Chapter 13opciones financieras

Chapter 13 Real-Options Analysis Financial Options

13.1 • 0.3 0.75 1.2967tu e eσ ∆ ×= = =

• 1 1 0.77121.2967

du

= = =

• Risk neutral probability

0.05 0.75 0.7712 0.5081

1.2967 0.7712

r te d equ d

∆ ×− −= = =

− −

• 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 • 0.4 1 1.4918tu e eσ ∆ ×= = =

• 1 1 0.67031.4918

du

= = =


0.05 0.6703 0.4638

1.4918 0.6703

r te d equ d

∆ − −= = =

− −

1 0.5362q− =

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

obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department,

Pearson Education, Inc., Upper Saddle River, NJ 07458.

• 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)

2Contemporary 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 obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical,

photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

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 = 0 $2X S− = • 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

2

call 0 1 20.08

201

2

2 1

( ) ( )

1.9 (0.2718) 2 ( 0.1283)$0.3246Mln( ) ( )

0.4ln(1.9 / 2) (0.08 )12 0.2718

0.4 10.2718 0.4 1 0.1283

fr T

f

C S N d Ke N d

N e N

S K r Td

T

d d T

σ

σ

σ

−

−

= −

= − −=

+ +=

+ += =

= − = − = −

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




2

call 0 1 20.18

201

2

2 1

( ) ( )

30 ( 0.0165) 40 ( 0.3629)$2.8610Mln( ) ( )

0.2ln(30 / 40) (0.06 )32 0.0165

0.2 30.0165 0.2 3 0.3629

fr T

f

C S N d Ke N d

N e N

S K r Td

T

d d T

σ

σ

σ

−

−

= −

= − − −=

+ +=

+ += =

= − = − − = −

−

Switching Options

13.7 A switching option is a special case of a put option.

• Compute the NPW of project B:

NPW $2 $1( / ,12%,10) $3.65MB P A= − + = • 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

3.0put

2

1

2

3.65 (0.2088) 4 ( 0.9093)

$0.85M0.5ln(4 3.65) (0.06 )5

2 0.90930.5 5

0.9093 1.118 0.2088

C e N N

d

d

−= − −

=

+ += =

= − = −




• 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:

Manufacturing and Distribution R&D Expenses

$73.42

$14.18109876543210

$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

call 0 1 20.08

2

1

2

( ) ( )

46.66 (0.2009) 80 ( 0.7991)$13.70M

0.5ln(46.66 / 80) (0.06 )42 0.2009

0.5 40.2009 0.5 4 0.7991

fr TC S N d Ke N d

N e N

d

d

−

−

= −

= − −=

+ += =

= − = −




• 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$0.35NPW $3 $0.08M0.12

= − + = −

0.35 0.35 0.35 0.35 0.35

. . . ∞...43210

$3

• 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 0.65 1.07 1.77 0.40 0.65

Monetary Value

0.24




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 1.55 1.13 0.43 1.80 1.55

Option value

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.

- 0.3 1 1.35tu e eσ ∆ ×= = =

- 1 1 0.741.35

du

= = =

- 0.06 1 0.74 0.53, 1 0.47

1.35 0.74

r te d eq qu d

∆ ×− −= = = − =

− −




11.55 10

13.5 Max(13.5×0.7+4,13.94) 13.94 Do not scale down

18.23 Max(18.23×0.7+4,18.22) 18.22 Do not scale down

10 Max(10×0.7+4,10.78) 11 Scale down

5.48 Max(5.48×0.7+4,7.59) 7.836 Scale down

Max(24.60×0.7+4,24.60) 24.60 Do not scale down

24.60

4.05

Max(7.4×0.7+4,7.4) 9.18 Scale down

7.4

Max(13.5×0.7+4,13.5) 13.5 Do not scale down

13.5

Max(7.4×0.7+4,8.94) 9.18 Scale down

7.4

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

• 0.15 1 1.1618tu e eσ ∆ ×= = =

• 1 1 0.80071.1618

du

= = =




• Tree with a one-year incremental period

) Option valuation


(b

134.99

116.18100

100

86.07

74.08

0.05 1 0.8007 0.6328

1.1618 0.8007

r te d equ d

∆ ×− −= = =

− −, 1 0.3672q− =

100

134.99 Max(134.9 ,134.99×1.3-20,134.99)

Max(100× 25,100×1.3-20,100)

74.08 Max(74.08 ,74.08×1.3-20,74.08)

ct

86.07 Max(86.07×0 6.07×1.3-20,101.24)

116.8 Max(116.8 ,116.8×1. 3.76)

tion open

102.46 Contract

.9+25,8

91.67 Contra

×0.9+25

115 Contract

100 133.76 Keep op

3-20,13×0.9+25

0.9+

155.48 Expand

9×0.9+25

116.31

∴ Option value = $116.31 - $100 = $16.31M




Compound Options

13.12 • Compound option parameters

0V 1I 2I 1T 2T 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.

- 0.5 1 1.6487tu e eσ ∆ ×= = =

- 1 1 0.60651.6487

du

= = =

- 0.06 1 0.6065 0.4369

1.6487 0.6065

r te d equ d

∆ ×− −= = =

− −

Combined OP = NPV + Option value = -$5 + $8.13 = $3.13

53.47 Max(29.77 - 10, 0) 19.77 Invest $10

88.15 59.90

Keep option open

32.43 9.66

Keep option open

11.93 0

Do not invest

Max(145.34 - 30, 0) 115.34 Invest $30

145.34

7.24

Max(19.67 - 30, 0) 0 Do not invest

19.67

Max(53.47 - 30, 0) 23.47 Invest $30

53.47

Max(3.97 - 10, 0) 0 Do not invest

19.67

32.43 8.13

Max(7.24 -30, 0) 0 Do not invest

First Option Second Option




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 • 0.3 1 1.3499tu e eσ ∆ ×= = =

• 1 1 0.74081.3499

du

= = =


0.05 1 0.7408 0.5097

1.3499 0.7408

r te d equ d

∆ ×− −= = =

− −, 1 0.490q 3− =

∴ American Option value = $6.64

368.94Max(100-368.94, 0) 0 Do not terminate

672.254Max(100-672.25, 0) 0 Do not terminate

150 6.64

202.48 Max(100-202.48, 1.85) 1.85 Keep option open

111.12 Max(100 -111.12, 12.32) 12.32 Keep option open

273.320

Keep option open

150 Max(100-150, 3.96) 3.96 Keep option open

82.32 Max(82.32-100, 22.31) 22.31 Keep option open

368.940

Keep option open

202.480

Keep option open


60.99 Max(100-60.99, 34.39) 31.09 Terminate

498.020

Keep option open


273.320

Keep option open

1500

Keep option open

45.18 Max(100-45.18, 49.94) 54.82 Terminate

60.99 Max(100-60.99, 0) 39.01 Terminate

111.12 Max(100-111.12, 0) 0 Do not terminate

202.48 Max(100-202.48, 0) 0 Do not terminate

33.47 Max(100-33.47, 0) 66.53 Terminate




(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) • 0.5 3 2.3774tu e eσ ∆ ×= = =

• 1 1 0.42062.3774

du

= = =


0.06 3 0.4206 0.3969

2.3774 0.4206

r te d equ d

∆ ×− −= = =

− −, 1 0.6031q− =

• One period lattice and option price: $5.17M

Max(85.59 - 72, 0) 15.59 Buy the stock

85.59

15.14

36 5.17

Max(15.14 - 72, 0) 0 Do not buy the stock

• Through the B-S model, the option price should be $6.54M.




Documents

Chapter 13opciones financieras