Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
Chapter 13 Real-Options Analysis Financial Options 13.1 •
Define the option parameters for the call option. S 0
K
T
r
σ
$80.38
$90
0.5
0.06
0.8
The value of the call option is $15.39 by Black-Scholes equation.
13.2 •
Define the option parameters for the put option. S 0
K
T
r
σ
$136.08
$160
0.167
0.05
0.6
The value of the put option is $28.39 by Black-Scholes equation.
13.3 •
Long call and short call
Page | 1
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
Long put and short put
•
13.4 Define the option parameters for this option.
•
S 0
K
T
r
σ
?
$52
0.75
0.05
0.6
To get a value value of call option option of $4 $4 by Black-Scholes Black-Scholes equation, equation, the current stock price should be $37.5 by Goal seek function in Excel.
∴
13.5 •
u
=
d = •
σ
e
1 u
=
Δt
=
e
0.3× 0.75
1 1.2967
=
= 1.2967
0.7712
Risk neutral probability q=
e
r Δt
−d
u − d
=
e
0.05 0.05×0.75 0.75
− 0.7712
1.2967 − 0.7712
=
0.5081
Page | 2
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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
•
100.88 Max(0, (100.88-60)) = $40.88 q
60 $9.81
77.80 $20.02 60 Max(0, (60-60)) = $0
1-q 46.27 $0.00
35.68 Max(0, (35.68-60)) = $0
European call option value = $9.81
13.6 =
σ
Δt
= 1.3499 ,
u
•
Risk neutral probability r t 0.05 1 − 0.7408 e −d e = q= u − d 1.3499 − 0.7408
e
=
0.3× 1
•
e
Δ
•
d =
1 u
=
1 1.3499
=
0.7408
×
=
0.5097
98.39
Tree valuation 72.89 Max(0,45-72.89) 0 Do not exercise
53.99
40
Max(3.34,45-53.99) 3.34 Do not exercise
$8.79
29.63 Max(14.24,45-29.63) 15.37 Do not exercise
40 Max(7.17,45-40) 7.17 Do not exercise
21.95 Max(20.86,45-21.95) 23.05 Exercise
American option value = $8.79
Max(0,45-98.39) 0 Do not exercise
53.99 Max(0,45-53.99) 0 Do not exercise
29.63 Max(0,45-29.63) 15.37 Exercise
16.26 Max(0,45-16.26) 28.74 Exercise
Page | 3
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
13.7 We may use the Goal Seek function in Excel to find the value of K by inputting the known parameters into the Black-Scholes equation. Or we may use a known result of put-call parity in financial option. The put-call parity shows the relationship between the price of a European call option and the price of a European put option when they have the same strike price and maturity date. −r T
c+Ke f
0.06(1)
35.15+Ke−
= p+S0 =13.95+90
K =$73.05 If the equation above does not hold, there are arbitrage opportunities. 13.8 Portfolio A long call with K = $40 A short put with K = $45 Two short calls with K = $35 Two stocks shorted at $40 Total
Premium $3 $4 $5
Payoff at stock price $60 $17 $4 ($40) ($40) ($59)
13.9 •
Intrinsic value = S0 − K = $2
•
Time premium = option premium – intrinsic value = $2
13.10 •
Invest $10,000 in the stocks
Stock Purchase Price
Initial Cost of Stock 400 shares
Stock, Price at Expiration
Value of Stock at Expiration
Payoff
$25
$10,000
$27
$10,800
$800
$25
$10,000
$30
$12,000
$2,000
$25
$10,000
$40
$16,000
$6,000
Page | 4
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
•
Invest $10,000 in the call options Option Price per Contract $4
Initial Cost of Options (2500)
$4 $4
Profit Per Option at Expiration
Total Profit of Options
Payoff
$0
$0
($10,000)
$3
$7,500
($2,500)
$13
$32,500
$22,500
$10,000 $10,000 $10,000
13.11 (a)
S0 = K= T= r= u= d=
volatility =
Discount rate per period
