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Principles ofOptions and Option Pricing
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We sent the first draft of our paper to the Journal of PoliticalEconomy and promptly got back a rejection letter. We thensent it to the Review of Economics and Statistics, where it
was also rejected.
Merton Miller and Eugene Famathen took an interest in thepaper and gave us extensive comments on it. They
suggested to the JPE that perhaps the paper was worthmore serious consideration. The journal then accepted the
paper.
- MILAN PATEL
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OutlineIntroduction
Option principles
Option pricing
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IntroductionInnovations in stock options have been
among the most important developments in
finance in the last 20 yearsThe cornerstone of option pricing is the
Black-Scholes Option Pricing Model
(OPM) Delta is the most important OPM progeny to
the portfolio manager
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Option PrinciplesWhy 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
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Why Options Are A Good IdeaOptions:
Give the marketplace opportunities to adjust
risk or alter income streams that wouldotherwise not be available
Provide financial leverage
Can be used to generate additional income frominvestment portfolios
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Why Options Are
A Good Idea (contd)The investment process is dynamic:
The portfolio managers needs to constantly
reassess and adjust portfolios with the arrival ofnew information
Options are more convenient and lessexpensive than wholesale purchases or sales
of stock
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What Options AreCall options
Put options
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Call OptionsA 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
premiumto the option seller in exchangefor the right to buy
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Practical Example
of A Call Option
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Put OptionsA 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
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Standardized
Option CharacteristicsAll exchange-traded options have
standardized expiration dates
The Saturday following the third Friday ofdesignated months for most options
Investors typically view the third Friday of themonth as the expiration date
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Standardized
Option Characteristics (contd)The striking priceof an option is the
predetermined transaction price
In multiples of $2.50 (for stocks priced $25.00or 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
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Standardized
Option Characteristics (contd)Puts and calls are based on 100 shares of the
underlying security
The underlying security is the security that theoption gives you the right to buy or sell
It is not possible to buy or sell odd lots ofoptions
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Where Options Come FromIntroduction
Opening and closing transactions
Role of the Options Clearing Corporation
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IntroductionIf 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
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Opening and
Closing TransactionsThe first trade someone makes in a
particular option is an opening transaction
An opening transaction that is the sale of anoption is called writing an option
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Opening and
Closing Transactions (contd)The trade that terminates a position by
closing it out is a closing transaction
Options have fungibilityMarket participants can reverse their positions by
making offsetting trades
E.g., the writer of an option can close out theposition by buying a similar one
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Opening and
Closing Transactions (contd)The owner of an option will ultimately:
Sell it to someone else
Let it expire or
Exercise it
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Role of the
Options Clearing CorporationThe 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
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OCC-Related
Information on the Web
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Where and How Options TradeOptions trade on four principal exchanges:
Chicago Board Options Exchange (CBOE)
American Stock Exchange (AMEX)
Philadelphia Stock Exchange
Pacific Stock Exchange
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Where and How
Options Trade (contd)AMEX and Philadelphia Stock Exchange
options trade via the specialist system
All orders to buy or sell a particular securitypass through a single individual (the specialist)
The specialist:Keeps an order book with standing orders from
investors and maintains the market in a fair andorderly fashion
Executes trades close to the current market price ifno buyer or seller is available
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Where and How
Options Trade (contd)CBOE and Pacific Stock Exchange options
trade via the marketmaker system
Competing marketmakers trade in a specificlocation on the exchange floor near the orderbook official
Marketmakers compete against one another for
the publics business
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Where and How
Options Trade (contd)Any given option has two prices at any
given time:
The bid priceis the highest price anyone iswilling to pay for a particular option
The asked priceis the lowest price at whichanyone is willing to sell a particular option
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The Option PremiumIntrinsic value and time value
The financial page listing
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Intrinsic Value and Time ValueThe price of an option has two components:
Intrinsic value:
For a call option equals the stock price minus thestriking price
For a put option equals the striking price minus the
stock price
Time valueequals the option premium minusthe intrinsic value
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Intrinsic Value and
Time Value (contd)An option with no intrinsic value is out of
the money
An option with intrinsic value is in themoney
If an options striking price equals the stockprice, the option is at the money
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The Financial Page ListingThe following slide shows an example from
the online edition of the Wall StreetJournal:
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
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The Financial Page Listing
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The Financial
Page Listing (contd)Investors identify an option by company,
expiration, striking price, and type of
option:
Disney JUN 22.50 Call
CompanyExpiration Striking
Price
Type
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The Financial
Page Listing (contd) 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
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The Financial
Page Listing (contd)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 anoption declines over time
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Sources of Profits and
Losses With OptionsOption exercise
Exercise procedures
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Option ExerciseAn American option can be exercised at
any time prior to option expiration
It is typically not advantageous to exerciseprior to expiration since this amount to
foregoing time value
European optionscan be exercised only atexpiration
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Exercise ProceduresThe owner of an option who decides to
exercisethe option:
Calls her broker Must put up the full contract amount for the
option
The premium is not a downpayment on the option
terms
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Exercise Procedures (contd)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
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Exercise Procedures (contd)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
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Profit and Loss DiagramsFor the Disney JUN 22.50 Call buyer:
-$0.25
$22.50
$0
Maximum loss
Breakeven Point = $22.75
Maximum profit
is unlimited
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Profit and Loss DiagramsFor the Disney JUN 22.50 Call writer:
$0.25
$22.50
$0
Maximum profit Breakeven Point = $22.75
Maximum loss
is unlimited
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Profit and Loss DiagramsFor the Disney JUN 22.50 Put buyer:
-$1.05
$22.50
$0
Maximum loss
Breakeven Point = $21.45
Maximum profit = $21.45
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Profit and Loss DiagramsFor the Disney JUN 22.50 Put writer:
$1.05
$22.50
$0
Maximum profitBreakeven Point = $21.45
Maximum loss = $21.45
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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
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Determinants of the
Option PremiumMarket factors
Accounting factors
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Market FactorsStriking 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 toexpiration, the higher the option premium
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Market Factors (contd)Current stock price
The higher the stock price, the higher the call
option premium and the lower the put optionpremium
Volatility of the underlying stock The great the volatility, the higher the call and
put option premium
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Market Factors (contd)Dividend yield on the underlying stock
Companies with high dividend yields have a
smaller call option premium than companieswith low dividend yields
Risk-free interest rate The higher the risk-free rate, the higher the call
option premium
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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 ratioThe number of shares covered by your options is
increased by the split ratio
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Black-Scholes
Option Pricing Model
The Black-Scholes OPM:
1 2
2
1
2 1
( ) ( )
ln( / ) ( / 2)
rtC S N d Ke N d
S K R t d
t
d d t
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Black-Scholes
Option Pricing Model (contd)
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
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Black-Scholes
Option Pricing Model (contd)
Variable definitions (contd):
= standard deviation of stock returns
N(x)= probability that a value less than x willoccur in a standard normal distribution
ln = natural logarithm
e = base of natural logarithm (2.7183)
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Black-Scholes
Option Pricing Model (contd)
Example
Stock ABC currently trades for $30. A call option on ABCstock 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, but should be the
call option premium for this option?
