Transcript Document

Chapters 14/15 – Part 1
Options: Basic Concepts
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Options
Call Options
Put Options
Selling Options
Reading The Wall Street Journal
Combinations of Options
Valuing Options
An Option-Pricing Formula
Investment in Real Projects and Options
Summary and Conclusions
0
Options Contracts: Preliminaries
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Option Definition.
Calls versus Puts
Call options
Put options.
Exercising the Option
Strike Price or Exercise Price
Expiration Date
European versus American options
1
Options Contracts: Preliminaries
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Intrinsic Value
Speculative Value
Option
Premium
=
Intrinsic
Value
+ Speculative
Value
2
Value of an Option at Expiration
Impact of leverage…
Stock price is $50. Buy 100 shares
Call strike is $50, price is $10. Buy 1 contract.
Put strike is $50, price is $10. Buy 1 contract.
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C=S–E
P=E-S
3
Call Option Payoffs
60
Option payoffs ($)
40
Buy a call
20
0
-20
0
10
20
30
40
50
60
70
80
90
100
Stock price ($)
Write a call
-40
-60
4
Put Option Payoffs
60
Option payoffs ($)
40
Buy a put
20
0
-20
-40
0
10
20
30
40
50
60
70
80
90
100
Stock price ($)
Write a put
-60
5
Call Option Payoffs
60
Option payoffs ($)
40
Buy a call
20
0
0
10
20
30
40
50
60
70
80
90
100
Stock price ($)
-20
-40
-60
Exercise price = $50
6
Call Option Payoffs
60
Option payoffs ($)
40
20
0
-20
0
10
20
30
40
50
60
70
80
90
100
Stock price ($)
Write a call
-40
-60
Exercise price = $50
7
Call Option Profits
60
Option profits ($)
40
Buy a call
20
0
-20
0
10
20
30
40
50
60
70
80
90
100
Stock price ($)
Write a call
-40
-60
Exercise price = $50; option premium = $10
8
Put Option Payoffs
60
Option payoffs ($)
40
Buy a put
20
0
0
10
20
30
40
50
60
70 80
90
100
Stock price ($)
-20
-40
-60
Exercise price = $50
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Put Option Payoffs
60
Option payoffs ($)
40
20
0
0
10
20
30
40
50
60
70 80
90
100
Stock price ($)
-20
-40
write a put
-60
Exercise price = $50
10
Option profits ($)
Put Option Profits
60
40
20
10
0
-10
-20
Write a put
0
10
20
30
Stock price ($)
40
50
60
70 80
Buy a put
90
100
-40
-60
Exercise price = $50; option premium = $10
11
Selling Options – Writing Options
The seller (or writer) of an
option has an obligation.
($)
Option profitsOption
profits ($)
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The purchaser of an
option has an option.
60
40
20
10
0
-10
-20
Buy a call
Write a put
0
10
20
30
Stock price ($)
40
50
60
70 80
Buy a put
90
100
Write a call
-40
-60
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Call Option Payoffs at Expiration (Δ exercise)
Buy a call
60
E=0
E=50
Option payoffs ($)
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
Stock price ($)
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Option Pricing Bounds at Expiration
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Upper bounds
 Call Options
 Put Options
Lower Bounds
 Call option intrinsic value