q= 1-q = w=
100 105 1.5 5% 1.354 0.739 35% 0.49 0.51 0.9632
183.35 135.41
78.35
36.74
100
100.00
17.23
73.85
0.00
0.00
54.54 17.23 = 0.9632*(0.49(36.74 )+ 0.51(0))
0.00 0= 0.9632(0.49(0) + 0.51(0))
Option value $
17.23
Page | 5
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
(b) S 0= K= T= r= u= d=
volatility =
Discount r ate per period
q= 1-q = w=
100 100 1 5% 1.191 0.839 35% 0.49 0.51 0.9876
201.38 169.05 141.91 119.12
0.00
141.91
0.00
119.12
0.00
0.00
3.73
100
100.00
100.00
0.00
10.43
83.95
83.95
7.43
0.00
17.18
70.47 27.06
17.18= 0.9876(0.46(7.43) + 0.54(27.06))
70.47
14.81
59.16 39.60
29.53
49.66 50.34
Option value=
$ 10.43
MAX(0, 100 - 49.66) = 50.34
Page | 6
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
(c) 100 100 1 5% 1.191 0.839 35% 0.49 0.51 0.9876
S 0= K= T= r= u= d=
volatility =
Discount rate per period
q= 1-q = w=
201.38 169.05 141.91 119.12
100
100.00
70.47
18.73
0.00
100.00
0.00
83.95
8.05
0.00
141.91
0.00
119.12
0.00
4.04
83.95
11.36
0.00
70.47
16.05
11.36 = 0.9876*(0.49(4.04 )+ 0.51(18.73))
59.16
29.53
$ 11.36
29.53
49.66
40.84
MAX(18.73, 100 - 83.95) = 18.73 Option value=
50.34 MAX(39.20, 100 - 59.16) = 40.84
MAX(0, 100 - 49.66) = 50.34
13.12 (a) • Define the option parameters for this call option. S 0
K
T
r
σ
$40
$40
1.167
0.06
0.4
The value of call option is $8.05 by Black-Scholes equation.
Page | 7
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
(b) • Define the option parameters for this put option. S 0
K
T
r
σ
$50
$55
1.5
0.06
0.2
The value of put option is $5.02 by Black-Scholes equation.
(c) • Define the option parameters for this put option. S 0
K
T
r
σ
$38
$40
0.25
0.06
0.6
The value of put option is $5.35 by Black-Scholes equation.
(d) •
Define the option parameters for this call option. S 0
K
T
r
σ
$100
$95
3
0.08
0.4
The value of call option is $38.27 by Black-Scholes equation.
13.13 •
The accumulated cost of the hedge at the end of year one is ($50,000 - $38,000)exp(0.06) = $12,742.
Let S be the market price: •
If S < $1.25, the put option is in the money and the payoff is $1,000,000(1.25 – S ) = $1,250,000 – 1,000,000 S . The sale of the coffee beans has a payoff of 1,000,000( S – 1) - $12,742 + $1,250,000 – 1,000,000 S = $237,258 Page | 8
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
•
From $1.25 to $1.40 neither option has a payoff and the profit is 1,000,000( S – 1) - $12,742 = 1,000,000S -1,012,742
•
If S > $1.40, the call option is in the money and the payoff is -$1,000,000( S – 1.40) = $1,400,000 – 1,000,000 S . The profit is 1,000,000( S -1) - $12,742 +$1,400,000 – 1,000,000 S = $387,258
•
Therefore, the range is $237,258 to $387,258.
Real-Options Analysis 13.14 • Define the real option parameters for delaying option. V 0
I
T
r
σ
$1.9 Million
$2 Million
1
0.08
0.4
The value of delaying is $0.32 Million by Black-Scholes equation. If the choice is to defer or cancel, the value of delaying is $0.32M as calculated. If the choice is to defer or upgrade now, then we have to subtract the conventional NPW and the answer is $0.32M – (-$0.1M) = $0.42M. 13.15 •
Define the real option parameters for license option. V 0
I
T
r
σ
$30 Million
$40 Million
3
0.06
0.2
The value of license is $2.86 Million by Black-Scholes equation. Here we assume that the option will be exercised in three years when exclusive mining rights expire.
