1. Chapter 13 Real-Options Analysis
Financial Options
13.1
• u = eσ ∆t
= e0.3× 0.75
= 1.2967
1 1
• d= = = 0.7712
u 1.2967
• Risk neutral probability
er ∆t − d e0.05×0.75 − 0.7712
q= = = 0.5081
u−d 1.2967 − 0.7712
• Tree valuation
100.88
q Max(0, (100.88-60)) = $40.88
q 77.80
($20.01) 1-q
60 60
9.79 q Max(0, (60-60)) = $0
1-q 46.27
($0.00) 1-q
35.68
Max(0, (35.68-60)) = $0
∴ European call option value = $9.79
13.2
• u = eσ ∆t
= e0.4× 1 = 1.4918
1 1
• d= = = 0.6703
u 1.4918
• Risk neutral probability
e r ∆t − d e0.05 − 0.6703
q= = = 0.4638
u−d 1.4918 − 0.6703
1 − q = 0.5362
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
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2. • Tree valuation:
Note to instructors: No mention is made about the dividend payment in
determining the option premium in the text. In financial option, any dividend
payment reduces the value of the option. The figures in yellow represent the
adjusted share prices after a 3% dividend payment, i.e., (1 – 0.03)($124.96) =
$121.21. If there is no dividend payment, we just use the original share prices in
determining the option premium.
Keep open 124.96
Keep open 86.35 Do not exercise
59.67 121.21 [0]
83.76
57.88 [0]
40 [ 5.34 ]
[ 12.04 ] Keep open 56.15
Exercise 38.8 Do not exercise
26.81 54.46 [0]
37.64
26.01 [ 10.47 ]
[ 18.99 ]
Exercise 25.23
17.43 Exercise
24.47 [ 20.53 ]
16.91
[ 28.09 ]
11.34
Exercise
11.00 [ 34.00 ]
∴ American option value = $12.04
13.3
Portfolio Premium Payoff at stock price $60
A long call with X = $40 $3 $20 - $3 = $17
A short put with X = $45 $4 $4 – 0 = $4
Two short call with X = $35 $5 ($5 - $25)×2 = ($40)
Two short stock at $40 ($40 - $60)×2 = ($40)
Total ($59)
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
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3. 13.4 Note to the instructors – The definitions of intrinsic value as well as time
value were not given in the text. In financial option, the option premium is
viewed as having two types of value. The intrinsic value is the value if you
exercise the option immediately. The time value is the value if you wait.
• Intrinsic value = X − S0 = $2
• Time premium = option premium – intrinsic value = $2
Real-Options Analysis
13.5
• Define the real option parameters for delaying option.
V I T r σ
$1.9 Million $2 Million 1 year 0.08 0.4
− rf T
Ccall = S0 N (d1 ) − Ke N (d 2 )
= 1.9 N (0.2718) − 2e −0.08 N (−0.1283)
= $0.3246M
ln( S0 K ) + (rf + σ 2 )T
2
d1 =
σ T
0.42
ln(1.9 / 2) + (0.08 + )1
= 2 = 0.2718
0.4 1
d 2 = d1 − σ T = 0.2718 − 0.4 1 = −0.1283
13.6
• Define the real option parameters for license option.
V* I T r σ
$30 Million $40 Million 3 0.06 0.2
* V = ($340 - $320) × 1.5M = 30M
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
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4. − rf T
Ccall = S0 N (d1 ) − Ke N (d 2 )
= 30 N (−0.0165) − 40e −0.18 N (−0.3629)
= $2.8610M
ln( S0 K ) + (rf + σ 2 )T
2
d1 =
σ T
0.22
ln(30 / 40) + (0.06 + )3
= 2 = −0.0165
0.2 3
d 2 = d1 − σ T = −0.0165 − 0.2 3 = −0.3629
Switching Options
13.7 A switching option is a special case of a put option.
• Compute the NPW of project B:
NPWB = −$2 + $1( P / A,12%,10) = $3.65M
• Define the real option parameters for the switching option and compute the put
option premium by using the Black-Scholes model.