= max [0, S - E]
 Put option intrinsic value

= max [0, E - S]
In-the-money / Out-of-the-money
Time premium/time decay
At expiration, an American call option is worth
the same as a European option with the same
characteristics.
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Reading The Wall Street Journal
Option/Strike Exp.
130 Oct
IBM
130 Jan
138¼
135 Jul
138¼
135 Aug
138¼
140 Jul
138¼
140 Aug
138¼
--Put---Call-Vol. Last Vol. Last
5¼
107
364 15¼
9¼
420
112 19½
4¾ 2431 13/16
2365
5½
94
9¼
1231
2¾
427
1¾
1826
7½
58
6½
2193
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Valuing Options
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The last section
concerned itself
with the value of
an option at
expiration.
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This section
considers the
value of an option
prior to the
expiration date.
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Option Value Determinants
Call Put
1.
2.
3.
4.
5.
Exercise price
Stock price
Interest rate
Volatility in the stock price
Expiration date
The value of a call option C0 must fall within
max (S0 – E, 0) < C0 < S0.
The precise position will depend on these factors.
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Varying Option Input Values
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Stock price:
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Call: as stock price increases call option
price increases
Put: as stock price increases put option price
decreases
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Strike price:
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Call: as strike price increases call option
price decreases
Put: as strike price increases put option price
increases
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Varying Option Input Values
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Time until expiration:
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Call & Put: as time to expiration increases call
and put option price increase
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Volatility:
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Call & Put: as volatility increases call & put value
increase
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Risk-free rate:
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Call: as the risk-free rate increases call option
price increases
Put: as the risk-free rate increases put option
price decreases
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Figure 15.1. Put and Call Option Prices
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Put Price
Call Price
15
10
5
0
12
8
11
6
11
4
11
2
11
8
6
4
2
0
0
11
10
10
10
10
10
98
96
94
92
90
88
86
84
82
0
80
Option Price ($)
20
Stock Price ($)
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Figure 15.2. Option Prices and Time to Expiration
35
30
Option Price ($)
Call Price
25
20
15
Put Price
10
5
0
0
3
6
9
12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
Time to Expiration (months)
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Figure 15.3. Option Prices and Sigma
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15
Call Price
10
Put Price
5
0
10
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
0
Option Price ($)
20
Sigma (%)
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Figure 15.4. Options Prices and Interest Rates
9
8
Call Price
Option Price ($)
7
6
5
4
Put Price
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Interest Rate (%)
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Option Value Determinants
1.
2.
3.
4.
5.
Exercise price
Stock price
Interest rate
Volatility in the stock price
Expiration date
Call Put
–
+
+
–
+
–
+
+
+
+
The value of a call option C0 must fall within
max (S0 – E, 0) < C0 < S0.
The precise position will depend on these factors.
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Market Value, Time Value and Intrinsic Value for
an American Call
The value of a call option C0 must fall within
max (S0 – E, 0) < C0 < S0.
Profit
CaT > Max[ST - E, 0]
Market Value
Time value
Intrinsic value
loss
Out-of-the-money
E
In-the-money
ST
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Combinations of Options
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Puts and calls can serve as the
building blocks for more complex
option contracts.
If you understand this, you can
become a financial engineer, tailoring
the risk-return profile to meet your
client’s needs.
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Protective Put Strategy: Buy a Put and Buy the
Underlying Stock: Payoffs at Expiration
Value at
expiration
Protective Put strategy has
downside protection and
upside potential
$50
Buy the
stock
Buy a put with an exercise
price of $50
$0
$50
Value of
stock at
expiration
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Protective Put Strategy Profits
Value at
expiration
$40
Buy the stock at $40
Protective Put
strategy has
downside protection
and upside potential
$0
$40 $50
-$40
Buy a put with
exercise price of
$50 for $10
Value of
stock at
expiration
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Covered Call Strategy
Value at
expiration
$40
Buy the stock at $40
Covered call
$10
$0
Value of stock at expiration
$30 $40 $50
-$30
-$40
Sell a call with
exercise price of
$50 for $10
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Long Straddle: Buy a Call and a Put
Value at
expiration
Buy a call with an
exercise price of
$50 for $10
$40
$30
$0
-$10
-$20
$30 $40 $50 $60
Buy a put with an
$70 exercise price of
$50 for $10
A Long Straddle only makes money if the
stock price moves $20 away from $50.
Value of
stock at
expiration
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Short Straddle: Sell a Call and a Put
Value at
expiration
$20
$10
$0
-$30
-$40
A Short Straddle only loses money if the stock
price moves $20 away from $50.
Sell a put with exercise price of
$50 for $10
Value of stock at
expiration
$30 $40 $50 $60 $70
Sell a call with an
exercise price of $50 for $10
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Put-Call Parity
S  P  C  Ee
C = Call option price
S = Current stock price
r = Risk-free rate
r T
P = Put option price
E = Option strike price
T = Time until option
expiration
Buy the stock, buy a put, and write a call; the sum of
which equals the strike price discounted at the risk-free rate
S  P  C  Ee
r T
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Put-Call Parity
Buy Stock & Buy Put
Position Value
Combination:
Long Stock &
Long Put
Long Stock
Long Put
Share Price
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Put-Call Parity
Buy Call & Buy Zero Coupon Risk-Free Bond @
Exercise Price
Position Value
Combination:
Long Stock &
Long Bond
Long Bond
Long Call
Share Price
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Put-Call Parity
Combination:
Long Stock &
Long Bond
Combination:
Long Stock &
Long Put
Position Value
Position Value
Long Stock
Long Bond
Long Call
Long Put
Share Price
Share Price
In market equilibrium, it must be the case that option prices
are set such that:
r T
S  P  C  Ee
Otherwise, riskless portfolios with positive payoffs exist.
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The Black-Scholes Model
Value of a stock option is a function of 6 input factors:
1. Current price of underlying stock.
2. Strike price specified in the option contract.
3. Risk-free interest rate over the life of the contract.
4. Time remaining until the option contract expires.
5. Price volatility of the underlying stock.
The price of a call option equals:
C  S N (d1 )  E e
r T
N (d 2 )
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Black-Scholes Model
C  S N (d1 )  E e
r T
N (d 2 )
Where the inputs are:
S = Current stock price
E = Option strike price
r = Risk-free interest rate
T = Time remaining until option expiration
 = Sigma, representing stock price volatility,
standard deviation
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Black-Scholes Model
C  S N (d1 )  E e
r T
N (d 2 )
Where d1 and d2 equal:
 
