c = max(SK,0)

Report
Real Options
Dr. Lynn Phillips Kugele
FIN 431
Options Review
• Mechanics of Option Markets
• Properties of Stock Options
• Introduction to Binomial Trees
• Valuing Stock Options: The Black-Scholes
Model
• Real Options
OPT-2
Mechanics of Options
Markets
OPT-3
Option Basics
• Option = derivative security
– Value “derived” from the value of the
underlying asset
• Stock Option Contracts
– Exchange-traded
– Standardized
• Facilitates trading and price reporting.
– Contract = 100 shares of stock
OPT-4
Put and Call Options
• Call option
– Gives holder the right but not the
obligation to buy the underlying asset at
a specified price at a specified time.
• Put option
– Gives the holder the right but not the
obligation to sell the underlying asset at
a specified price at a specified time.
OPT-5
Options on Common Stock
1.
2.
3.
4.
5.
•
Identity of the underlying stock
Strike or Exercise price
Contract size
Expiration date or maturity
Exercise cycle
American or European
6. Delivery or settlement procedure
OPT-6
Option Exercise
• American-style
– Exercisable at any time up to and
including the option expiration date
– Stock options are typically American
• European-style
– Exercisable only at the option expiration
date
OPT-7
Option Positions
• Call positions:
– Long call = call “holder”
• Hopes/expects asset price will increase
– Short call = call “writer”
• Hopes asset price will stay or decline
• Put Positions:
– Long put = put “holder”
• Expects asset price to decline
– Short put = put “writer”
• Hopes asset price will stay or increase
OPT-8
Option Writing
• The act of selling an option
• Option writer = seller of an option
contract
– Call option writer obligated to sell the
underlying asset to the call option holder
– Put option writer obligated to buy the
underlying asset from the put option holder
– Option writer receives the option premium
when contract entered
OPT-9
Option Payoffs & Profits
Notation:
•
•
•
•
S0 = current stock price per share
ST = stock price at expiration
K = option exercise or strike price
C = American call option premium per share
• c = European call option premium
• P = American put option premium per share
• p = European put option premium
• r = risk free rate
• T = time to maturity in years
OPT-10
Option Payoffs & Profits
Call Holder
Payoff to Call Holder
(S - K)
0
if S >K
if S < K
= Max (S-K,0)
Profit to Call Holder
Payoff - Option Premium
Profit =Max (S-K, 0) - C
OPT-11
Option Payoffs & Profits
Call Writer
Payoff to Call Writer
- (S - K)
0
if S > K = -Max (S-K, 0)
if S < K = Min (K-S, 0)
Profit to Call Writer
Payoff + Option Premium
Profit = Min (K-S, 0) + C
OPT-12
Payoff & Profit Profiles for Calls
Call Payoff and Profit
K = $20 c = $5.00
Stock
Call Holder
Price
Payoff
Profit
0
$0
-$5
$10
$0
-$5
$20
$0
-$5
$30
$10
$5
$40
$20
$15
Payoff:
Profit:
Max(S-K,0)
Max (S-K,0) – c
Call Writer
Payoff
Profit
$0
$5
$0
$5
$0
$5
-$10
-$5
-$20
-$15
-Max(S-K,0)
-[Max (S-K, 0)-p]
OPT-13
Payoff & Profit Profiles for Calls
Call Payoff and Profit
$25
$20
Call Holder
$15
$10
$$
$5
$0
0