Page | 9
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
Growth Options 13.16 • Define the real option parameters for this option. V0
T
r
$1 Million
2
0.06
∴
The best cutting policy: • It is most profitable when we cut the trees at year 2. • Keeping option open/waiting is better than cutting trees from year 0 to year 1. ∴ Option Value of the investment opportunity: $0.62M 0 Grow th rate Rent Cost
0.4
1
2
1.6
1.5
0.4
unit: $M
0.06 2 1 1.25 0.8 0 0.58 0.42
r T= dT u= d= v= q= 1-q =
3.75=2(1.25)(1.5) 2= 1(1.25)1.6 3.75 2
1 1.62
1
3. 75
2.60
w ait
2.00
Cutting
Cu tt in g
2.40
w ait
Cutting
1.28
2. 40
1.52
w ait
1.28
Cutting
Cu tt in g
1.54
1.62=EXP(-0.06*1)*(2.6*0.58+1.52*0.42)-0.4: Wait
1: Cutting
1. 54 1.52=EXP(-0.06*1)*(2.4*0.58+1.54*0.42)-0.4: Wait
1.28: Cutting
Option value = $
0.62
Page | 10
Cu tt in g
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
13.17
•
Define the real option parameters for deferral option. V 0
I 1
I 2
T
r
σ
$60,000
$38,588
$60,638
2
0.06
0.2
∴ The value of postponing the construction decision for two years: $349,743
0
2
r T= dT = u= d=
38,588 60,638 0.2 0.06 2 2 1.33 0.75
q= 1-q =
0.65 0.35
I1 = I2 =
volatility
38,588=35,000(1.05)(1.05) 60,638=55,000(1.05)(1.05)
410,263=MAX(79,614-38,588,0)*10 569,289=MAX(79,614-60,638,0)*30
79,614 410,263 10 stores mall open 60,000
569,289 30 stores mal l open
349,743
45,218 66,308 10 stores mal l open
0 30 stores mall open 349,743=EXP(-0.06*2)*(569,289*0.65+66,308*0.35)
Page | 11
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
Switching Options 13.18 • The NPW of project B: PW (12%) B •
= −$2 + $1( P /
A,12%,10)
= $3.65 M
Define the real option parameters for switching option. V 0
I
T
r
σ
$4 Million
$3.65 Million
5
0.06
0.5
The value of switching is $0.85 Million by Black-Scholes equation for put option. •
Therefore, the total value is: SNPW = Value of project A + Option to switch to project B SNPW
= $4 + $0.85 = $4.85 M
R&D Options 13.19 • Assuming MARR = 12%, the cash flow diagram transforms to:
$73.42
0 14.18
1
2
3
4
5
R&D Expenses
6
7
8
9
10
Manufacturing and Distribution $80
•
Define the real option parameters for R&D option.
Page | 12
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
V 0
I
T
r
σ
$46.66 Million
$80 Million
4
0.06
0.5
The value of option today is $13.70 Million by Black-Scholes equation for a call option.
•
Therefore, the total value is: Option value = SNPW - Cost for R&D = $13.7 − $14.18 = −$0.48 M < 0 The firm should not embark on the project as the required R&D expenditure already exceeds by $0.48M.
Abandonment Options 13.20 • Standard NPW approach
0.35 0.35 0.35 0.35
0.35
... 0
1
2
3
4
.
.
.
