CHAPTER
TWENTY-FOUR
OPTIONS
1
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
2
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
3
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
4
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
5
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
6
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)
7
OPTION TRADING
• THE MECHANICS OF EXCHANGE
TRADING
– Use of specialist
– Use of market makers
8
THE VALUATION OF OPTIONS
• VALUATION AT EXPIRATION (E)
– FOR A CALL OPTION
E
-100
value
of
option
0
100
stock price
200
9
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
10
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
11
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
12
THE VALUATION OF OPTIONS
• VALUATION AT EXPIRATION
– FOR A PUT OPTION
value 100
of
the
option
E=100
0
stock price
13
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
14
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
15
THE VALUATION OF OPTIONS
• PROFITS AND LOSSES ON CALLS AND PUTS
PROFITS
PROFITS
CALLS
100
0
PUTS
p
P
0
LOSSES
100
LOSSES
16
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
17
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 )
18
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 )
19
THE BINOMIAL OPTION
PRICING MODEL (BOPM)
• WHAT DOES BOPM DO?
– it estimates the fair value of a call or a put
option
20
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
21
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)
22
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
23
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
24
THE BINOMIAL OPTION
PRICING MODEL (Price Tree)
$125 P0=25
Annual Analysis:
$100
$80 P0=$0
Semiannual Analysis:
$125 P0=65
$111.80
$100
$100 P0=0
$89.44
$80
t=0
t=.5T
t=T
P0=0
25
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
26
THE BINOMIAL OPTION
PRICING MODEL (BOPM)
• Summary
Security
Payoff:
Up state
Payoff:
Down state
Stock
Bond
Call
$125.00
108.33
25.00
$ 80.00
108.33
0.00
Current
Price
$100.00
$100.00
???
27
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
28
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
29
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
30
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
31
BOPM: REPLICATING
PORTFOLIOS
• INTERPRETATION
– Investor replicates payoffs from the call by
• Short selling the bonds: $41.03
• Purchasing .5556 shares of Widget
32
BOPM: REPLICATING
PORTFOLIOS
Portfolio
Component
Stock
Loan
Net Payoff
Payoff In
Up State
.5556 x $125
= $6 9.45
-$41.03 x 1.0833
= -$44.45
$25.00
Payoff In
Down State
.5556 x $80
= $ 44.45
-$41.03 x 1.0833
= -$ 44.45
$0.00
33
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
34
BOPM: REPLICATING
PORTFOLIOS
• MORE GENERALLY
V0  N S PS  Nb Pb
where
V0
Pd
Pb
Nd
Nb
= the value of the option
= the stock price
= the risk free bond price
= the number of shares
= the number of bonds
35
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
36
THE HEDGE RATIO
• THE HEDGE RATIO
Pou  Pod
h
Psu  Psd
where
P = the end-of-period price
o = the option
s = the stock
u = up
d = down
37
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 )
38
THE HEDGE RATIO
– the value of a call option
V0 = h Ps - B
where h =
B=
the hedge ratio
the current value of a short
bond position in a portfolio
that replicates the payoffs of
the call
39
PUT-CALL PARITY
• Relationship of hedge ratios:
hp = h c - 1
where hp = the hedge ratio of a call
hc = the hedge ratio of a put
40
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
41
PUT-CALL PARITY
• FORMULA:
PP + PS = PC + E / eRT
where PP and PC denote the current
market prices of the put and the call
42
THE BLACK-SCHOLES
MODEL
• What if the number of periods before
expiration were allowed to increase
infinitely?
43
THE BLACK-SCHOLES
MODEL
• The Black-Scholes formula for valuing a
call option
E
Vc  N (d1 ) Ps  RT N (d 2 )
e
where
ln( Ps / E )  ( R  .5 )T
d1 
 T
2
44
THE BLACK-SCHOLES
MODEL
ln( Ps / E )  ( R  .5 )T
d2 
 T
2
and where Ps = the stock’s current market price
E = the exercise price
R = continuously compounded risk
free rate
T = the time remaining to expire
 = risk (standard deviation of the
stock’s annual return)
45
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
46
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
47
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
48
THE BLACK-SCHOLES
MODEL
• LIMITATIONS OF B/S MODEL:
– It only applies to
• European-style options
• stocks that pay NO dividends
49