$10
$20
$30
$40
-$5
-$10
-$15
Call Writer
-$20
-$25
Stock Price
Call Holder Payoff
Call Holder Profit
Call Writer Payoff
Call Writer Profit
OPT-14
Payoff & Profit Profiles for Calls
Payoff
Profit
Call Holder
Profit
0
Call Writer
Profit
Stock Price
OPT-15
Option Payoffs and Profits
Put Holder
Payoffs to Put Holder
0
(K - S)
if S > K
if S < K
= Max (K-S, 0)
Profit to Put Holder
Payoff - Option Premium
Profit = Max (K-S, 0) - P
OPT-16
Option Payoffs and Profits
Put Writer
Payoffs to Put Writer
0
if S > K
-(K - S) if S < K
= -Max (K-S, 0)
= Min (S-K, 0)
Profits to Put Writer
Payoff + Option Premium
Profit = Min (S-K, 0) + P
OPT-17
Payoff & Profit Profiles for Puts
Put Payoff and Profit
K = $20 p = $5.00
Stock
Put Holder
Price
Payoff
Profit
0
$20
$15
$10
$10
$5
$20
$0
-$5
$30
$0
-$5
$40
$0
-$5
Payoff:
Profit:
Max(K-S,0)
Max (K-S,0) – p
Put Writer
Payoff
Profit
-$20
-$15
-$10
-$5
$0
$5
$0
$5
$0
$5
-Max(K-S,0)
-[Max (K-S, 0)-p]
OPT-18
Payoff & Profit Profiles for Puts
Put Payoff and Profit
$25
$20
Put Holder
$15
$10
$$
$5
$0
0
$10
$20
$30
$40
-$5
-$10
-$15
Put Writer
-$20
-$25
Stock Price
Put Holder Payof
Put Holder Profit
Put Writer Payoff
Put Writer Profit
OPT-19
Payoff & Profit Profiles for Puts
Profits
Put Writer
Profit
0
Put Holder
Profit
Stock Price
OPT-20
Option Payoffs and Profits
CALL
PUT
Holder: Payoff
(Long) Profit
Max (S-K,0)
Max (S-K,0) - C
“Bullish”
Max (K-S,0)
Max (K-S,0) - P
“Bearish”
Writer: Payoff
(Short) Profit
Min (K-S,0)
Min (K-S,0) + C
“Bearish”
Min (S-K,0)
Min (S-K,0) + P
“Bullish”
OPT-21
Long Call
Long Call Profit = Max(S-K,0) - C
Call option premium (C) = $5, Strike price (K) = $100.
30 Profit ($)
20
10
70
0
-5
80
90
100
Terminal
stock price (S)
110 120 130
OPT-22
Short Call
Short Call Profit = -[Max(S-K,0)-C] = Min(K-S,0) + C
Call option premium (C) = $5, Strike price (K) = $100
Profit ($)
5
0
-10
110 120 130
70
80
90 100
Terminal
stock price (S)
-20
-30
OPT-23
Long Put
Long Put Profit = Max(K-S,0) - P
Put option premium (P) = $7, Strike price (K) = $70
30 Profit ($)
20
10
0
-7
Terminal
stock price ($)
40
50
60
70
80
90 100
OPT-24
Short Put
Short Put Profit = -[Max(K-S,0)-P] = Min(S-K,0) + P
Put option premium (P) = $7, Strike price (K) = $70
Profit ($)
7
0
40
50
Terminal
stock price ($)
60
70
80
90 100
-10
-20
-30
OPT-25
Properties of Stock Options
OPT-26
Notation
c
p
S0
ST
K
T

D
r
= European call option price (C = American)
= European put option price (P = American)
= Stock price today
=Stock price at option maturity
= Strike price
= Option maturity in years
= Volatility of stock price
= Present value of dividends over option’s life
= Risk-free rate for maturity T with
continuous compounding
OPT-27
American vs. European Options
An American option is worth at least
as much as the corresponding
European option
Cc
Pp
OPT-28
Factors Influencing Option
Values
Effect on Option Value
European
Call
Put
Input Factor
Underlying stock price
S
Strike price of option contract
K
Time remaining to expiration
T
Volatility of the underlying stock price
σ
Risk-free interest rate
r
Dividend
D
+
?
+
+
-
+
?