n
$3 $0.35 = − $0.08M 0.12 e0.06 − d 1 = 1.6487, d = = 0.6065, and q = = 0.4369
PW (12%) = −$3 + u
•
=
e
0.5
u
u − d
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
Page | 13
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
- Option valuations Time
0 2.92
1 4.81 1.77
2 7.94 2.92 1.07
3 13.09 4.81 1.77 0.65
4 21.58 7.94 2.92 1.07 0.40
0.50
0.23 0.77
0.06 0.42 1.13
0.00 0.12 0.69 1.55
0.00 0.00 0.23 1.13 1.80
Monetary Value
Option value
5 35.57 13.09 4.81 1.77 0.65 0.24 0.00 0.00 0.00 0.43 1.55 1.96
* It is optimal to exercise early at the shaded positions. Option premium = $0.50 – (-0.08) = $0.58M
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. o
u
o
d =
o
=
q=
σ
e
1
Δt
=
=
e
0.3× 1
= 1.35
1 = 0.74 1.35
u e r Δt − d u − d
=
e0 .06×1 − 0.74
1.35 − 0.74
=
0.53
Page | 14
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
0
1
K=
4 0.3 0.06 1.35 0.74 0.53
volatility r u= d= q=
13.50 14.7989 scale down
0.74 7.41 9.9265 scale down
•
3
=max(18.22*0.8+4,EXP(-0.06)*(24.6*0.53+14.8*(1-0.53))) =18.8
1.35 10 11.7671
2
24.60 max(24.6*0.8+4,24.6) 1.35 24.60 =24.6 18.22 Do not scale down 1.35 18.8003 0.74 Do not scale down 13.50 max(13.5*0.8+4,13.5) 0.74 1.35 14.80 =14.8 10 scale down 1.35 12.0000 0.74 scale down 7.41 0.74 1.35 9.93 5.49 scale down 8.3905 0.74 scale down 4.07 7.25 scale down
From the result of the tree we can get the SNPW: SNPW = NPV + Option value = $11.77M Option value = $11.77M - $10M = $1.77M
Expansion-Contraction Options 13.22 (a) Binomial lattice tree •
σ
u=e
=
e
0.15× 1
= 1.1618
1 = 0.8607 u 1.1618 • Tree with incremental period one year •
d =
1
Δt
=
134.99
116.18 100 100 86.07 74.08
Page | 15
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
(b) Option valuation •
Risk neutral probability e
q =
r Δt
−
d
u − d
=
e
0.05× 1
− 0.8607
1.1618 − 0.8607
=
0.6329 , 1 − q = 0.3671
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) 115 Contract
100 116.31
86.07 Max(86.07×0.9+25,86.07×1.3-20,101.24) 102.46 Contract
74.08 Max(74.08×0.9+25,74.08×1.3-20,74.08) 91.67 Contract
Option value = SNPW – NPV = $116.31 - $100 = $16.31M
Compound Options 13.23 • Compound option parameters
•
V 0
I 1
I 2
T 1
T 2
$32.43
$10
$30
1
3
r 0.06
σ
0.5
Decision tree for a scale-down option through one-year time increment. - u = e t = e0.5 1 = 1.6487 1 1 = 0.6065 - d = = u 1.6487 0.06 1 e r t − d e − 0.6065 = = 0.4369 - q= u − d 1.6487 − 0.6065 σ
Δ
Δ
×
×
Page | 16
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
145.34 Max(145.34 - 30, 0) 115.34 Invest $30
88.15 59.90 Keep option open
53.47 Max(53.47 - 30, 0) 23.47 Invest $30
53.47 Max(29.77 - 10, 0) 19.77 Invest $10
32.43 9.66 Keep option open
32.43 8.13
Max(19.67 - 30, 0) 0 Do not invest
19.67 Max(3.97 - 10, 0) 0 Do not invest
19.67
11.93 0 Do not invest
7.24 Max(7.24 -30, 0) 0 Do not invest
First Option
Second Option
SNPW is $8.13 and this exceeds the initial cost $5. Initiate the phase I.
