CHAPTER NINETEEN OPTIONS. TYPES OF OPTION CONTRACTS n WHAT IS AN OPTION? Definition: a type of contract between two investors where one grants the other.

CHAPTER NINETEEN

OPTIONS

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

CALL OPTIONS

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

specifies four conditions

•FOUR CONDITIONSthe company whose shares can be boughtthe number of shares that can be boughtthe purchase price for the shares known as

the exercise or strike pricethe date when the right expires

CALL OPTIONS

Role of Exchangeexchanges 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

PUT OPTIONS

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

specifies four conditionsthe company whose shares can be soldthe number of shares that can be soldthe selling price for those shares known

as the exercise or strike pricethe date the right expires

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

OPTION TRADING

TRADING ACTIVITY•currently option trading takes place in

the following locations:the Chicago Board Options Exchange

(CBOS)the American Stock Exchangethe Pacific Stock Exchangethe Philadelphia Stock Exchange

(especially currency options)

OPTION TRADING

THE MECHANICS OF EXCHANGE TRADING•Use of specialist

•Use of market makers

THE VALUATION OF OPTIONS VALUATION AT EXPIRATION

•FOR A CALL OPTION-100

100 200stock price

value

of

option

E

0

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

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

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

THE VALUATION OF OPTIONS VALUATION AT EXPIRATION

•FOR A PUT OPTION

100valueofthe option

stock price

E=1000

THE VALUATION OF OPTIONS VALUATION AT EXPIRATION

•FOR A CALL OPTIONif the strike price is greater than $100,

the option is worthless at expiration

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

THE VALUATION OF OPTIONS PROFITS AND LOSSES ON CALLS AND PUTS

100

100

p P

PROFITS PROFITS

00

CALLS PUTS

LOSSES LOSSES

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

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 )

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 )

THE BINOMIAL OPTION PRICING MODEL (BOPM) WHAT DOES BOPM DO?

•it estimates the fair value of a call or a put option

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

THE BINOMIAL OPTION PRICING MODEL (BOPM) EXAMPLE: CALL OPTIONS

•ASSUMPTIONS:price of Widget stock = $100at current t: t=0after one year: t=Tstock sells for either

$125 (25% increase)$ 80 (20% decrease)

THE BINOMIAL OPTION PRICING MODEL (BOPM) EXAMPLE: CALL OPTIONS

•ASSUMPTIONS: Annual riskfree rate = 8% compounded

continuouslyInvestors cal lend or borrow through an

8% bond

THE BINOMIAL OPTION PRICING MODEL (BOPM) Consider a call option on Widget

Let the exercise price = $100the exercise date = Tand the exercise value:

If Widget is at $125 = $25

or at $80 = 0

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:

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

THE BINOMIAL OPTION PRICING MODEL (BOPM) SummarySecurity Payoff: Payoff: Current

Up state Down state Price

Stock $125.00 $ 80.00 $100.00Bond 108.33 108.33 $100.00Call 25.00 0.00 ???

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

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

BOPM: REPLICATING PORTFOLIOS

•COMPOSITION OF THE REPLICATING PORTFOLIO:Consider a portfolio with Ns shares of Widget

and Nb risk free bonds

•In the up stateportfolio payoff =

125 Ns + 108.33 Nb = $25

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

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

BOPM: REPLICATING PORTFOLIOS INTERPRETATION

•Investor replicates payoffs from the call byShort selling the bonds: $41.03Purchasing .5556 shares of Widget

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

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

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

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

THE HEDGE RATIO

THE HEDGE RATIO

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

sdsu

odou

PP

PPh

THE HEDGE RATIO

THE HEDGE RATIO•to replicate a call option

h shares must be purchasedB is the amount borrowed by short

selling bonds

B = PV(h Psd - Pod )

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

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

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

PUT-CALL PARITY

FORMULA:

PP + PS = PC + E / eRT

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

THE BLACK-SCHOLES MODEL What if the number of periods

before expiration were allowed to increase infinitely?

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

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)

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

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

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

THE BLACK-SCHOLES MODEL LIMITATIONS OF B/S MODEL:

•It only applies to European-style optionsstocks that pay NO dividends

END OF CHAPTER 19