+
+
American
Call
Put
+
+
+
+
-
+
+
+
+
OPT-29
Effect on Option Values
Underlying Stock Price (S) & Strike Price (K)
• Payoff to call holder: Max (S-K,0)
– As S , Payoff increases; Value increases
– As K , Payoff decreases; Value decreases
• Payoff to Put holder: Max (K-S, 0)
– As S , Payoff decreases; Value decreases
– As K , Payoff increases; Value increases
OPT-30
Option Price Quotes
Calls
MSFT (MICROSOFT CORP)
$ 25.98
July 2008 CALLS
Strike
Last Sale
Bid
Ask
Vol
Open Int
15.00
10.85
10.95
11.10
10
85
17.50
10.54
8.45
8.55
0
33
20.00
6.00
6.00
6.05
4
729
22.50
3.60
3.55
3.65
195
3891
24.00
2.30
2.24
2.27
422
2464
25.00
1.50
1.45
1.48
3190
10472
26.00
0.83
0.83
0.85
2531
15764
27.50
0.31
0.29
0.31
2554
61529
OPT-31
Option Price Quotes
Puts
MSFT (MICROSOFT CORP)
$ 25.98
July 2008 PUTS
Strike
Last Sale
Bid
Ask
Vol
Open Int
15.00
0.01
0.00
0.01
0
2751
17.50
0.01
0.00
0.02
0
2751
20.00
0.01
0.01
0.02
0
5013
22.50
0.03
0.03
0.04
13
4788
24.00
0.11
0.11
0.12
50
25041
25.00
0.25
0.24
0.25
399
7354
26.00
0.45
0.45
0.47
10212
51464
27.50
0.80
0.82
0.84
2299
39324
OPT-32
Effect on Option Values
Time to Expiration = T
• For an American Call or Put:
– The longer the time left to maturity, the greater
the potential for the option to end in the
money, the grater the value of the option
• For a European Call or Put:
– Not always true due to restriction on exercise
timing
OPT-33
Option Price Quotes
MSFT (MICROSOFT CORP)
STRIKE = $25.00
CALLS
Last Sale
July 2008
1.42
August 2008
1.80
October 2008
2.36
January 2009
3.10
Bid
1.45
1.85
2.43
3.15
Ask
1.48
1.87
2.46
3.20
Vol
355
257
41
454
Open Int
10472
927
3309
59244
PUTS
July 2008
August 2008
October 2008
January 2009
Bid
0.45
0.80
1.39
2.06
Ask
0.47
0.82
1.41
2.08
Vol
419
401
215
2524
Open Int
51464
1591
25323
155877
Last Sale
0.47
0.81
1.43
2.09
25.98
OPT-34
Effect on Option Values
Volatility = σ
• Volatility = a measure of uncertainty about
future stock price movements
– Increased volatility increased upside
potential and downside risk
• Increased volatility is NOT good for the
holder of a share of stock
• Increased volatility is good for an option
holder
– Option holder has no downside risk
– Greater potential for higher upside payoff
OPT-35
Effect on Option Values
Risk-free Rate = r
• As r :
–Investor’s required return increases
–The present value of future cash
flows decreases
= Increases value of calls
= Decreases value of puts
OPT-36
Effect on Option Values
Dividends = D
• Dividends reduce the stock price
on the ex-div date
–Decreases the value of a call
–Increases the value of a put
OPT-37
Upper Bound for Options
• Call price must be ≤ stock price:
c ≤ S0
C ≤ S0
• Put price must be ≤ strike price:
p≤K P≤K
p ≤ Ke-rT
OPT-38
Upper Bound for a Call Option Price
Call option price must be ≤ stock price
• A call option is selling for $65; the
underlying stock is selling for $60.
• Arbitrage: Sell the call, Buy the stock.
– Worst case: Option is exercised; you pocket
$5
– Best case: Stock price < $65 at expiration, you
keep all of the $65.
