1 Chapter 17 Principles of Options and Option Pricing Portfolio Construction, Management, & Protection, 5e, Robert A. Strong Copyright ©2009 by South-Western,

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Chapter 17

Principles of Options and Option Pricing

Portfolio Construction, Management, & Protection, 5e, Robert A. StrongCopyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved.

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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, on his journal article with Myron Scholes that

gave birth to the Black-Scholes Option Pricing Model

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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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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

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

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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 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 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

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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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Intrinsic Value and Time Value Option “prices” are referred to as option

“premiums” 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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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 down payment 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 (cont’d)

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 (cont’d)

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 (cont’d)

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

• Market Factors & Accounting Factors

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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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 greater 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:

tdd

t

tRKSd

dNKedNSC Rt

12

2

1

21

)2/()/ln(

)()(

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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, what should be the 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 Microsoft 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:

97.5$

11.19$08.25$

774.025$836.030$

)()()25.0)(05.0(

21

dNKedNSC Rt

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Using Microsoft 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

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 (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

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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 investors 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:

Part A Part B

)()( 21 dNKedNSC Rt

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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 the same:

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

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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 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

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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