GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.127

A model for determining whether a firm should exercise
multiple real options individually or simultaneously

Michi Nishihara

April 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanA model for determining whether a ﬁrm should exercise
multiple real options individually or simultaneously∗
Michi NISHIHARA†

Abstract
We develop a model for determining whether a ﬁrm should exercise two real optionsindividually or simultaneously. The simultaneous exercise of both options has positivesynergy, such as economies of scale, scope, and networks, while separate exercise of each
option beneﬁts from project ﬂexibility. This tradeoﬀ determines the optimal exercise
policy. We investigate the static and dynamic management of multiple real options. A ﬁrm
under static management determines the type of exercise of real options ex ante; on theother hand, a ﬁrm under dynamic management makes the decision at the time of exercise.
The analysis reveals the gap between the two styles of managing. Most importantly, wehighlight the advantage of dynamic management over static management, particularly
for weakly correlated markets. We also explain empirical implications regarding a ﬁrm’s
entry into several countries and regions in Asia.
JEL Classiﬁcations Code: C61, G13, G31.
Keywords: multiple real options, optimal stopping, exercise region, entry into Asia.

∗†

This version: March 25, 2010. This work was supported by KAKENHI 20710116.
Corresponding Author, Address: Graduate School of Economics, Osaka University, 1-7 Machikaneyama-cho,

Toyonaka, Osaka 560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Tel: 81-6-6850-5277

1Introduction

The global ﬁnancial crisis which began in 2007 has increased uncertainty about the future
market demand in many industries throughout the world. It has become increasingly
important for project managers to take into account both uncertainty and ﬂexibility inthe future. The real options approach, in which option pricing theory is applied to capitalbudgeting decisions, better enables us to ﬁnd an optimal investment strategy and undertake project valuation in this environment than is possible under more classical methods.
The early literature has investigated a real option that has a rather simple payoﬀ
structure, assuming that dynamics of project value follow a one-dimensional stochastic
process (e.g., (Dixit and Pindyck 1994)). Naturally, the studies have been developed
into a more complicated real options analysis on the basis of a multidimensional process(e.g., (Geltner, et al 1996, Loubergé, et al 2002, Cortazar, et al 2008, Martzoukos 2009,
Nishihara 2010)).1 For example, (Geltner, et al 1996) investigates land development
timing with an alternative land use choice, while (Loubergé, et al 2002) investigatestiming in switching methods of nuclear waste disposal. These multidimensional models
focus primarily on the nature of a single real option that has a complex payoﬀ structure
individually.
However, a ﬁrm typically possesses a collection of real options at the same time. Because exercising multiple real options, unlike ﬁnancial options, has the potential to yieldsynergy, such as economies of scales, scopes, and networks, a ﬁrm faces the problem ofwhether to exercise multiple options individually or simultaneously. To our knowledge,
this paper is the ﬁrst work that attempts to capture the nature of this problem.2 Several
papers (e.g., (Meier, et al 2001, Luehrman 2004, Wang and Hwang 2007)) investigate
the management of a portfolio of multiple real options in the context of project portfolio choice. For example, (Meier, et al 2001) proposes both static and dynamic zero-one
optimization models for a portfolio of real options, and (Luehrman 2004) presents a conceptual framework for strategic management of real options. However, there is a large gap
between these studies and the real options literature on the basis of a multidimensional
stochastic process. Indeed, these papers tend to be positioned in the context of portfolio
optimization rather than in the context of real options. This paper ﬁlls the gap by investigating the problem of how to manage multiple real options in terms of a multidimensional

1

Another stream of real options development is combined with game theory. Strategic interactions among

several ﬁrms are investigated in (Grenadier 1996, Grenadier 2002, Lambrecht and Perraudin 2003, Nishihara andFukushima 2008), while agency problems in a single ﬁrm are investigated in (Mauer and Sarkar 2005, Grenadierand Wang 2005, Shibata and Nishihara 2010).
2
Although (Trigeorgis 1993) investigates the nonadditivity of the value of multiple real options, he does notconsider the problem of whether multiple real options are exercised individually or simultaneously. In addition,
the analysis is based on a one-dimensional process.