V I T r σ
$4 Million $3.65 Million 5 0.06 0.5
Cput = 3.65e −3.0 N (0.2088) − 4 N (−0.9093)
= $0.85M
0.52
ln(4 3.65) + (0.06 + )5
d1 = 2 = 0.9093
0.5 5
d 2 = 0.9093 − 1.118 = −0.2088
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
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5. • Determine the combined option value
Combined Option Value = Value of project A + Option to switch to project B
$4 + $0.85 = $4.85M
R&D Options
13.8
• Assuming MARR = 12%, the cash flow diagram transforms to:
$73.42
0 1 2 3 4 5 6 7 8 9 10
$14.18
R&D Expenses Manufacturing and Distribution
$80
• Define the real option parameters for R&D option.
V I T r σ
$46.66 Million $80 Million 4 0.06 0.5
− rf T
Ccall = S0 N (d1 ) − Ke N (d 2 )
= 46.66 N (0.2009) − 80e −0.08 N (−0.7991)
= $13.70M
0.52
ln(46.66 / 80) + (0.06 + )4
d1 = 2 = 0.2009
0.5 4
d 2 = 0.2009 − 0.5 4 = −0.7991
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
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6. • Therefore, the total value is:
Combined option premium = Cost for R&D + Option value
= −$14.18 + $13.7 = −$0.48M
The maximum amount the firm should spend on R&D for this project is equal to
$13.7M.
Abandonment Options
13.9
• Standard NPV approach
0.35 0.35 0.35 0.35 0.35
...
0 1 2 3 4 . . . ∞
$3
$0.35
NPW0 = −$3 + = −$0.08M
0.12
• Abandon Option value through the binomial tree
- Option parameters
V 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
1.77 2.92 4.81 7.94 13.09
1.07 1.77 2.92 4.81
Monetary Value
0.65 1.07 1.77
0.40 0.65
0.24
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
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7. 0.50 0.23 0.06 0.00 0.00 0.00
0.77 0.42 0.12 0.00 0.00
1.13 0.69 0.23 0.00
Option value
1.55 1.13 0.43
1.80 1.55
1.96
* Early abandonment decisions could occur in the shaded periods.
∴ Combined option value = -0.08 + 0.50 = 0.42M
Scale-Down Options
13.10
• Scale down option parameters
V 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.
- u = eσ ∆t = e0.3× 1 = 1.35
1 1
- d= = = 0.74
u 1.35
er ∆t − d e0.06×1 − 0.74
- q= = = 0.53, 1 − q = 0.47
u−d 1.35 − 0.74
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be
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obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical,
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8. 24.60
Max(24.60×0.7+4,24.60)
24.60
Do not scale down
18.23
Max(18.23×0.7+4,18.22)
18.22
Do not scale down 13.5
13.5 Max(13.5×0.7+4,13.5)
13.5
Max(13.5×0.7+4,13.94) Do not scale down
13.94
10
Do not scale down
10 Max(10×0.7+4,10.78)
11
11.55 Scale down 7.4
7.4 Max(7.4×0.7+4,7.4)
9.18
Max(7.4×0.7+4,8.94)
Scale down
9.18
Scale down 5.48
Max(5.48×0.7+4,7.59)
7.836
Scale down 4.05
Max(4.05×0.7+4,4.05)
6.835
Scale down
• From the result of the tree we can get the combined option value as follows:
Combined option value = NPW + Option value = $11.55
∴ Option value = $11.55 - $10 = $1.55M
Expansion-Contraction Options
13.11
(a) Binomial lattice tree
• u = eσ ∆t = e0.15× 1 = 1.1618
1 1
• d= = = 0.8007
u 1.1618
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
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9. • Tree with a one-year incremental period
134.99
116.18
100
100
86.07
74.08
(b) Option valuation
• Risk neutral probability
e r ∆t − d e0.05×1 − 0.8007
q= = = 0.6328 , 1 − q = 0.3672
u−d 1.1618 − 0.8007
134.99
Max(134.99×0.9+25,134.99×1.3-20,134.99)
155.48
Expand
116.8
Max(116.8×0.9+25,116.8×1.3-20,133.76)
133.76
Keep option open 100
Max(100×0.9+25,100×1.3-20,100)
100 115
116.31 Contract
86.07
Max(86.07×0.9+25,86.07×1.3-20,101.24)
102.46 74.08
Contract
Max(74.08×0.9+25,74.08×1.3-20,74.08)
91.67
Contract
∴ Option value = $116.31 - $100 = $16.31M
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be
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obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical,
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10. Compound Options
13.12
• Compound option parameters
V0 I1 I2 T1 T2 r σ
$32.43 $10 $30 1 3 0.06 0.5
• Decision tree for a scale-down option with a one-year time increment.
