Transcript c = max(SK,0)
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. Identity of the underlying stock 2. Strike or Exercise price 3. Contract size 4. Expiration date or maturity 5. 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: • S 0 • S T = current stock price per share = 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 Price 0 $10 $20 $30 $40 Call Holder Payoff $0 $0 $0 $10 $20 Profit -$5 -$5 -$5 $5 $15
Payoff: Profit: Max(S-K,0) Max (S-K,0) – c
Call Writer Payoff $0 Profit $5 $0 $0 -$10 -$20 $5 $5 -$5 -$15
-Max(S-K,0) -[Max (S-K, 0)-p]
OPT-13
Payoff & Profit Profiles for Calls
Call Payoff and Profit
$25 $20 $15 $10 $5 $0 -$5 -$10 -$15 -$20 -$25 0 $10 Call Holder $20 $30 Call Writer $40
Stock Price
Call Holder Payoff Call Writer Payoff Call Holder Profit Call Writer Profit
OPT-14
Payoff & Profit Profiles for Calls
Profit Payoff Call Holder Profit 0 Stock Price Call Writer Profit OPT-15
Option Payoffs and Profits
Put Holder
Payoffs to Put Holder
0 if S > K (K - S) 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 Price 0 $10 $20 $30 $40 Put Holder Payoff $20 $10 Profit $15 $5 $0 $0 $0 -$5 -$5 -$5
Payoff: Profit: Max(K-S,0) Max (K-S,0) – p
Put Writer Payoff -$20 -$10 Profit -$15 -$5 $0 $0 $0 $5 $5 $5
-Max(K-S,0) -[Max (K-S, 0)-p]
OPT-18
Payoff & Profit Profiles for Puts
Put Payoff and Profit
$25 $20 $15 -$10 -$15 -$20 -$25 $10 $5 $0 -$5 0 Put Holder $10 Put Writer $20 $30 $40 Put Holder Profit
Stock Price
Put Writer Payoff Put Holder Payof Put Writer Profit
OPT-19
Payoff & Profit Profiles for Puts
Profits 0 Put Writer Profit Put Holder Profit Stock Price OPT-20
Option Payoffs and Profits
Holder: Payoff (Long) Profit Writer: Payoff (Short) Profit CALL PUT Max (S-K,0) Max (K-S,0) Max (S-K,0) - C Max (K-S,0) - P “Bullish” “Bearish” Min (K-S,0) Min (S-K,0) Min (K-S,0) + C Min (S-K,0) + P “Bearish” “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 0 -5 70 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) = $100 5 0 -10 Profit ($) 70 80 90 100 110 120 130 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 40 50 60 70 80 90 Terminal stock price ($) 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) = $70 7 0 Profit ($) 40 50 60 70 -10 80 90 Terminal stock price ($) 100 -20 -30
OPT-25
Properties of Stock Options
OPT-26
c p S
0
S T K T
D r
Notation
= 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
Input Factor Underlying stock price Strike price of option contract Time remaining to expiration Volatility of the underlying stock price Risk-free interest rate Dividend
S K T
σ
r D
Effect on Option Value European Call Put + + + ?
+ + + ?