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Black-Scholes
Option Pricing Model (contd)
Example (contd)
Solution: We must first determine d1 and d2:2
1
2
ln( / ) ( / 2)
ln(30 / 25) 0.05 (0.45 / 2) 0.25
0.45 0.25
0.1823 0.03780.978
0.225
S K R t d
t
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Black-Scholes
Option Pricing Model (contd)
Example (contd)
Solution (contd):
2 1
0.978 (0.45) 0.25
0.978 0.225
0.753
d d t
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Black-Scholes
Option Pricing Model (contd)
Example (contd)
Solution (contd): The next step is to find the normalprobability values for d1 and d2. Using ExcelsNORMSDIST function yields:
1
2
( ) 0.836
( ) 0.774
N d
N d
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Black-Scholes
Option Pricing Model (contd)
Example (contd)
Solution (contd): The final step is to calculate the optionpremium:
1 2
(0.05)(0.25)
( ) ( )
$30 0.836 $25 0.774$25.08 $19.11
$5.97
rtC S N d Ke N d
e
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Using Excels
NORMSDIST Function
The Excel portion below shows the input
and the result of the function:
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Development and
Assumptions of the Model
Introduction
The stock pays no dividends during the options
life
European exercise terms
Markets are efficient
No commissions
Constant interest rates
Lognormal returns
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Introduction
Many of the steps used in building theBlack-Scholes OPM come from:
Physics
Mathematical shortcuts
Arbitrage arguments
The actual development of the OPM iscomplicated
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The Stock Pays no Dividends
During the Options 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
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The Stock Pays no Dividends
During the Options Life
The OPM can be adjusted for dividends:
Discount the future dividend assumingcontinuous compounding
Subtract the present value of the dividend fromthe stock price in the OPM
Compute the premium using the OPM with theadjusted stock price
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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
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Markets Are Efficient
The OPM assumes markets are
informationally efficient
People cannot predict the direction of themarket or of an individual stock
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No Commissions
The OPM assumes market participants do
not have to pay any commissions to buy or
sellCommissions paid by individual can
significantly affect the true cost of an option
Trading fee differentials cause slightly differenteffective option prices for different market
participants
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Constant Interest Rates
The OPM assumes that the interest rateR inthe model is known and constant
It is common use to use the discount rate ona U.S. Treasury bill that has a maturityapproximately equal to the remaining life of
the option This interest rate can change
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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
I i ht I t th
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Insights Into the
Black-Scholes Model
Divide the OPM into two parts:
1 2( ) ( )rt
C S N d Ke N d
Part A Part B
I i ht I t th
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Insights Into the
Black-Scholes Model (contd)
Part A is the expected benefit from
acquiring the stock:
Sis the current stock price and the discountedvalue 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
I i ht I t th
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Insights Into the
Black-Scholes Model (contd)
Part B is the present value of the exercise
price on the expiration day:
N(d2) is the actual probability the option will bein the money on expiration day
I i ht I t th
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Insights Into the
Black-Scholes Model (contd)
The value of a call option is the difference
between the expected benefit from
acquiring the stock and paying the exerciseprice on expiration day
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Delta
Delta is the change in option premiumexpected from a small change in the stock
price, all other things being equal:
where the first partial derivative of the call premium
with respect to the stock price
C
S
C
S
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Delta (contd)
Delta allows us to determine how many
options are needed to mimic the returns of
the underlying stock
Delta is exactly equal toN(d1)
E.g., ifN(d1) is 0.836, a $1 change in the priceof the underlying stock price leads to a change
in the option premium of 84 cents
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Theory of Put/Call Parity
The following variables form an interrelatedsecurities complex:
Price of a put
Price of a call
The value of the underlying stock
The riskless rate of interest
If put/call parity does not hold, arbitrage ispossible
Th f
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Theory of
Put/Call Parity (contd)
The put/call parity relationship:
(1 )where price of a call
price of a put
option striking pricerisk-free interest rate
time until expiration in years
T
KC P S
R
C
P
K
R
T
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Stock Index Options
A stock index option is 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
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Stock Index Options (contd)
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 strikingprice and the index level