OPT-39
Upper Bound for a Put Option Price
Put option price must be ≤ strike price
• Put with a $50 strike price is selling for $60
• Arbitrage: Sell the put, Invest the $60
– Worse case: Stock price goes to zero
• You must pay $50 for the stock
• But, you have $60 from the sale of the put (plus
interest)
– Best case: Stock price ≥ $50 at expiration
• Put expires with zero value
• You keep the entire $60, plus interest
OPT-40
Lower Bound for European Call Prices
Non-dividend-paying Stock
c  Max(S0 –Ke –rT,0)
Portfolio
Portfolio A:
1 European call
+ Ke-rT cash
If S > K
Stock
Option
Cash
Total
A
S-K
K
S
B
S
S
Portfolio B:
1 share of stock
If S < K
Stock
Option
Cash
Total
S
0
K
K
S
OPT-41
Lower Bound for European Put Prices
Non-dividend-paying Stock
p  Max(Ke -rT–S0,0)
Portfolio
Portfolio C:
1 European put
+ 1 share of stock
Portfolio D:
Ke-rT cash
If S > K
Stock
Put Option
Cash
Total
If S < K
Stock
Put Option
Cash
Total
C
S
0
D
K
K
S
S
K-S
K
K
S
OPT-42
Put-Call Parity
No Dividends
• Portfolio A: European call + Ke-rT in cash
• Portfolio C: European put + 1 share of
stock
• Both are worth max(ST , K ) at maturity
• They must therefore be worth the same
today:
c + Ke -rT = p + S0
OPT-43
9.43
Put-Call Parity
American Options
• Put-Call Parity holds only for European
options.
• For American options with no
dividends:
S0  K  C  P  S0  Ke
 rT
OPT-44
Introduction to
Binomial Trees
OPT-45
A Simple Binomial Model
(Cox, Ross, Rubenstein, 1979)
• A stock price is currently $20
• In three months it will be either $22 or $18
Stock Price = $22
Stock price = $20
Stock Price = $18
OPT-46
A Call Option
A 3-month European call option on the
stock has a strike price of $21.
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=?
Stock Price = $18
Option Price = $0
OPT-47
Setting Up a Riskless Portfolio
• Consider the Portfolio:
Long D shares
Short 1 call option
22D – 1
18D
• Portfolio is riskless when:
22D – 1 = 18D or D = 0.25
OPT-48
Valuing the Portfolio
Risk-Free Rate = 12%
• Assuming no arbitrage, a riskless portfolio must
earn the risk-free rate.
• The riskless portfolio is: Long 0.25 shares
Short 1 call option
• The value of the portfolio in 3 months is
22  0.25 – 1 = 4.50
or 18 x 0.25 = 4.50
• The value of the portfolio today is
4.5e – 0.120.25 = 4.3670
OPT-49
Valuing the Option
Description
Value
Portfolio
Long 0.25 shares
Short 1 call
$4.367
Shares
=0.25 x $20
$5.000
Call option = $5.000 – 4.367
$0.633
OPT-50
Generalization – 1-Step Tree
S0u
ƒu
S
0
S0d
ƒd
ƒ
Stock price
Option price
u> 1
d<1
At t=0
After move
up
After move
down
S0
f
S0u
fu
S0d
fd
OPT-51
Generalization
(continued)
• Consider the portfolio that is long D shares and
short 1 derivative
S0uD – ƒu
S0dD – ƒd
• The portfolio is riskless when:
S0uD – ƒu = S0d D – ƒd or
ƒu  f d
D
S0 u  S0 d
OPT-52
Generalization
(continued)
• Value of the portfolio at time T is:
S0u D – ƒu
• Cost to set up the portfolio today:
S0D – f
= Value of the portfolio today today:
S0 D – f = (S0u D – ƒu )e–rT
• Hence ƒ = S0 D(1ue-rT)+ƒu e–rT
OPT-53
Generalization
(continued)
• Substituting for D we obtain
ƒ = e–rT[ p ƒu + (1 – p )ƒd ]
where
e d
p
u d
rT
OPT-54
Risk-Neutral Valuation
• p and (1 – p ) can be interpreted as the
risk-neutral probabilities of up and
down movements
• The value of a derivative is its expected
payoff in a risk-neutral world
discounted at the risk-free rate.