- u = eσ ∆t = e0.5× 1 = 1.6487
1 1
-d= = = 0.6065
u 1.6487
e r ∆t − d e0.06×1 − 0.6065
- q= = = 0.4369
u−d 1.6487 − 0.6065
145.34
Combined OP = NPV + Option value
Max(145.34 - 30, 0)
= -$5 + $8.13 = $3.13 115.34
Invest $30
88.15
59.90
Keep option open
53.47
53.47 Max(53.47 - 30, 0)
23.47
Max(29.77 - 10, 0) Invest $30
19.77
Invest $10 32.43
32.43 9.66
Keep option open
8.13 19.67
19.67 Max(19.67 - 30, 0)
0
Max(3.97 - 10, 0)
Do not invest
0
Do not invest 11.93
0
Do not invest
7.24
Max(7.24 -30, 0)
0
Do not invest
First Option Second Option
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
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11. Short Case Studies
ST 13.1
(a) American option value
• Option parameters
S K T ∆t r σ
$150 $100 5 1 year 0.05 0.3
• u = eσ ∆t
= e0.3× 1 = 1.3499
1 1
• d= = = 0.7408
u 1.3499
• Risk neutral probability
e r ∆t − d e0.05×1 − 0.7408
q= = = 0.5097 , 1 − q = 0.4903
u−d 1.3499 − 0.7408
672.25
∴ American Option value = $6.64 4 ax(100-672.25, 0)
M
0
498.02 Do not terminate
0
Keep option open 368.94
368.94 Max(100-368.94, 0)
0 0
Keep option open 273.32 Do not terminate
0
273.32 Keep option open
0 202.48
Keep option open 202.48 Max(100-202.48, 0)
202.48 0 0
Max(100-202.48, 1.85) Keep option open 150 Do not terminate
0
1.85 150 Keep option open 111.12
Keep option open Max(100-150, 3.96)
150 Max(100-111.12, 0)
3.96 111.12
6.64 Keep option open Max(100-111.12, 8.48) 0
111.12 Do not terminate
8.48 82.32
Max(100 -111.12, 12.32) Keep option open Max(100-82.32, 18.19)
12.32 82.32
18.19
Keep option open Max(82.32-100, 22.31) Keep option open 60.99
22.31 60.99 Max(100-60.99, 0)
Keep option open Max(100-60.99, 34.39) 39.01
31.09 Terminate
Terminate 45.18
Max(100-45.18, 49.94) 33.47
54.82 Max(100-33.47, 0)
Terminate 66.53
Terminate
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
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12. (b) European option value is $6.09 by B-S equation
ST 13.2
(a) Since $4 M is lower than the option price, it is a good investment for Merck.
• To give a range for the option value, first use one period lattice.
V K T ∆t r σ
$36M $72M 3 3 years 0.06 0.5
($30×1.2M) ($60×1.2M)
• u = eσ ∆t = e0.5× 3 = 2.3774
1 1
• d= = = 0.4206
u 2.3774
• Risk neutral probability
er ∆t − d e0.06×3 − 0.4206
q= = = 0.3969 , 1 − q = 0.6031
u−d 2.3774 − 0.4206
• One period lattice and option price: $5.17M
85.59
Max(85.59 - 72, 0)
15.59
Buy the stock
36
5.17
15.14
Max(15.14 - 72, 0)
0
Do not buy the stock
• Through the B-S model, the option price should be $6.54M.
Contemporary Engineering Economics, Fourth Edition, by Chan S. Park. ISBN 0-13-187628-7.
© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be
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