1

stochastic model.
Our model assumes that a ﬁrm has two business opportunities in which it may invest.
A ﬁrm is able to decide whether two projects are to be carried on individually or simultaneously. Investing in each project individually yields project ﬂexibility, while initiating
both projects simultaneously yields positive synergy, including economies of scale, scope,
networks, etc. Taking account of this tradeoﬀ, a ﬁrm determines the optimal type ofinvestment.
This paper distinguishes two styles of management. One is static management. A
ﬁrm under static management determines whether it exercises options individually or
simultaneously ex ante. This style is likely to apply to a ﬁrm which takes a top-down
approach to the management decision. The managerial ﬂexibility also depends on thetype of project. A project which requires advance preparation contingent on the type ofinvestment forces a ﬁrm to make the management decision ex ante. Static management
is related to the static optimization approach to a project portfolio choice.
The second style is dynamic management. A ﬁrm under dynamic management iscapable of deciding whether it invests in projects individually or simultaneously at thetime of investment. In comparison to static management, this style is likely to apply
to a ﬁrm in which the management decision can be made ﬂexibly and with a bottom-up
approach. It is presumed for dynamic management that a project does not require advance
preparation depending on the type of investment. Dynamic management is closely related
to the dynamic optimization approach to the evaluation of an option on multiple assets.
In the model, we reveal the nature of static and dynamic management as well as the
gap arising between the two. Our results regarding the exercise region of multiple options
under dynamic management can be positioned as an extension of the previous ﬁndings by(Geltner, et al 1996, Broadie and Detemple 1997, Detemple 2006, Nishihara 2010). In thecomparative statics, we focus on the eﬀects of a correlation among the project values. We
demonstrate that a lower correlation among the values gives a ﬁrm the incentive to invest
individually rather than simultaneously. This ﬁnding is contrasted with (Childs, Ott, andTriantis 1998), which shows that a higher correlation increases the value of sequential
development rather than parallel development.3 The diﬀerence results from the model
assumptions. They focus on the mutually exclusive case in which a ﬁrm invests in thedevelopment stage of two projects and then may select only a single project to implement.
In contrast, we investigate the inclusive case in which a ﬁrm can receive proﬁts from bothprojects. Further, and more importantly, we ﬁnd that a weaker correlation increases theadvantage of dynamic management over static management. This is principally because aweaker correlation increases the possibility that an ex ante choice of the investment typeturns out suboptimal ex post.
3

Their analysis is restricted within static management.

2

The model applies to the strategic decision concerning market entry into several countries and regions. Below, we focus on a situation in which a ﬁrm expands business into
several countries and regions in Asia. Recently, an increasing number of corporations are
driven by the need to expand business to Asia’s markets, primarily because Asia’s rapidly
growing population will potentially generate the largest markets in the world. For instance, UNIQLO, the Japanese casual wear brand which has already launched operationsin China in 2002, Hong Kong and South Korea in 2005, and Singapore in 2009, announced
its plans to enter markets in Indonesia, Thailand, and Malaysia within a couple of years.
In expanding business into Asia, a ﬁrm must take careful consideration of the diversity which is characteristic of Asia. Even within the same country, the dynamics ofthe economy vary across regions. In addition to the economies, there are a wide varietyof languages, ethnicities, cultural and religious prescriptions, and business practices in
Asia. Naturally, a ﬁrm entering Asia’s markets faces many risks that diﬀer among countries and regions. For example, Indonesia has a risk of political instability, while China’s
information control greatly aﬀects Internet businesses.
The paper demonstrates that the heterogeneity of market risk in Asia increases the
incentive for a ﬁrm to enter each market individually, depending on country-speciﬁc andregion-speciﬁc risks, rather than a simultaneous entry into the whole market. This argument supports the overseas expansion strategies of many ﬁrms, including UNIQLO.
Further, and more importantly, we highlight the advantage of dynamic corporate management over static management for weakly correlated markets. In our view, the dynamic
management capability will be a major success determinant for a business in Asia.
The paper is organized as follows. Section 2 presents the properties of the option
value and the exercise policy under static management. Then, Section 3 presents those of
dynamic management and reveals a gap between the two styles. Section 4 shows further
properties of static and dynamic management in numerical examples. Section 5 concludes
the paper.