• Expected payoff from option:
pfu  ( 1  p ) f d
OPT-55
Irrelevance of Stock’s Expected
Return
• Expected return on the underlying stock
is irrelevant in pricing the option
– Critical point in ultimate development of
option pricing formulas
• Not valuing option in absolute terms
• Option value = f(underlying stock price)
OPT-56
Original Example Revisited
Risk-Neutral = No Arbitrage
S0
ƒ
S0u = 22
ƒu = 1
S0d = 18
ƒd = 0
• Since p is a risk-neutral probability
20e0.12 0.25 = 22p + 18(1 – p ); p = 0.6523
• Alternatively, use the formula
e rT  d e 0.120.25  0.9
p

 0.6523
ud
1.1  0.9
OPT-57
Valuing the Option
S0
ƒ
S0u = 22
ƒu = 1
S0d = 18
ƒd = 0
Value of the option:
= e–0.12 x 0.25 [0.65231 + 0.34770]
= 0.633
OPT-58
Relevance of Binomial model
• Stock price only having 2 future price
choices appears unrealistic
• Consider:
– Over a small time period, a stock’s price can
only move up or down one tick size (1 cent)
– As the length of each time period approaches
0, the Binomial Model converges to the BlackScholes Option Pricing Model.
OPT-59
Valuing Stock Options:
The Black-Scholes Model
OPT-60
BSOPM
Black-Scholes (-Merton) Option Pricing Model
• “BS” = Fischer Black and Myron Scholes
– With important contributions by Robert Merton
• BSOPM published in 1973
• Nobel Prize in Economics in 1997
• Values European options on non-dividend
paying stock
OPT-61
BSOPM Assumptions
 m = expected return on the stock
  = volatility of the stock price
 Therefore in time Δt:
μ Δt = mean of the return
 Dt = standard deviation
and: DS ~  ( mDt ,  2 Dt )
S
OPT-62
The Lognormal Property
• Assumptions → ln ST is normally distributed with
mean:
lnS0  ( m   2 / 2 )T
and standard deviation:
 T
• Because the logarithm of ST is normal, ST is
lognormally distributed
OPT-63
The Lognormal Property
continued

l nST ~  l nS0  ( m   2 2 )T ,  2T

or

ST
ln
~  ( m   2 2 )T ,  2T
S0

where  m,v] is a normal distribution with
mean m and variance v
OPT-64
The Lognormal Distribution
E ( ST )  S0 e mT
var( ST )  S0 e
2
2 mT
(e
 2T
 1)
Restricted to positive values
OPT-65
The Expected Return
Expected value of the stock price
S0emT
Expected return on the stock with
continuous compounding
m – 2/2
Arithmetic mean of the returns
over short periods of length Δt
Geometric mean of returns
m
m – 2/2
OPT-66
Concepts Underlying Black-Scholes
• Option price and stock price depend on
same underlying source of uncertainty
• A portfolio consisting of the stock and the
option can be formed which eliminates
this source of uncertainty (riskless).
– The portfolio is instantaneously riskless
– Must instantaneously earn the risk-free
rate
OPT-67
Assumptions Underlying BSOPM
1. Stock price behavior corresponds to the
lognormal model with μ and σ constant.
2. No transactions costs or taxes. All securities
are perfectly divisible.
3. No dividends on stocks during the life of the
option.
4. No riskless arbitrage opportunties.
5. Security trading is continuous.
6. Investors can borrow & lend at the risk-free
rate.
