Chapter 13: Real-Options Analysis
Financial Options
13.1
•Define the option parameters for the call option.
S0K T r
σ
$120 $150 0.5 0.06 0.45
The value of the call option is $6.97 by Black-Scholes equation.
Stock price (S
0
) or Project Value (V
0
)
120
Exercise Price or Investment Cost (K)
150
Volatility (σ)
0.45
d
1
–0.447892596
d
2–0.766090647
Black-Scholes Model
13.2
•Define the option parameters for the put option.
S0K T r
σ
0.06
The value of the put option is $7.20 by Black-Scholes equation.
Stock price (S
0
) or Project Value (V
0
)
68.5
Exercise Price or Investment Cost (K)
70
Put Value
7.20
d
1
0.068307205
d
2–0.176886596
Black-Scholes Model
13.3
• Long call and short call
• Long put and short put
13.4
• Define the option parameters for this option.
S0 K T r
σ
13.5
•
0.3 0.75
1.2967
t
u e e
σ
∆ ×
= = =
• Tree valuation
100.88
Max(0, (100.88-60)) = $40.88
13.6
•
0.3 1 1.3499
t
u e e
σ
∆ ×
= = =
,
1 1 0.7408
1.3499
du
= = =
• Risk neutral probability
98.39
• Binomial lattice valuation
53.99
Max(3.34,45–53.99)
Max(14.24,45–29.63)
Max(7.17,45–40)
Max(20.86,45–21.95)
53.99
29.63
16.26
72.89
Max(0,45–72.89)
Max(0,45–98.39)
0
Do not exercise
1 – q
Max(0, (60-60)) = $0
13.7
• Put-call parity:
13.8
Portfolio Premium Payoff at stock price $60
A long call with K = $40
$3
$17
13.9
• Intrinsic value =
0
max[ ,0] $2KS−=
• Time premium = option premium – intrinsic value = $2
13.10
• Invest $10,000 in the stocks
Stock Purchase
Price
Initial
Cost of
Stock 100
shares
Stock, Price
at Expiration
Value of Stock
at Expiration
Payoff
$100
$10,000
$95
$9,500
($500)
• Invest $10,000 in the call options
Option
Price
per
Contract
Initial Cost of
Options (10
contacts)
Profit Per
Option
at
Expiration
Total Profit of
Options
Payoff
$1000
$10,000
$0
$0
($10,000)
$1000
$10,000
$5
$1000
$10,000
$100
$10,000
$1,500
13.11
(a)
S
0
=100
K= 105
183.35
135.41
78.35
36.74
100 100.00
(b)
S
0
=100
K= 100
T= 1
169.05 0.00
141.91 0.00 141.91
119.12 0.00 119.12 0.00
(c)
S0=100
K= 100
T= 1
r= 5%
169.05
0.00
141.91
0.00 141.91
119.12
0.00 119.12 0.00
100
4.04 100.00 0.00 100.00
13.12
(a)
• Define the option parameters for this call option.
S0 K T r
σ
(b)
• Define the option parameters for this put option.
S0 K T r
σ
(c)
• Define the option parameters for this put option.
S0 K T r
σ
(d)
• Define the option parameters for this call option.
S0 K T r
σ
13.13
• The accumulated cost of the hedge at the end of year one is
0.06
($50, 000 $38,000) $12,742e−=
Let S be the market price:
Real–Options Analysis
13.14
• Define the real option parameters for delaying option.
V0 I T r
σ
13.15
• Define the real option parameters for license option.
V0 I T r
σ
Growth Options
13.16
• Define the real option parameters for this option.
(a) Optimal Cutting Policy – Wait until Year 3.
0 1 2 3
r0.06
(b) Optimal Cutting Policy – Wait until Year 3.
1.6 1.5 1.4
0123
r0.06
∆t16.5625
u1.25
d0.8
Deferral Options
13.17
• Define the real option parameters for deferral option.
V0 I1 I2 T r
σ
The value of postponing the construction decision for two years: $349,743
0 2
I1 =38,588
I2 =60,638
σ = 0.2
38,588=35,000(1.05)(1.05)
60,638=55,000(1.05)(1.05)
∴
Switching Options
13.18
• The NPW of project B:
B
PW(12%) $2 $1( / ,12%,10) $3.65M=−+ =PA
• Define the real option parameters for switching option.
V0 I T r
σ
R&D Options
13.19
• Assuming MARR = 12%, the cash flow diagram transforms to:
0
1
2
3
4
5
6
7
8
9
10
$14.18
$80
$73.42
• Define the real option parameters for R&D option.
V0 I T r
σ
$46.66 Million $80 Million 4 0.06 0.5
Abandonment Options
13.20
• Standard NPW approach
• Abandon Option value through the binomial tree
– Option parameters
V
0
I
T
r
σ
$2.92 Million $2.2 Million 5 0.06 0.5
– Option valuations
Time
0
1
2
3
4
5
2.92
4.81
7.94
13.09
21.58
35.57
Option value
1.55
1.13
0.43
1.80
1.55
1.96
* It is optimal to exercise early at the shaded positions.
Scale–Down Options
13.21
• Scale down option parameters
V
0
I
T
r
σ
$10 Million
$4 Million
3
0.06
0.3
• Decision tree for a scale–down option through one-year time increment.
0.65
1.07
1.77
0.40
0.65
0.50
0.23
0.06
0.00
0.00
0.00
0.77
0.42
0.12
0.00
0.00
0 1 2 3
K = 4
σ =
0.3
r =
0.06
=max(18.22*0.8+4,EXP(–0.06)*(24.6*0.53+14.8*(1–0.53))) =18.8
• From the result of the tree we can get the flexible NPV:
Expansion–Contraction Options
13.22
(a) Binomial lattice tree
86.07
74.08
(b) Option valuation
• Risk neutral probability
134.99
134.99
Max(134.99
×
0.9+25,134.99
×
1.3–20,134.99)
155.48
Expand
74.08
Max(86.07
×
0.9+25,86.07
×
1.3–20,101.24)
102.46
Contract
116.8
100
Max(116.8
0.9+25,116.8
1.3–20,133.76)