2



S
 T
ln
  r 
E 
2 
d1 
2
 T
d 2  d1   2T
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Black-Scholes Models
Remembering put-call parity, the value of a put,
given the value of a call equals:
S  P  C  Ee
r T
P  C  S  Ee
r T
Also, remember at expiration:
C SE
P ES
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The Black-Scholes Model
Find the value of a six-month call option on the
Microsoft with an exercise price of $150
The current value of a share of Microsoft is $160
The interest rate available in the U.S. is r = 5%.
The option maturity is 6 months (half of a year).
The standard deviation of the underlying asset is
30% per annum.
Before we start, note that the intrinsic value of the
option is $10—our answer must be at least that
amount.
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The Black-Scholes Model
Assume S = $160, X = $150, T = 6 months, r = 5%,
and  = 30%, calculate the value of a call.
First calculate d1 and d2
ln( S / E )  (r  .5σ 2 )T
d1 
 T
ln( 160 / 150)  (.05  .5(0.30) 2 ).5
d1 
 0.5282
0.30 .5
Then d2,
d 2  d1   T 
d 2  0.52815  0.30 .5  0.31602
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The Black-Scholes Model
C0  S  N(d1 )  Ee rT  N(d 2 )
d1  0.5282
d 2  0.31602
N(d1) = N(0.52815) = 0.7013
N(d2) = N(0.31602) = 0.62401
C0  $160  0.7013  150e .05.5  0.62401
C0  $20.92
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Another Black-Scholes Example
Assume S = $50, X = $45, T = 6 months, r = 10%,
and  = 28%, calculate the value of a call and a put.


2


0
.
28
50
 0.50
ln
  0.10 
45 
2 
d1 
 0.884
0.28 0.50
d2  0.884  0.28 0.50  0.686
From a standard normal probability table, look up N(d1) =
0.812 and N(d2) = 0.754 (or use Excel’s “normsdist” function)
C  50 (0.812)  45 e 0.10 (0.50 ) (0.754)  $8.32
P  $8.32  $50  $45e 0.10 (0.50 )  $1.125
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Real Options
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Real estate developer buys 70 acres in a rural area. He
plans on building a subdivision when the population
from the city expands this direction. If growth is less
than anticipated, the developer thinks he can sell the
land to a country club to build a golf course on the
property.
The development option is a ______ option.
The golf course option is a _______ option.
How would these real options change the standard NPV
analysis?
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Collar: Buy a Put, Buy the Stock, Sell the Call
Value at
expiration
Buy the stock at $80
Collar
$49.33
$42.11
$2.76
Value of
stock at
expiration
$0
$0.67
-$27.91
$120
$50
-$80
Buy a put with exercise
price of $50 for $0.67
$80
Sell a call with
exercise price of
$120 for $2.76
NTS
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