7. The short-term rate of interest, r, is constant.
OPT-68
Notation
• c and p = European option prices
(premiums)
• S0 = stock price
• K = strike or exercise price
• r = risk-free rate
• σ = volatility of the stock price
• T = time to maturity in years
OPT-69
Formula Functions
• ln(S/K) = natural log of the "moneyness" term
• N(x) = the probability that a normally
distributed variable with a mean of zero
and a standard deviation of 1 is less than x
• N(d1) and N(d2) denote the standard normal
probability for the values of d1 and d2.
• Formula makes use of the fact that:
N(-d1) = 1 - N(d1)
OPT-70
The Black-Scholes Formulas
c  S0 N ( d 1 )  K e
 rT
N( d 2 )
p  K e  rT N ( d 2 )  S0 N ( d 1 )
w here:
d1 
ln(S0 / K )  ( r   2 / 2 )T
 T
d 2  d1   T
OPT-71
BSOPM
Example
Given:
S0 = $42
K = $40
d1 
r = 10%
T = 0.5
σ = 20%
ln(S0 / K )  ( r   2 / 2 )T
 T
ln(42 40 )  ( 0.10  0.20 2 2 )  0.5
d1 
 0.7693
0.20 0.50
d 2  d1   T
d 2  0.7693  0.20 0.50  0.6278
OPT-72
BSOPM
Call Price Example
d1 = 0.7693
N(0.7693) = 0.7791
d2 = 0.6278
N(0.6278) = 0.7349
c  S0 N ( d 1 )  K e  rT N ( d 2 )
c  40(0.7791) - 42e
c  $4.76
-.10.5
(0.7349)
OPT-73
BSOPM
Put Price Example
d1 = 0.7693
N(-0.7693) = 0.2209
d2 = 0.6278
N(-0.6278) = 0.2651
pKe
 rT
p  42 e
N ( d 2 )  S0 N ( d 1 )
 .10 .50
( 0.2651 )  40 ( 0.2209 )
p  $0.81
OPT-74
BSOPM in Excel
• N(d1):
=NORMSDIST(d1)
Note the “S” in the function
“S” denotes “standard normal”
~ Φ(0,1)
=NORMDIST() → Normal distribution
Mean and variance must be specified
OPT-75
Properties of Black-Scholes Formula
• As S0 →
Call  Fwd with delivery = K
Almost certain to be exercised
d1 and d2 → very large
N(d1) and N(d2) → 1.0
N(-d1) and N(-d2) → 0
c = max(S-K,0)
c  S0 N( d 1 )  K e  rT N( d 2 )
p = max(K-S,0)
p  K e  rT N( d 2 )  S0 N( d 1 )
c → S0 – Ke-rT
p→0
OPT-76
Properties of Black-Scholes Formula
• As S0 →0
d1 and d2 → very large & negative
N(d1) and N(d2) → 0
N(-d1) and N(-d2) → 1.0
c = max(S-K,0)
c  S0 N( d 1 )  K e  rT N( d 2 )
p = max(K-S,0)
p  K e  rT N( d 2 )  S0 N( d 1 )
c→0
p → Ke-rT – S0
OPT-77
Real Options
Real Options
• Examples
– Option to vary output / production
– Option to delay investment
– Option to expand / contract
– Option to abandon
• Use the same option valuation
approach for non-financial assets
– Assume underlying asset is traded
– Price as any financial asset
OPT-79
Example
• 1 year lease on a gold mine (T)
– Extract up to 10,000 oz
– Cost of extraction is $270 per oz (K)
– Current market price of gold is $300 per oz (S)
– Volatility of gold prices is 22.3% per annum (σ)
– Interest rate is 10% per annum
• Continuously compounded = ln(1.1) = 9.53% (r)
OPT-80
Options Approach
T
K
S
r
σ
1
$270.00
$300.00
9.53%
22.3%
c  S0 N ( d 1 )  Ke  rT N ( d 2 )
p  Ke  rT N ( d 2 )  S0 N ( d 1 )
where :
S/K
ln(S/K)
σ^2
Sqrt(T)
1.1111
0.1054
0.0497
1.0000
d1 (num)
d1 (den)
d1
N(d1)
0.2255
0.2230
1.0113
0.8441
d2
N(d2)
0.7883
0.7847
d1 
ln( S0 / K )  ( r   2 / 2 )T
 T
d 2  d1   T
European Call
c (component 1) =
c (component 2)=
c (per oz)=
Option Value (c * 10,000)
$
$
253.22
$192.62
60.599
$605,993
OPT-81
Option to Expand
• At t=1, we can expand production for t=2.
