Chap 5
Numerical Methods
for Simple Options
Numerical Methods for Simple
Options
NPV is forced to treat future courses of action as
mutually exclusive, ROA can combine them into a
single value with a decision rule for choosing among
them.
 We explore the effect of dividend payments by the
underlying risky asset on American call options such as
a deferral or an expansion option.
 They would never be exercised early unless the
underlying risky asset dropped in value, for example,
when it pays a dividend.

Methodology for modeling the stochastic
process of the underlying asset
A normal distribution is symmetric and has values in
the left tail that go to minus infinity.
 The multiplicative process, the additive process is also
recombining.
 To preserve the recombining property of the event trees
that describe the evolution of the price of the underlying
through time, we usually assume that dividends paid are
proportional to value in a multiplicative tree, and are
constant and additive in an additive tree.

The multiplicative process and constant for the additive
process.
 Also, the limiting distributions ( for the predividend
values ) are still lognormal for the multiplicative
process and normal for the additive process.
 Finally, the obvious result of paying dividends is that
the values at the end branches are lower than they were
in the no-dividend case.
 As the project generates free cash flows that can be
dividended out, the remaining value is reduced by the
amount of the free cash flow ( dividend ).

Abandonment options valued
indirectly
If one has the right, but not be the obligation, to rid
oneself of a risky asset at a fixed (predetermined) price,
it is called an abandonment option.
 Abandonment options are important in research and
development, in exploration and development of natural
resources,
 Abandonment option analysis not only provides an
estimate of the value of optimal abandonment, but it
also indicates when abandonment should be
implemented.


In merger and acquisition situations, abandonment is
equivalent to being able to bail out of an investment at a
floor price (900)– the estimated exercise price of the
abandonment option.
V0  puV0 e  kt  (1  p)dV0 e  kt
 (0.15/ 4)
 (0.15/ 4)
1,000  p(1.06184)1000e
 (1  p)(0.9418)1,000e
1  p(1.06184)(0.9631944)  (1  p)(0.9418)(0.9631944)
1  1.022755 p  0.9071023  0.9071023 p
p  0.803246
payoff  MAX [Vt , X ]
replicating portfolio :
mdV0  B  Value of put at node C
State E : m(duV0 )  (1  rf ) B  1, 000
State F : [m(d V0 )  (1  rf ) B  900]
2
mdV0 (u  d )  Cdu  Cdd
(1, 000  900)
m
 0.88433
941.76(1.0618  0.9417)
m(duV0 )  (1  rf ) B  Cdu
C du  mduV0
B
(1  r f )


 C du  C dd 
C du  
 duV0 
 dV0 (u  d ) 


B
(1  r f )
 C du (u  d )  (C du  C dd )u 


u

d



(1  r f )
 uC dd  dC du 
 ud




(1  r f )
 1.06184(900)  0.94176(1,000)


(1.06184  0.94170)


B
(1.01258)
 13.896 

 115.72
0.12008 



=114.28
(1.01258) 1.01258
m(dV0 )  B  0.88433(941.76)  114.28
 832.83  114.28
 947.10
Value of abandonment option 
Value of project with flexibility minus
the value of project without flexibility
Value of abandonment option
 $1, 002.16  $1, 000
 $2.16
Expectedpayoffs
Value at node C 
(1  RAR)

0.8032(1,000)  (1  0.8032)(900)
947.08 
(1  RAR )

803.2  117.12
947.08 
(1  RAR )
980.32
3.5097%
RAR 
1 
947.08
quarter
 The
risk-adjusted rate of return (RAR) varies
depending on where we are in the decision tree
because the riskless of outcomes changes as well.
 Recall that this is the major reason that
decision-tree analysis does not work.
 DTA inappropriately assumes a constant
discount rate.
MAX [0, X  V ]
[m(dV0 )  Pd ](1  rf )  m(udV0 )  Pud
[m(dV0 )  Pd ](1  rf )  m(d V0 )  Pdd
2
m(udV0 )  Pud  m(d V0 )  Pdd
Pdd  Pdu
(13.08  0)
m

dV0 (u  d ) [941.76(1.0618  0.9418)]
2
13.08

 0.11567
113.08
(mVd  Pd )(1  rf )  muVd  Pud
(mdV0  Pd )(1  rf )  mduV0  Pud
since Vd  dV0
(mV0  P0 )(1  rf )  muV0  Pu
since Pd  dP0 and Pud = dPu
 u  (1  r f ) 
 (1  r f )  d  

 Pd  
 Pu 
 u  d 
 u  d  
P0 
(1  r f )
u  (1  r f )
1.0618  (1  0.12458)
q

 0.4096
ud
1.0618  0.9418
(1  r f )  d
1 q 
 1  0.4096  0.5904
ud
Pd
qPdd  (1  q) Pud 


(1  rf )
[0.4096(13.08)  0.5904(0)]

(1.01258)
5.357

 5.29
1.01258
Pu
qPud  (1  q) Puu  [0.4096(0)  0.5904(0)]



0
(1  rf )
(1.01258)
Valuing the option to contract (shrink)
a project
The right to sell off some capacity, thereby shrinking
the scale of operations, is an American put option that
we call the option to contract.
 But introduce an option to contract the scale of
operations (and therefore its value) by 50 percent by
selling assets (plant and equipment) worth $450 after
taxes.
An alternative that is similar, would be to scale back
operations and sublet equipment and facilities to
another company.

