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CHAPTER TWENTY-FOUR

OPTIONS

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TYPES OF OPTION CONTRACTS

• WHAT IS AN OPTION?– Definition: a type of contract between two

investors where one grants the other the right to buy or sell a specific asset in the future

– the option buyer is buying the right to buy or sell the underlying asset at some future date

– the option writer is selling the right to buy or sell the underlying asset at some future date

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CALL OPTIONS

• WHAT IS A CALL OPTION CONTRACT?– DEFINITION: a legal contract that specifies four

conditions

– FOUR CONDITIONS• the company whose shares can be bought

• the number of shares that can be bought

• the purchase price for the shares known as the exercise or strike price

• the date when the right expires

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CALL OPTIONS

• Role of Exchange• exchanges created the Options Clearing Corporation

(CCC) to facilitate trading a standardized contract (100 shares/contract)

• OCC helps buyers and writers to “close out” a position

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PUT OPTIONS

• WHAT IS A PUT OPTION CONTRACT?– DEFINITION: a legal contract that specifies

four conditions• the company whose shares can be sold

• the number of shares that can be sold

• the selling price for those shares known as the exercise or strike price

• the date the right expires

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OPTION TRADING

• FEATURES OF OPTION TRADING– a new set of options is created every 3 months– new options expire in roughly 9 months– long term options (LEAPS) may expire in up to

2 years– some flexible options exist (FLEX)– once listed, the option remains until expiration

date

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OPTION TRADING

• TRADING ACTIVITY– currently option trading takes place in the

following locations:• the Chicago Board Options Exchange (CBOS)

• the American Stock Exchange

• the Pacific Stock Exchange

• the Philadelphia Stock Exchange (especially currency options)

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OPTION TRADING

• THE MECHANICS OF EXCHANGE TRADING– Use of specialist– Use of market makers

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THE VALUATION OF OPTIONS

• VALUATION AT EXPIRATION (E)– FOR A CALL OPTION

-100

100 200stock price

value

of

option

E

0

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THE VALUATION OF OPTIONS

• VALUATION AT EXPIRATION– ASSUME: the strike price = $100– For a call if the stock price is less than $100,

the option is worthless at expiration– The upward sloping line represents the intrinsic

value of the option

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THE VALUATION OF OPTIONS

• VALUATION AT EXPIRATION– In equation form

IVc = max {0, Ps, -E}where

Ps is the price of the stock

E is the exercise price

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THE VALUATION OF OPTIONS

• VALUATION AT EXPIRATION– ASSUME: the strike price = $100– For a put if the stock price is greater than $100,

the option is worthless at expiration– The downward sloping line represents the

intrinsic value of the option

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THE VALUATION OF OPTIONS

• VALUATION AT EXPIRATION– FOR A PUT OPTION

100valueofthe option

stock price

E=1000

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THE VALUATION OF OPTIONS

• VALUATION AT EXPIRATION– FOR A CALL OPTION

• if the strike price is greater than $100, the option is worthless at expiration

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THE VALUATION OF OPTIONS

– in equation form

IVc = max {0, - Ps, E}where

Ps is the price of the stock

E is the exercise price

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THE VALUATION OF OPTIONS• PROFITS AND LOSSES ON CALLS AND PUTS

100

100

p P

PROFITS PROFITS

00

CALLS PUTS

LOSSES LOSSES

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THE VALUATION OF OPTIONS

• PROFITS AND LOSSES– Assume the underlying stock sells at $100 at

time of initial transaction– Two kinked lines = the intrinsic value of

the options

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THE VALUATION OF OPTIONS

• PROFIT EQUATIONS (CALLS)

C = IVC - PC

= max {0,PS - E} - PC

= max {-PC , PS - E - PC }This means that the kinked profit line for the call is

the intrinsic value equation less the call premium

(- PC )

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THE VALUATION OF OPTIONS

• PROFIT EQUATIONS (CALLS)

P = IVP - PP

= max {0, E - PS} - PP

= max {-PP , E - PS - PP }This means that the kinked profit line for the put is

the intrinsic value equation less the put premium

(- PP )

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THE BINOMIAL OPTION PRICING MODEL (BOPM)

• WHAT DOES BOPM DO?– it estimates the fair value of a call or a put

option

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THE BINOMIAL OPTION PRICING MODEL (BOPM)

• TYPES OF OPTIONS– EUROPEAN is an option that can be exercised

only on its expiration date– AMERICAN is an option that can be exercised

any time up until and including its expiration date

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THE BINOMIAL OPTION PRICING MODEL (BOPM)

• EXAMPLE: CALL OPTIONS– ASSUMPTIONS:

• price of Widget stock = $100

• at current t: t=0

• after one year: t=T

• stock sells for either$125 (25% increase)

$ 80 (20% decrease)

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THE BINOMIAL OPTION PRICING MODEL (BOPM)

• EXAMPLE: CALL OPTIONS– ASSUMPTIONS:

• Annual riskfree rate = 8% compounded continuously

• Investors cal lend or borrow through an 8% bond

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THE BINOMIAL OPTION PRICING MODEL (BOPM)

• Consider a call option on WidgetLet the exercise price = $100

the exercise date = T

and the exercise value:

If Widget is at $125 = $25

or at $80 = 0

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THE BINOMIAL OPTION PRICING MODEL (Price Tree)

t=0 t=.5T t=T

$125 P0=25

$80 P0=$0$100

$100

$111.80

$89.44

$125 P0=65

$100 P0=0

$80 P0=0

Annual Analysis:

Semiannual Analysis:

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THE BINOMIAL OPTION PRICING MODEL (BOPM)

• VALUATION– What is a fair value for the call at time =0?

