
acquiring the stock and paying the exercise
price on expiration day
70
Delta
 Delta
is the change in option premium
expected from a small change in the stock
price, all other things being the same:
C

S
C
where
 the first partial derivative of the call premium
S
with respect to the stock price
71
Delta (cont’d)
 Delta
allows us to determine how many
options are needed to mimic the returns of
the underlying stock
 Delta
is exactly equal to N(d1)
• e.g., if N(d1) is 0.836, a $1 change in the price
of the underlying stock price leads to a change
in the option premium of 84 cents
72
Theory of Put/Call Parity
 The
following variables form an interrelated
securities complex:
•
•
•
•
Price of a put
Price of a call
The value of the underlying stock
The riskless rate of interest
73
Theory of
Put/Call Parity (cont’d)
 The
put/call parity relationship:
K
CPS
T
(1  R)
where C  price of a call
P  price of a put
K  option striking price
R  risk-free interest rate
T  time until expiration in years
74
Stock Index Options
 Stock
index options are the option
exchanges’ most successful innovation
• e.g., the S&P 100 index option
 Index
options have no delivery mechanism
• All settlements are in cash
75
Stock Index Options (cont’d)
 The
owner of an in-the-money index call
receives the difference between the closing
index level and the striking price
 The
owner of an in-the-money index put
receives the difference between the striking
price and the index level
76