CHAPTER NINETEEN OPTIONS. TYPES OF OPTION CONTRACTS n WHAT IS AN OPTION? Definition: a type of...
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Transcript of CHAPTER NINETEEN OPTIONS. TYPES OF OPTION CONTRACTS n WHAT IS AN OPTION? Definition: a type of...
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