CHAPTER TWENTYFOUR

1
CHAPTER TWENTY-FOUR

• OPTIONS

2
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

3
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

4
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

5
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

6
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

7
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)

8
OPTION TRADING

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

9
THE VALUATION OF OPTIONS

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

E
-100
value of option
0
200
100
stock price
10
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

12
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

100
value of the option
E100
0
stock price
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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

PROFITS
PROFITS
CALLS
PUTS
100
p
P
0
100
0
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)
• PC 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)
• PP 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 )

20
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 t0
• after one year tT
• 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 Widget
• Let 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)
Annual Analysis
125 P025
100
80 P00
Semiannual Analysis
125 P065
111.80
100
100 P00
89.44
80 P00
t0
t.5T
tT
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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)

• Summary
• Security 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
Portfolio Component
Payoff In Down State
Payoff In Up State
.5556 x 125 6 9.45
.5556 x 80 44.45
Stock
-41.03 x 1.0833 -44.45
-41.03 x 1.0833 - 44.45
Loan
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

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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 price
• o the option
• s the stock
• u up
• d down

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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

40
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

41
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

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THE BLACK-SCHOLES MODEL
and where Ps the stocks current market
price E the exercise price R
continuously compounded risk free rate T
the time remaining to expire s risk
(standard deviation of the stocks 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 s 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
  options value
• The higher the exercise price, the lower the
  options value
• The longer the time to expiration, the higher the
  options value

48
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
  options value
• The greater the risk, the higher the options
  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