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
$ 25.98
MSFT (MICROSOFT CORP) July 2008 CALLS Strike 15.00
Last Sale 10.85
Bid 10.95
17.50
20.00
22.50
24.00
25.00
26.00
27.50
10.54
6.00
3.60
2.30
1.50
0.83
0.31
8.45
6.00
3.55
2.24
1.45
0.83
0.29
Ask 11.10
8.55
6.05
3.65
2.27
1.48
0.85
0.31
Vol 10 0 4 195 422 3190 2531 2554 Open Int 85 33 729 3891 2464 10472 15764 61529 OPT-31
Option Price Quotes
Puts
$ 25.98
MSFT (MICROSOFT CORP) July 2008 PUTS Strike Last Sale Bid 15.00
17.50
20.00
22.50
24.00
25.00
26.00
27.50
0.01
0.01
0.01
0.03
0.11
0.25
0.45
0.80
0.00
0.00
0.01
0.03
0.11
0.24
0.45
0.82
Ask 0.01
0.02
0.02
0.04
0.12
0.25
0.47
0.84
Vol 0 0 0 13 50 399 10212 2299 Open Int 2751 2751 5013 4788 25041 7354 51464 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 July 2008 Last Sale 1.42
August 2008 October 2008 January 2009 1.80
2.36
3.10
PUTS July 2008 August 2008 October 2008 January 2009 Last Sale 0.47
0.81
1.43
2.09
Bid 1.45
1.85
2.43
3.15
Bid 0.45
0.80
1.39
2.06
25.98
Ask 1.48
1.87
2.46
3.20
Ask 0.47
0.82
1.41
2.08
Vol 355 257 41 454 Open Int 10472 927 3309 59244 Vol 419 401 215 2524 Open Int 51464 1591 25323 155877 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 ≤ S 0 C ≤ S 0
• 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(S
0
–Ke
–rT
,0)
Portfolio A: 1 European call + Ke -rT cash Portfolio B: 1 share of stock
If S > K Stock Option Cash Total If S < K Stock Option Cash Total A Portfolio B S S - K K S S 0 K K S S OPT-41
Lower Bound for European Put Prices Non-dividend-paying Stock Portfolio C:
p
Max(Ke
-rT
–S
0 ,0) Portfolio
1 European put + 1 share of stock
If S > K Stock Put Option Cash Total C S 0 S
Portfolio D:
D K K
Ke -rT cash
If S < K Stock Put Option Cash Total 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( S T , K )
at maturity • They must therefore be worth the same today:
c + Ke
-rT
= p + S 0
Put-Call Parity American Options • Put-Call Parity holds only for European options.
• For American options with no dividends:
S 0
K
C
P
S 0
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 = $20 Stock Price = $22 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 22 D – 1 18 D • Portfolio is riskless when: 22 D – 1 = 18 D 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
Portfolio Description Long 0.25 shares Short 1 call Shares =0.25 x $20 Call option = $5.000 – 4.367
Value $4.367
$5.000
$0.633
OPT-50
Generalization – 1-Step Tree
Stock price Option price
S 0 u
ƒ
u S 0
ƒ
At
t=0
S 0 f S 0 d
ƒ
d
After move up
S 0 u f u u> 1 d < 1
After move down
S 0 d f d
OPT-51
Generalization
(continued) • Consider the portfolio that is long D shares and short 1 derivative
S 0 u
D
– ƒ
u S 0 d
D
– ƒ
d
• The portfolio is riskless when:
S 0 u
D
– ƒ
u
=
S 0 d
D
– ƒ
d
or D
ƒ u
S 0 u
f d S 0 d
OPT-52
Generalization
(continued) • Value of the portfolio at time
T
is:
S 0 u
D – ƒ
u
• Cost to set up the portfolio today:
S 0
D –
f
= Value of the portfolio today today:
S 0
D –
f =
(
S 0 u
D – ƒ
u
)
e
–
rT
• Hence ƒ =
S 0
D(1
ue -rT )+
ƒ
u e
–
rT
OPT-53
Generalization
(continued) • Substituting for D we obtain
ƒ = e –rT [ p ƒ
u
+ (1 – p )ƒ
d ]
where
p
e rT u
d d
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:
pf u
( 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
S 0 u
ƒ
u
= 22 = 1
S 0
ƒ
S 0 d
ƒ
d
= 18 = 0
• Since
p
is a risk-neutral probability 20
e
0.12 0.25 = 22
p
+ 18(1 –
p
);
p
= 0.6523
• Alternatively, use the formula
p
e rT u
d d
e
0.12
0.25
1 .
1
0 .
9 0 .
9
0 .
6523
OPT-57
Valuing the Option
S 0 u
ƒ
u
= 22 = 1
S 0
ƒ
S 0 d
ƒ
d
= 18 = 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 Black Scholes 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
D
t
= mean of the return = standard deviation and: D
S
~
( S
m D
t ,
2
D
t )
OPT-62
The Lognormal Property
• Assumptions →
ln S T
mean:
ln S
0
(
m is normally distributed with
2
/
2
) T
and standard deviation :
T
• Because the logarithm of
S T
lognormally distributed is normal,
S T
is
OPT-63
The Lognormal Property
continued
ln S T
or
~
ln S 0
(
m
2 ln S T S 0 ~
(
m
2 2 ) T ,
2 ) T ,
2 T
2 T
where
m , v
] is a normal distribution with mean
m
and variance
v
OPT-64
The Lognormal Distribution
E ( S T )
S 0 e
m
T
var
( S T )
S 0 2 e 2
m
T ( e
2 T
1 )
Restricted to positive values
OPT-65
The Expected Return
Expected value of the stock price Expected return on the stock with continuous compounding Arithmetic mean of the returns over short periods of length Δt Geometric mean of returns
S
m – 0
e
m
T
2 /2 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)
r S K σ T 0
= stock price = strike or exercise price = risk-free rate = volatility of the stock price = 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(-d 1 ) = 1 - N(d 1 )
OPT-70
The Black-Scholes Formulas
c p
S 0 N ( d 1 )
K e
rT N ( d 2 K e
rT N (
d 2 )
S 0 N ( )
d 1 )
where :
d 1 d 2
ln( S 0 / K )
( r
T
2 / 2 ) T
d 1
T
OPT-71
BSOPM Example Given: S 0 = $42 K = $40 r = 10% T = 0.5
σ = 20%
d 1 d 1
ln( S 0 ln( 42 / K )
( r
2 / 2 ) T
T 40 )
( 0 .