State E : mudV0  B (1  rf )  1, 000.00
State F : [md V0  B (1  rf )  893.46]
2
106.54
m
 0.942165
113.08
mudV0  B(1  rf )  1, 000
0.942165(1, 000)  B(1  0.01258)  1, 000
(1, 000  942.165) 57.116
B

 55.079
1.01258
1.01258
mdV0  B  0.942165(941.76)  55.079  942.4
Value of the option to expand ( when the
underlying does not pay dividends )
If projects turn out better than expected, it is often
desirable to invest in expanding them.
 The extra investment is the exercise price of an
expansion option – an American call.
 The value tree for the underlying shifts upward to
reflect the benefits of the expansion.
 And introduce the option to expand the scale of
operations, at an expanse of $100, for a benefit that
increases the value of operations by 10 percent

Valuing combinations of simple
options
1.An American put, the option to abandon for a
salvage value of $900.
2.A contraction option, also an American put, to
shrink the scale of operations by 50 percent by
selling assets for $450.
3.The option to expand the scale of the project by
10 percent at a cost of $100, an American call.
State D : mu V0  B (1  rf )  1,140.25
2
State E : [mudV0  B (1  rf )  1, 000.00]
muV0 (u  d )  Cu  Cd
140.25
m
 1.1
(1,127.50  1,000)
muV0  B  1.1(1,061.84)  98.76  $1,069.26
MAX [1, 069.26,1.1(1, 061.84)  100, 1, 061.84 / 2   450,900]
 MAX [1, 069.26, 1068.02, 980.92, 900]
[Unexercised , expand , contract , abandon]
MAX [947.06, 1.1(941.76)  100, 941.76 / 2  450, 900]
MAX []947.06, 935.94, 920.00, 900]
MAX [Unexercised , expand , contract , abandon]
Value of the option to abandon
$2.16
Value of the option to contract
$1.07
Value of the option to expand
Value of the combination of options
$4.24
$6.44
While it is true that the simple sum of the values of the
three separate options is not equal to the value of their
combination, we must look more carefully and note that
the option to contract was never used in the
combination – in other words it was valueless in the
combination because it was dominated by the other two
options.
 Therefore, the value of the combination, $6.51 is equal
to the simple sum of the values of the (undominated)
options, namely the abandonment option and the option
to expand.
 The sum of their values is $6.49, within rounding error,
the same as the value of the combination.

Net present value and real options analysis handle
mutually exclusive alternatives differently
 The
cash flow of a project right now is $100.
 At the end of year 1, it goes up by 20 percent to
$120 or down by 16.67 percent to $83.3 with
similar percentage changes in year 2.
 At the end of the second year, the company has
two mutually exclusive alternatives.
 In
one branch of the decision tree, it can spend
$700 to lock in its annual level of cash flows
forever.
 In a mutually exclusive branch, it can spend an
additional $120 to test-market a new version of
the product (thereby forgoing a year of cash
flows) to find out, as of year 4 that, with 50-50
probability, the perpetual cash flows will be
either 50 percent higher, or 33 13 percent lower.
 Then, in year 4, the company can, at a cost of
$700, lock in the perpetual cash flows or it can
abandon the project.
 Additionally,
the cost of capital is 10 percent,
the risk-free rate is 10 percent, and the initial
outlay required to commence the project is $400.
0.727(120)  0.272(83.3)
PV  100 
1.1
0.529(884)  0.396(400)  0.074(69.4)

 722
2
(1.1)
NPV  PV  Investment  722  400  322
E (CF , year 4)  0.529(0.5)1, 676  0.529(0.5)355  0.396(0.5)950 
0.396(0.5)33.3  0.074(0.5)444.0  749.2
E (CF , year 3)  0
E (CF , year 2)  0.529(24)  0.396( 20)  0.074( 50.6)=1.02
E (CF , year 1)  0.727(120)  0.272(83.3)  110
 Discounting
these expected cash flows at 10
percent, adding the initial cash inflow of $100,
and subtracting the initial investment of $400
results in an NPV of $312.55.
At node E : m400  (1  rf ) B  400
At node F : m69.4  (1  rf ) B  133.2
At node C : 0.807(281.7)  $73.57  $300.9
NPV with flexibility  $728.84  $400  $328.84
Node B : muV0  (1  rf ) B  1, 068.02
Node C : [mdV0  (1  rf ) B  941.76]
mV0 (u  d )  Cu  Cd
m1, 000(1.06184  0.4176)  126.26
126.26
m
 1.0515
120.08
mV0  B  1.0515(1000)  47.89  $1,003.6
500 500
500
NPV  1, 000 
 2  5
1.1
1.1
1.1
 1, 000  454.55  413.22  375.66  341.51  310.46
 1, 000  1,895.40  895.40
500 500
500
PV  500 
 2  4
1.1 1.1
1.1
 500  454.55  413.22  375.66  341.51
 2, 084.94
 At
time zero the present value of the project is
$1,895, it can move up or down by 20 percent,
its expected present value at the end of the first
year is $2,085, and we have assumed a 10
percent discount rate.
 Using these facts, we can solve for the objective
probability of the up movements as follows :
puVu  (1  p)Vd
E ( PV1 )
PV0 

1  WACC
1  WACC
p (2,274)  (1  p)(1,579)
1,895 
1.1
1.1(1,895)  1,579 505.5
p

 0.727
2,274  1,579
695
p6,000  (1  p)4,167
PV0 
 $5,000
1.1
5,000(1.1)  4,167
p
 0.727
6,000  4,167