Page | 17
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13.24
Option parameters V 0
I 1
I 2
T
r
σ
$19.5M
$3M
$12M
3
0.05
0.25
0
Phase 1 1
I=
3,000,000
Phase 2 2
3
12,000,000
12,000,000 29,281,500=MAX(41,281,500-12,000,000, 0) 2,700,000: Ab andon (Selling p rice of the land)
volatility= r T= dT u= d= v= q= 1-q=
0.25 0.05 3.00 1.00 1.28 0.78 0.00 0.54 0.46
20,735,312=EXP(-0.05*1)*(29,281,500*0.54+13,038,496*0.46) 20,150,065=MAX(32,150,065-12,000,000, 0) 2,700,000: Ab andon (Selling p rice of the land)
41,281,500
25,038,496 14,180,447 22,038,496 2,700,000 15,186,615 5,382,658 12,186,615 2,700,000
19,500,000 16,646,312
Keep Option Invest 3M Abandon
Keep Option Invest 3M Abandon
5,382,658=EXP(-0.05*1)*(8,085,247*0.54+2,817,955*0.46) 12,186,615=MAX(15,186,615-3,000,000, 0) 2,700,000: Ab andon (Selling price o f the land)
Op tio n valu e=
32,150,065 20,735,312 20,150,065 2,700,000 19,500,000 8,085,247 7,500,000 2,700,000 11,827,348 2,817,955 2,700,000
Keep Option Invest 12M Abandon
29,281,500 Inve st 12M 2,700,000 Abandon 25,038,496
Keep Option Invest 12M Abandon
13,038,496 Inve st 12M 2,700,000 Abandon 15,186,615
Keep Option Invest 12M Abandon
3,186,615 Inve st 12M 2,700,000 Abandon 9,211,148
Invest 12M 2,700,000 Abandon
16,646,312
Page | 18
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
Short Case Studies ST 13.1 (a) American put option value •
Option parameters V 0
K
T
Δt
r
σ
$150
$100
2
1 year
0.05
0.3
σ
Δt
e0.3×
1
= 1.3499 ,
•
u=e
•
Risk neutral probability q=
e
r Δt
=
−d
u − d
=
d =
e0.05×1 − 0.7408
1.3499 − 0.7408
1 u
=
=
1 1.3499
=
0.7408
0.5097 , 1 − q = 0.4903
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
Short Case Studies ST 13.1 (a) American put option value •
Option parameters V 0
K
T
Δt
r
σ
$150
$100
2
1 year
0.05
0.3
σ
Δt
e0.3×
1
= 1.3499 ,
•
u=e
•
Risk neutral probability q=
e
r Δt
=
−d
u − d
=
e
0.05×1
− 0.7408
1.3499 − 0.7408
0 K=
volatility r u d q
= = = =
d =
1 u
=
=
1 1.3499
=
0.5097 , 1 − q = 0.4903
1
2
100 0.3 0.05 1.35 0.74 0.51 1.35 1.35
150 3.84
0.7408
202.48 0.00
0.74
0.74 1.35
111.12 8.24
273.32 0.00
150.00 0.00
0.74
82.32 17.68
American put option value = $3.84
Page | 19
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
(b) Expansion Option value •
Option parameters V 0
K
$400 σ
Δt
0.35 × 1
= 1.42 ,
•
u = e
•
Risk neutral probability q=
=
e
er Δt − d u − d
=
r= u= d= q=
Δt
r
σ
2
1 year
0.07
0.35
d =
1 u
=
1 1.42
=
0.70
0.51, 1 − q = 0.49
0
volatility
T
1
2
0.35 0.07 1.42 0.70 0.51
805.50 1.42
1.42
400 593.17
567.63 658.20
0.70
400.00
902.16 0.70
1.42
281.88 201.00 353.89
1361.00
550.00
0.70
198.63 198.63
Total option value = $593.18 Real option premium = $598.13 - $400 = $198.13
Page | 20
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
ST 13.2 (a) Since $4 M is lower than the option price, it is a good investment for Merck Co. •
To give a range for the option value, first using one period lattice. V 0
K
$36M $72M ($30 ×1.2M) ($60 ×1.2M) σ
Δt
0.5× 3
u=e
•
d =
•
Risk neutral probability
u
q=
•
=
e
1 2.3774
e r Δt − d u − d
=
=
e
=
Δt
r
σ
3
3 years
0.06
0.5
2.3774
•
1
=
T
0.4206
0.06×3
− 0.4206
2.3774 − 0.4206
=
0.3969 , 1 − q = 0.6031
One period lattice and option price: 4.50M 85.59 Max(85.59 - 72, 0) 13.59 Buy the stock 36 4.50 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.
Page | 21
Contemporary Engineering Economics, Fifth Edition, by Chan S. Park. ISBN: 0-13-611848-8 © 2011 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 any 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.
Note: We can think that the price from the one step binomial tree is the lower bound and B-S being the upper bound. So the option price is definitely higher than the suggested price.
(b) With the agreement, Merck has a chance to buy Genetics with a lower price than the prevailing market price when the project is successful. Otherwise they just lose $4M. To make this concept simple, a basic example is demonstrated below: •
Profit/loss analysis Success Fail
Buy stock Gain $37.59M Loss $32.86M
Aforesaid agreement Gain $9.59M Loss $4M
The “Buy stock” option has a higher risk than the option.
Page | 22