2

Static management

2.1

Model

Consider a ﬁrm that plans two projects (denoted by projects i = 1 and 2) in which toinvest. The risk-adjusted values of the projects, X(t) = (X1 (t), X2 (t)), are random andfollow a bidimensional time-homogeneous diﬀusion process
dXi (t) = µi (Xi (t))dt + σi (Xi (t))dBi (t),(1)

where (B1 (t), B2 (t)) is a bidimensional Brownian Motion (BM) with correlation coeﬃcientρ satisfying |ρ| < 1. Coeﬃcients µi (Xi (t)) and σi (Xi (t))(> 0) denote the risk-adjusted

3

growth rate and volatility of the project value, respectively. The ﬁrm chooses between
individual and simultaneous investment. Investing in project i individually requires anirreversible capital expenditure of Ii (> 0), while simultaneous investment in both projects
requires an irreversible capital expenditure of I1,2 (> 0). Assume that max(I1 , I2 ) <I1,2 < I1 + I2 . This assumption means that simultaneous investment has positive synergy,
including economies of scale, scope, networks, etc. Mathematically, the model is built
on the ﬁltered probability space (Ω, F, P ; Ft ) generated by (B1 (t), B2 (t)). The set Ft
represents the set of available information in time t, and the ﬁrm ﬁnds the optimal policy
under this information. The ﬁrm’s real options are perpetual. The risk-free rate is aconstant r(> 0).

2.2

Valuation of each option

To begin, we evaluate the option to invest in a single project i individually. For Xi (0) = xi ,the option value is equal to the value function of the time-homogeneous optimal stopping
problem as follows:
Vi (xi ) = sup Exi [e−rτ (Xi (τ ) − Ii )],
τ ∈T

(2)

where T denotes the set of all stopping times τ and E xi [·] is the expectation conditional on
Xi (0) = xi . Note that (2) corresponds to a perpetual American call option. Under someplausible assumptions (for details, see (Peskir and Shiryaev 2006)) the optimal stopping
time τi for problem (2) becomes τi = inf{t ≥ 0 | X(t) ∈ Si }, where the stopping region Si
is deﬁned by
Si = {x ∈ R2 | Vi (xi ) = xi − Ii }.

(3)

The optimal policy is that a ﬁrm makes investment in project i as soon as X(t) hits Si .
Next, consider simultaneous investment in both projects. For X(0) = x = (x1 , x2 ),the option value is equal to the value function of the time-homogeneous optimal stopping
problem as follows:
V1,2 (x) = sup Ex [e−rτ (X1 (τ ) + X2 (τ ) − I1,2 )].
τ ∈T

(4)

Note that (4) corresponds to a perpetual American basket option. Under some plausible
assumptions, the optimal stopping time τ1,2 for problem (4) can be expressed as τ1,2 =inf{t ≥ 0 | X(t) ∈ S1,2 }, where the stopping region S1,2 is deﬁned by
S1,2 = {x ∈ R2 | V1,2 (x) = x1 + x2 − I1,2 }.

(5)

The optimal policy is that a ﬁrm makes simultaneous investment in projects 1 and 2 assoon as X(t) hits S1,2 .
In general, the value functions Vi (xi ), V1,2 (x) and the stopping regions Si , S1,2 cannot
be derived in any closed form. It is well known that, for X(t) following either a geometric4

Brownian motion (GBM) or a Brownian motion (BM) with a drift, Vi (xi ) and Si canbe derived in closed forms (see (Dixit and Pindyck 1994)). First, consider the case of a
GBM. Assume that µi (Xi (t)) = µi Xi (t), σi (Xi (t)) = σi Xi (t), µi < r and Xi (0) = xi > 0
for i = 1, 2. Then we have
 µ ¶βi
 xi(x∗i − Ii ) (0 < xi < x∗i )x∗i
Vi (xi ) = x −I(xi ≥ x∗i )
i
i

(6)

Si = {x ∈ R2++ | xi ≥ x∗i },

(7)

and

where βi = 1/2 − µi /σi2 +q(µi /σi2 − 1/2)2 + 2r/σi2 (> 1), and the investment thresholdx∗i is deﬁned by
x∗i =

βi
Ii .
βi − 1(8)