10
0 .
20 2 0 .
20 d 2
d 1
T d 2
0 .
7693
0 .
20 0 .
50 0 .
50 2 )
0 .
5
0 .
6278
0 .
7693
OPT-72
BSOPM Call Price Example
d 1 = 0.7693
d 2 = 0.6278
N(0.7693) = 0.7791
N(0.6278) = 0.7349
c
S 0 N ( d 1 )
K e
rT N ( d 2 )
c
40(0.7791) 42e .10
.5
(0.7349) c
$4.76
OPT-73
BSOPM Put Price Example
d 1 = 0.7693
d 2 = 0.6278
N(-0.7693) = 0.2209
N(-0.6278) = 0.2651
p
p
K e
rT N (
d 2 )
S 0 N (
d 1 ) 42 e
.
10
.
50 ( 0 .
2651 )
40 ( 0 .
2209 ) p
$ 0 .
81
OPT-74
BSOPM in Excel
• N(d 1 ): =NORM
S
DIST(d 1 ) 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
S
0 →
c = max(S-K,0) p = max(K-S,0)
Call Fwd with delivery = K Almost certain to be exercised d 1 and d 2 → very large N(d 1 ) and N(d 2 ) → 1.0
N(-d 1 ) and N(-d 2 ) → 0
c p
S 0 N ( d 1 )
K e
rT N ( d 2 K e
rT N (
d 2 )
) S 0 N (
d 1 ) c p → S
0 →
0
– Ke -rT
OPT-76
Properties of Black-Scholes Formula • As
S
0 →
0
c = max(S-K,0) p = max(K-S,0)
d 1 and d 2 → very large & negative N(d 1 ) and N(d 2 ) → 0 N(-d 1 ) and N(-d 2 ) → 1.0
c p
S 0 N ( d 1 )
K e
rT N ( d 2 K e
rT N (
d 2 )
) S 0 N (
d 1 ) c
→
0
p
→
Ke -rT – S
0 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
T K S r σ S/K ln(S/K) σ^2 Sqrt(T) d1 (num) d1 (den) d1 N(d1) d2 N(d2)
Options Approach
1 $270.00
$300.00
9.53% 22.3%
c p
S 0 N ( d 1 )
Ke
rT N ( d 2
Ke
rT N (
d 2 )
S 0 N ( )
d 1 )
1.1111
0.1054
0.0497
1.0000
where :
d 1 d 2
ln( S 0 / K )
( r
T
2
d 1
T / 2 ) T
0.2255
0.2230
1.0113
0.8441
European Call 0.7883
0.7847
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
T K S r σ S/K ln(S/K) σ^2 Sqrt(T) d1 (num) d1 (den) d1 N(d1) d2 N(d2)
Multiple Options
2 $270.00
$300.00
9.53% 22.3%
c p
S 0 N ( d 1 )
Ke
rT N ( d 2
Ke
rT N (
d 2 )
S 0 N ( )
d 1 )
1.1111
0.1054
0.0497
1.4142
where :
d 1 d 2
ln( S 0 / K )
( r
T
2
d 1
T / 2 ) T
0.3457
0.3154
1.0961
0.8635
European Call 0.7808
0.7825
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 =
$300
e
D
t
22.3% u = = 1
d
= 1/
u =
1.250
0.80
$375 x 1.25
= $469 $300 x 1.25
= $375 $375 x 0.80
= $300 $375 x 1.25
= $469 $300 x 0.80
= $240 $375 x 0.80
= $300 OPT-84
T K S(1) r σ S/K ln(S/K) σ^2 Sqrt(T) d1 (num) d1 (den) d1 N(d1) d2 N(d2)
If Expansion and S 1 = 375
1 $280.00
$375.00
9.53% 22.3% 1.3393
0.2921
0.0497
1.0000
0.4123
0.2230
1.8489
0.9678
1.6259
0.9480
c p
S 0 N ( d 1 )
Ke
rT N ( d 2 Ke
rT N (
d 2 )
S 0 N ( )
d 1 )
where :
d 1
ln( S 0 / K )
( r
T
2 / 2 ) T d 2
d 1
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 S 1 = 375
T K S(1) r σ 1 $270.00
$375.00
9.53% 22.3%
c p
