1 Chapter 17 Principles of Options and Option Pricing.
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Transcript of 1 Chapter 17 Principles of Options and Option Pricing.
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Chapter 17
Principles of Options and Option Pricing
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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
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Outline Introduction Option principles Option pricing
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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
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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
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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
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Why Options Are A Good Idea (cont’d)
The investment process is dynamic:• The portfolio managers 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
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What Options Are Call options Put options
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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
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Practical Example of A Call Option
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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
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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
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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
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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
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Where Options Come From Introduction Opening and closing transactions Role of the Options Clearing Corporation
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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
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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
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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
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Opening and Closing Transactions (cont’d)
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 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
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OCC-Related Information on the Web
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Where and How Options Trade Options 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 (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
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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
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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
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The Option Premium Intrinsic value and time value The financial page listing
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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
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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
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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
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The Financial Page Listing
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The Financial Page Listing (cont’d)
Investors identify an option by company, expiration, striking price, and type of option:
Disney JUN 22.50 Call
CompanyExpiration Striking
PriceType
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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
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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
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Sources of Profits and Losses With Options
Option exercise Exercise procedures
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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
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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 downpayment on the option
terms
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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
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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
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Profit and Loss Diagrams For the Disney JUN 22.50 Call buyer:
-$0.25
$22.50
$0
Maximum loss
Breakeven Point = $22.75Maximum profit is unlimited
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Profit and Loss Diagrams For 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 Diagrams For 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 Diagrams For 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 Premium
Market factors Accounting factors
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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
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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 great the volatility, the higher the call and
put option premium
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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
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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 ratio– The 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 td
t
d d t
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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
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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)
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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, but should be the call option premium for this option?
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Black-Scholes Option Pricing Model (cont’d)
Example (cont’d)
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 0.250.1823 0.0378
0.9780.225
S K R td
t
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Black-Scholes Option Pricing Model (cont’d)
Example (cont’d)
Solution (cont’d):
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 (cont’d)
Example (cont’d)
Solution (cont’d): The next step is to find the normal probability values for d1 and d2. Using Excel’s NORMSDIST function yields:
1
2
( ) 0.836
( ) 0.774
N d
N d
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Black-Scholes Option Pricing Model (cont’d)
Example (cont’d)
Solution (cont’d): The final step is to calculate the option premium:
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 Excel’s 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 option’s
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 the
Black-Scholes OPM come from:• Physics• Mathematical shortcuts• Arbitrage arguments
The actual development of the OPM is complicated
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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
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The Stock Pays no Dividends During the Option’s Life
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
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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 the
market 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 sell
Commissions paid by individual can significantly affect the true cost of an option• Trading fee differentials cause slightly different
effective option prices for different market participants
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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
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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
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Insights Into the Black-Scholes Model
Divide the OPM into two parts:
1 2( ) ( )rtC S N d Ke N d
Part A Part B
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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
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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
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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
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Delta Delta is the change in option premium
expected 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
SC
S
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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
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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
If put/call parity does not hold, arbitrage is possible
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Theory of Put/Call Parity (cont’d)
The put/call parity relationship:
(1 )
where price of a call
price of a put
option striking price
risk-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 (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