The option value V1,2 (x) and the stopping region S1,2 can not be derived in any closed
forms, because the sum of GBMs, X1 (t) + X2 (t), does not follow a GBM. Instead, thefollowing properties are well known (e.g., (Broadie and Detemple 1997, Detemple 2006)):(Convexity of the value function) V1,2 (x) is a convex function.
(Convexity of the stopping region) S1,2 is a convex set.
(Monotonicity of the stopping region) x ∈ S1,2 ⇒ x0 ∈ S1,2 (∀x01 ≥ x1 , ∀x02 ≥ x2 ).
Next, suppose that X(t) follows a BM with a drift. Assume that µi (Xi (t)) = µi , σi (Xi (t)) =σi for i = 1, 2. Then we have the option value(∗∗∗∗
e−γi (xi −xi ) (x∗∗i − Ii ) (xi < xi )Vi (xi ) =xi − Ii(xi ≥ x∗∗i )(9)and the stopping region

where γi = −µi /σi2 +qSi = {x ∈ R2 | xi ≥ x∗∗
i },

(10)

(µi /σi2 )2 + 2r/σi2 (> 0), and investment threshold x∗∗
i is deﬁned

byx∗∗
i = Ii +

1
.
γi

(11)

The option value V1,2 (x) and the stopping region S1,2 can also be derived in closed forms,
because the sum of BMs, Y (t) = X1 (t) + X2 (t), follows
qdY (t) = (µ1 + µ2 )dt + σ12 + σ22 + 2ρσ1 σ2 dBY (t),where BY (t) denotes another BM. Deﬁnesµ¶2µ1 + µ22rµ1 + µ2++ 2(> 0),
γ1,2 = − 22222
σ1 + σ2 + 2ρσ1 σ2
σ1 + σ2 + 2ρσ1 σ2
σ1 + σ2 + 2ρσ1 σ2

5

(12)

(13)

where σ12 + σ22 + 2ρσ1 σ2 6= 0. We have the option value(
V1,2 (x) =∗∗∗∗e−γ1,2 (x1,2 −(x1 +x2 )) (x∗∗1,2 − I1,2 ) (x1 + x2 < x1,2 )(x1 + x2 ≥ x∗∗
1,2 )x1 + x2 − I1,2

(14)

and the stopping region
S1,2 = {x ∈ R2 | x1 + x2 ≥ x∗∗
1,2 },(15)

where the investment threshold x∗∗
1,2 is deﬁned byx∗∗
1,2 = I1,2 +

1
γ1,2

.

(16)

The following proposition shows the comparative statics with respect to the correlationcoeﬃcient ρ.
Proposition 1 Assume that X(t) follows a BM with a drift.
(Monotonicity of the value function) V1,2 (x) monotonically increases with ρ.
(Monotonicity of the stopping region) S1,2 monotonically decreases with ρ.
Proof By ∂γ1,2 /∂ρ < 0 and ∂V1,2 (x)/∂γ1,2 ≤ 0, we have
∂V1,2 (x)∂V1,2 (x) ∂γ1,2
=
≥ 0,
∂ρ
∂γ1,2 ∂ρand∂x∗∗
1 ∂γ1,2
1,2
=− 2
< 0.
∂ρ
γ1,2 ∂ρ¤

Proposition 1 can be interpreted as follows. The sum of two project values shows ahigher volatility as the correlation between two projects increases. An increase in volatility enhances the option value, as well as the investment threshold. The properties of
Proposition 1 tend to hold for a more general diﬀusion X(t), though it is hard to prove
the properties mathematically. We will check the properties numerically for a GBM in
Section 4.

2.3

Static management of real options

This section considers static management. A ﬁrm under static management decides
whether the projects are launched individually or simultaneously ex ante. For X(0) = x,
ﬁrm value under static management is evaluated by
VM (x) = max{ V1 (x1 ) + V2 (x2 ) ,{z}|

V1,2 (x)| {z }

individual investment simultaneous investment

6

}.

(17)

The value V1 (x1 ) + V2 (x2 ) corresponds to the value of individual investment, while thevalue V1,2 (x) corresponds to the value of simultaneous investment. When X(0) = x liesin
S1,2,M = {x ∈ R2 | V1,2 (x) ≥ V1 (x1 ) + V2 (x2 )},(18)