S 0 N ( d 1 )
Ke
rT N ( d 2 Ke
rT N (
d 2 )
S 0 N ( )
d 1 )
S/K ln(S/K) σ^2 Sqrt(T) 1.3889
0.3285
0.0497
1.0000
where :
d 1
ln( S 0 / K )
( r
T
2 / 2 ) T d 2
d 1
T
d1 (num) d1 (den) d1 N(d1) 0.4487
0.2230
2.0120
0.9779
European Call d2 N(d2) 1.7890
0.9632
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
T K S(1) r σ S/K ln(S/K) σ^2 Sqrt(T) d1 (num) d1 (den) d1 N(d1) d2 N(d2)
If Expansion and S 1 = 240
1 $280.00
$240.00
9.53% 22.3%
c p
S 0 N ( d 1 )
Ke
rT N ( d 2
Ke
rT N (
d 2 )
S 0 N ( )
d 1 )
0.8571
-0.1542
0.0497
1.0000
where :
d 1
ln( S 0 / K )
( r
T
2 / 2 ) T d 2
d 1
T
-0.0340
0.2230
-0.1524
0.4394
European Call -0.3754
0.3537
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 S 1 = 240
T K S(1) r σ 1 $270.00
$240.00
9.53% 22.3%
c p
S 0 N ( d 1 )
Ke
rT N ( d 2 Ke
rT N (
d 2 )
S 0 N ( )
d 1 )
S/K ln(S/K) σ^2 Sqrt(T) 0.8889
-0.1178
0.0497
1.0000
where :
d 1
ln( S 0 / K )
( r
T
2 / 2 ) T d 2
d 1
T
d1 (num) d1 (den) d1 N(d1) 0.0024
0.2230
0.0107
0.5043
European Call d2 N(d2) -0.2123
0.4159
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 $300 $270 REAL OPTIONS RECAP S = $300 S = $375 $300 $270 375 280 375 270 1 10,000 2 20,000 1 12,500 1 10,000 $605,993 $1,688,588 $1,519,953 $1,302,864 $192,945 RO 1 RO 2 RO 3 RO 3 240 280 S = $240 240 270 1 12,500 1 10,000 RO 4 $189,299 RO 4 OPT-89
S = price/oz K = extraction cost/oz Time Ounces per year Option Value Tab
Conclusions
$300 $270 REAL OPTIONS RECAP S = $300 S = $375 1 10,000 $300 $270 2 20,000 375 280 1 12,500 375 270 1 10,000 240 280 1 S = $240 12,500 $605,993 $1,688,588 $1,519,953 $1,302,864 $192,945 RO 1 RO 2 RO 3 RO 3 RO 4 240 270 1 10,000 $189,299 RO 4 Value of Option to Expand Minus cost to expand $217,089 ($150,000) $67,089 $60,991 PV =
• If S 1 = 375: – Value of option to expand = $217,089 – Subtracting cost of expansion and discounting to
t=0
– Value = $60,991 • If S 1 = 240, net value is negative
$3,646 ($150,000) ($146,354) OPT-90
Probability of Up Movement
• Know that
u
=
e
D
t
and
d
= 1/
u p
( 1
r f u
)
d
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
S = price/oz K = extraction cost/oz Time Ounces per year REAL OPTIONS RECAP S = $300 $300 $270 1 10,000 $300 $270 2 20,000 375 280 S = $375 375 270 1 12,500 1 10,000 Option Value $605,993 $1,688,588 $1,519,953 $1,302,864 Value of Option to Expand Minus cost to expand
d e
D
t
= 1/
u = p
( 1
u r
)
d
d
1.250
0.80
0.667
PV = PV * p $217,089 ($150,000) $67,089 $60,991 $40,680.75
2-yr + Option $1,729,269 OPT-92