• Two Possible Future States– The “Up State” when p = $125

– The “Down State” when p = $80

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THE BINOMIAL OPTION PRICING MODEL (BOPM)

• SummarySecurity Payoff: Payoff: Current

Up state Down state Price

Stock $125.00 $ 80.00 $100.00

Bond 108.33 108.33 $100.00

Call 25.00 0.00 ???

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BOPM: REPLICATING PORTFOLIOS

• REPLICATING PORTFOLIOS– The Widget call option can be replicated – Using an appropriate combination of

• Widget Stock and

• the 8% bond

– The cost of replication equals the fair value of the option

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BOPM: REPLICATING PORTFOLIOS

• REPLICATING PORTFOLIOS– Why?

• if otherwise, there would be an arbitrage opportunity– that is, the investor could buy the cheaper of the two

alternatives and sell the more expensive one

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BOPM: REPLICATING PORTFOLIOS

– COMPOSITION OF THE REPLICATING PORTFOLIO:

• Consider a portfolio with Ns shares of Widget• and Nb risk free bonds

– In the up state• portfolio payoff =

125 Ns + 108.33 Nb = $25

– In the down state 80 Ns + 108.33 Nb = 0

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BOPM: REPLICATING PORTFOLIOS

– COMPOSITION OF THE REPLICATING PORTFOLIO:

• Solving the two equations simultaneously

(125-80)Ns = $25

Ns = .5556

Substituting in either equation yields

Nb = -.4103

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BOPM: REPLICATING PORTFOLIOS

• INTERPRETATION – Investor replicates payoffs from the call by

• Short selling the bonds: $41.03

• Purchasing .5556 shares of Widget

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BOPM: REPLICATING PORTFOLIOS

PortfolioComponent

Payoff InUp State

Payoff InDown State

Stock

Loan

.5556 x $125= $6 9.45

.5556 x $80= $ 44.45

-$41.03 x 1.0833= -$44.45

-$41.03 x 1.0833= -$ 44.45

Net Payoff $25.00 $0.00

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BOPM: REPLICATING PORTFOLIOS

• TO OBTAIN THE PORTFOLIO– $55.56 must be spent to purchase .5556 shares

at $100 per share– but $41.03 income is provided by the bonds

such that

$55.56 - 41.03 = $14.53

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BOPM: REPLICATING PORTFOLIOS

• MORE GENERALLY

where V0 = the value of the option

Pd = the stock price

Pb = the risk free bond price

Nd = the number of shares

Nb = the number of bonds

bbSS PNPNV 0

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THE HEDGE RATIO

• THE HEDGE RATIO– DEFINITION: the expected change in the

value of an option per dollar change in the market price of an underlying asset

– The price of the call should change by $.5556 for every $1 change in stock price

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THE HEDGE RATIO• THE HEDGE RATIO

where P = the end-of-period priceo = the options = the stocku = upd = down

sdsu

odou

PP

PPh

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THE HEDGE RATIO

• THE HEDGE RATIO– to replicate a call option

• h shares must be purchased

• B is the amount borrowed by short selling bonds

B = PV(h Psd - Pod )

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THE HEDGE RATIO

– the value of a call option

V0 = h Ps - B

where h = the hedge ratio

B = the current value of a short bond position in a

portfolio that replicates the payoffs of the call

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PUT-CALL PARITY

• Relationship of hedge ratios:

hp = hc - 1

where hp = the hedge ratio of a call

hc = the hedge ratio of a put

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PUT-CALL PARITY

– DEFINITION: the relationship between the market price of a put and a call that have the same exercise price, expiration date, and underlying stock

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PUT-CALL PARITY

• FORMULA:

PP + PS = PC + E / eRT

where PP and PC denote the current market prices of the put and the call

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THE BLACK-SCHOLES MODEL

• What if the number of periods before expiration were allowed to increase infinitely?

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THE BLACK-SCHOLES MODEL

• The Black-Scholes formula for valuing a call option

where

)()( 21 dNe

EPdNV

RTsc

T

TREPd s

)5.()/ln( 2

1

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THE BLACK-SCHOLES MODEL

T

TREPd s

)5.()/ln( 2

2

and where Ps = the stock’s current market priceE = the exercise priceR = continuously compounded risk

free rateT = the time remaining to expire = risk (standard deviation of the

stock’s annual return)

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THE BLACK-SCHOLES MODEL

• NOTES:– E/eRT = the PV of the exercise price where

continuous discount rate is used

– N(d1 ), N(d2 )= the probabilities that outcomes of less will occur in a normal distribution with mean = 0 and = 1

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THE BLACK-SCHOLES MODEL

• What happens to the fair value of an option when one input is changed while holding the other four constant?– The higher the stock price, the higher the

option’s value– The higher the exercise price, the lower the

option’s value– The longer the time to expiration, the higher the

option’s value

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THE BLACK-SCHOLES MODEL

• What happens to the fair value of an option when one input is changed while holding the other four constant?– The higher the risk free rate, the higher the

option’s value– The greater the risk, the higher the option’s

value

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THE BLACK-SCHOLES MODEL

• LIMITATIONS OF B/S MODEL:– It only applies to

• European-style options

• stocks that pay NO dividends