• Up-front capital investment (at t=1) of
$150k
• With the new investment, we can mine up
to 12,500 oz per year, at a per unit cost of
$280 per oz.
• How much would you pay at t=0 for this
option?
OPT-82
Multiple Options
T
K
S
r
σ
2
$270.00
$300.00
9.53%
22.3%
c  S0 N ( d 1 )  Ke  rT N ( d 2 )
p  Ke  rT N ( d 2 )  S0 N ( d 1 )
where :
S/K
ln(S/K)
σ^2
Sqrt(T)
1.1111
0.1054
0.0497
1.4142
d1 (num)
d1 (den)
d1
N(d1)
0.3457
0.3154
1.0961
0.8635
d2
N(d2)
0.7808
0.7825
d1 
ln( S0 / K )  ( r   2 / 2 )T
 T
d 2  d1   T
European Call
c (component 1) =
c (component 2)=
c (per oz)=
Option Value (c * 20,000)
$
$
259.05
$174.62
84.429
$1,688,588
OPT-83
σ=
t=
22.3%
1
 Dt
e
u=
d = 1/u
=
=
1.250
0.80
$375 x 1.25
= $469
$300 x 1.25
= $375
$375 x 0.80
= $300
$300
$375 x 1.25
= $469
$300 x 0.80
= $240
$375 x 0.80
= $300
OPT-84
If Expansion and S1 = 375
T
K
S(1)
r
σ
1
$280.00
$375.00
9.53%
22.3%
c  S0 N ( d 1 )  Ke  rT N ( d 2 )
p  Ke  rT N ( d 2 )  S0 N ( d 1 )
where :
S/K
ln(S/K)
σ^2
Sqrt(T)
1.3393
0.2921
0.0497
1.0000
d1 (num)
d1 (den)
d1
N(d1)
0.4123
0.2230
1.8489
0.9678
d2
N(d2)
1.6259
0.9480
d1 
ln( S0 / K )  ( r   2 / 2 )T
 T
d 2  d1   T
European Call
c (component 1) =
c (component 2)=
c (per oz)=
Option Value (c * 12,500)
$
$
362.91
$241.31
121.596
$1,519,953
OPT-85
If No Expansion and S1 = 375
T
K
S(1)
r
σ
1
$270.00
$375.00
9.53%
22.3%
c  S0 N ( d 1 )  Ke  rT N ( d 2 )
p  Ke  rT N ( d 2 )  S0 N ( d 1 )
where :
S/K
ln(S/K)
σ^2
Sqrt(T)
1.3889
0.3285
0.0497
1.0000
d1 (num)
d1 (den)
d1
N(d1)
0.4487
0.2230
2.0120
0.9779
d2
N(d2)
1.7890
0.9632
d1 
ln( S0 / K )  ( r   2 / 2 )T
 T
d 2  d1   T
European Call
c (component 1) =
c (component 2)=
c (per oz)=
Option Value (c * 10,000)
$
$
366.71
$236.42
130.286
$1,302,864
OPT-86
If Expansion and S1 = 240
T
K
S(1)
r
σ
1
$280.00
$240.00
9.53%
22.3%
c  S0 N ( d 1 )  Ke  rT N ( d 2 )
p  Ke  rT N ( d 2 )  S0 N ( d 1 )
where :
S/K
ln(S/K)
σ^2
Sqrt(T)
0.8571
-0.1542
0.0497
1.0000
d1 (num)
d1 (den)
d1
N(d1)
-0.0340
0.2230
-0.1524
0.4394
d2
N(d2)
-0.3754
0.3537
d1 
ln( S0 / K )  ( r   2 / 2 )T
 T
d 2  d1   T
European Call
c (component 1) =
c (component 2)=
c (per oz)=
Option Value (c * 12,500)
$
$
105.46
$90.03
15.436
$192,945
OPT-87