a ﬁrm chooses simultaneous investment ex ante and initiates both projects at the time τ1,2
Otherwise, it chooses individual investment ex ante and executes project i individually at
the time τi . A favorable characteristic of static management is its simplicity, though thevalue is lower than that of dynamic management. Indeed, we can derive the value andthe optimal exercise policy in the manner described in Section 2.2. It should be notedthat the static management approach resembles project portfolio selection models. For
example, the option value maximization method in (Meier, et al 2001) aims to maximize
statically a value of a portfolio of real options.
Let us explore the nature of static management. As will be seen numerically in Section
4, S1,2,M dose not satisfy either monotonicity or convexity. Instead, we can show that
S1 ∩S2 ⊂ S1,2,M . Indeed, for any x ∈ S1 ∩S2 , we have V1 (x1 )+V2 (x2 ) = x1 −I1 +x2 −I2 <x1 + x2 − I1,2 ≤ V1,2 (x). We can also derive the boundary of S1,2,M for a suﬃciently
large xi . For simplicity, assume that X(t) follows either a GBM or a BM with a drift.
For a suﬃciently large xi , we have V1 (x1 ) + V2 (x2 ) = xi − Ii + Vj (xj ) (j 6= i) andV1,2 (x) = x1 + x2 − I1,2 because x lies in Si ∩ S1,2 . There exists a unique solution xˆj <x∗j (or x∗∗j ) to Vj (xj ) = xj − I1,2 + Ii because of 0 < I1,2 − Ii < Ij . Then, the boundaryof S1,2,M coincides with a line xj = xˆj . In the region xj ≥ xˆj a ﬁrm chooses simultaneous
investment, while in the region xj < xˆj it chooses individual investment. By proposition
1, we can show the comparative statics with respect to the correlation coeﬃcient ρ.
Proposition 2 Assume that X(t) follows a BM with a drift.
(Monotonicity of the value function) VM (x) monotonically increases with ρ.
(Monotonicity of the simultaneous investment region) S1,2,M monotonically increases with ρ.
Proof By proposition 1, we have ∂V1,2 (x)/∂ρ ≥ 0. Then, by (17) we have the monotonicity of VM (x) with respect to ρ. Because V1 (x1 ) + V2 (x2 ) is independent of ρ, we also
have the monotonicity of S1,2,M with respect to ρ.

¤

Proposition 2 leads to the straightforward result that a ﬁrm is more likely to makesimultaneous investment in strongly correlated markets. This result can account for theoverseas expansion strategies of many ﬁrms entering several countries and regions in Asia.
A ﬁne example is UNIQLO, the Japanese casual wear brand. UNIQLO has been operatingin China since 2002, but has not yet planned to enter India. On the other hand, it hasplanned to enter Indonesia and Malaysia almost simultaneously. This is because Indonesia
and Malaysia have much in common, while China and India have few similarities.

7

More generally, there are a wide variety of risks that diﬀer among countries and regions in Asia. Therefore, it is commonly believed that a ﬁrm should market diﬀerent
products which meet country-speciﬁc and region-speciﬁc demands. We complement theconventional argument in terms of the timing of market entry. Diversity, which is a major
characteristic in Asia, provides the incentive for a ﬁrm to enter each market separately.
Note that the properties of Proposition 2, like Proposition 1, tend to hold for a moregeneral diﬀusion X(t). Relevantly, (Childs, Ott, and Triantis 1998) investigates a model
where a ﬁrm invests in the development stage of two projects and then may select only asingle project to implement. The model compares the values of developing the projects insequence or in parallel. Because of the assumption of mutual exclusion, their result is in
opposition to ours. In their analysis, a ﬁrm chooses sequential development rather thanparallel development, when projects have highly correlated values.

3

Dynamic management

3.1

Dynamic management of real options

This section considers dynamic management. A ﬁrm under dynamic management is capable of determining whether it initiates projects individually or simultaneously at the timeof investment. In comparison to static management, a ﬁrm requires managerial ﬂexibility.
For X(0) = x, ﬁrm value under dynamic management is evaluated by
VD (x) = sup Ex [e−rτ max{V1 (X1 (τ )) + V2 (X2 (τ )),
|
{z
}
τ ∈T

V1,2 (X(τ ))| {z }}]

option to invest individually option to invest simultaneously= sup E [e
x

−rτ

τ ∈T

VM (X(τ )) ],| {z }(19)

static management

or equivalently,
VD (x) = sup Ex [e−rτ max{X1 (τ ) + V2 (X2 (τ )) − I1 , X2 (τ ) + V1 (X1 (τ )) − I2 ,
|
{z
} |
{z
}
τ ∈T
individual investment in 1

individual investment in 2

X1 (τ ) + X2 (τ ) − I1,2 }].
|
{z
}

(20)

simultaneous investment

In (20), Xi (τ ) + Vj (Xj (τ )) − Ii (i 6= j) is composed of the value of individual investment inproject i at the time τ , Xi (τ ) − Ii , and the value of the option to invest in project j(6= i)
individually, Vj (Xj (τ )). In (20), X1 (τ ) + X2 (τ ) − I1,2 represents the value of simultaneous
investment in both projects at the time τ . Under some plausible assumptions, the optimal
stopping time τD for problem (20) can be expressed as τD = inf{t ≥ 0 | X(t) ∈ S1,D ∪S2,D ∪ S1,2,D }, where the stopping region Si,D are deﬁned by
Si,D = {x ∈ R2 | VD (x) = xi + Vj (xj ) − Ii }8

