new systematic integration modelling and coupling
DESCRIPTION
management magnetismTRANSCRIPT
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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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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