If No Expansion and S1 = 240
T
K
S(1)
r
σ
1
$270.00
$240.00
9.53%
22.3%
c  S0 N ( d 1 )  Ke  rT N ( d 2 )
p  Ke  rT N ( d 2 )  S0 N ( d 1 )
where :
S/K
ln(S/K)
σ^2
Sqrt(T)
0.8889
-0.1178
0.0497
1.0000
d1 (num)
d1 (den)
d1
N(d1)
0.0024
0.2230
0.0107
0.5043
d2
N(d2)
-0.2123
0.4159
d1 
ln( S0 / K )  ( r   2 / 2 )T
 T
d 2  d1   T
European Call
c (component 1) =
c (component 2)=
c (per oz)=
Option Value (c * 10,000)
$
$
121.02
$102.09
18.930
$189,299
OPT-88
Real Options Recap
S = price/oz
K = extraction cost/oz
Time
Ounces per year
Option Value
Tab
REAL OPTIONS RECAP
S = $300
S = $375
$300
$300
375
375
$270
$270
280
270
1
2
1
1
10,000
20,000
12,500
10,000
$605,993
RO 1
$1,688,588 $1,519,953 $1,302,864
RO 2
RO 3
RO 3
S = $240
240
280
1
12,500
240
270
1
10,000
$192,945
$189,299
RO 4
RO 4
OPT-89
Conclusions
S = price/oz
K = extraction cost/oz
Time
Ounces per year
Option Value
REAL OPTIONS RECAP
S = $300
S = $375
$300
$300
375
375
$270
$270
280
270
1
2
1
1
10,000
20,000
12,500
10,000
$605,993
Tab
RO 1
$1,688,588 $1,519,953 $1,302,864
RO 2
RO 3
PV =
$217,089
($150,000)
$67,089
$60,991
Value of Option to Expand
Minus cost to expand
S = $240
240
280
1
12,500
240
270
1
10,000
$192,945
$189,299
RO 4
RO 4
RO 3
$3,646
($150,000)
($146,354)
• If S1 = 375:
– Value of option to expand = $217,089
– Subtracting cost of expansion and discounting to t=0
– Value = $60,991
• If S1 = 240, net value is negative
OPT-90
Probability of Up Movement
 Dt
• Know that u = e
and d = 1/u
p
(1r f )d
u d
• In our example u=1.25 and d=.8, thus p = 0.677
• Option value = .667*$60,991 = $40,680.75
• Total lease value = 2-yr without expansion +
value of option to expand:
$1,688,588 + $40,681 = $1,729,269
OPT-91
Probability of Up Movement
REAL OPTIONS RECAP
S = $300
S = price/oz
$300
$300
K = extraction cost/oz
$270
$270
Time
1
2
Ounces per year
10,000
20,000
Option Value
$605,993
 Dt
d = 1/u
p
(1r f )d
=
=
=
1.250
0.80
0.667
375
280
1
12,500
375
270
1
10,000
$1,688,588 $1,519,953 $1,302,864
Value of Option to Expand
Minus cost to expand
u=e
S = $375
PV =
$217,089
($150,000)
$67,089
$60,991
PV * p
$40,680.75
2-yr +
Option
$1,729,269
u d
OPT-92

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