(j 6= i)for i = 1, 2, and the stopping region S1,2,D is deﬁned by
S1,2,D = {x ∈ R2 | VD (x) = x1 + x2 − I1,2 }.
We ﬁrst explore the nature of dynamic management for a general diﬀusion X(t). The
following proposition shows the properties of the value function VD (x) and the stopping
regions Si,D and S1,2,D .
Proposition 3
VM (x) ≤ VD (x), Si,D ⊂ Si , S1 ∩ S2 ⊂ S1,2,D ⊂ S1,2 ∩ S1,2,M
Proof Clearly, VM (x) ≤ VD (x) follows from (19). For any x ∈ S1,D , we have
V1 (x1 ) + V2 (x2 ) ≤ VD (x) = x1 + V2 (x2 ) − I1 .
Then, we have V1 (x1 ) ≤ x1 − I1 , which implies x1 ∈ S1 . Hence, we have S1,D ⊂ S1 .
Similarly, we can show S2,D ⊂ S2 , S1,2,D ⊂ S1,2 , and S1,2,D ⊂ S1,2,M . For any x ∈ S1 ∩ S2 ,we have
VD (x) ≤ sup Ex1 [e−rτ (X1 (τ ) − I1 )] + sup Ex2 [e−rτ (X2 (τ ) − I1,2 + I1 )]
τ ∈T
τ ∈T
|
{z
}
≤

=V1 (x1 )=x1 −I1
x1 − I1 + sup Ex2 [e−rτ (X2 (τ ) − I2 )] + sup Ex2 [e−rτ (I1 + I2 − I1,2 )]
τ ∈T
τ ∈T

|

{z

=V2 (x2 )=x2 −I2}≤ x1 − I1 + x2 − I2 + I1 + I2 − I1,2
= x1 + x2 − I1,2 ,where the last inequality implies x ∈ S1,2,D , and, hence, we have S1 ∩ S2 ⊂ S1,2,D .

¤For x ∈ S1,D ∪ S2,D ∪ S1,2,D , VM (x) agrees with VD (x), while for x ∈/ S1,D ∪ S2,D ∪S1,2,D VD (x) is strictly larger than VM (x). This gap measures the signiﬁcance of the
dynamic management capability. Note that, although dynamic management may require
higher costs associated with the diﬃculty of the optimal exercise policy when comparedwith static management, the model does not assume any extra costs arising in dynamic
management.
We now focus on a case where X(t) follows either a GBM or a BM with a drift toshow detailed properties of dynamic management. Before deriving the results, we need
the following lemma.
Lemma 1 Assume that X(t) follows either a GBM or a BM with a drift.
0 ≤ Vi (x0i ) − Vi (xi ) ≤ x0i − xi

9(x0i ≥ xi ).

Proof We can easily check that 0 < dVi (xi )/dxi ≤ 1 holds for all xi . Then, the statement¤

follows from the mean value theorem.

Using Lemma 1, we can show the following properties of the value function VD (x) andthe stopping regions Si,D and S1,2,D .
Proposition 4 Assume that X(t) follows either a GBM or a BM with a drift.
(Convexity of the value function) VD (x) is a convex function.
(Convexity of the simultaneous exercise region) S1,2,D is a convex set.
(Monotonicity of the simultaneous exercise region) x ∈ S1,2,D ⇒ x0 ∈ S1,2,D (∀x01 ≥x1 , ∀x02 ≥ x2 ).
(Semi-monotonicity of the individual exercise regions) x ∈ Si,D ⇒ x0 ∈ Si,D (∀x0i ≥xi , x0j = xj (j 6= i)).
(Behavior on the indiﬀerence lines) x1 +V2 (x2 )−I1 = x2 +V1 (x1 )−I2 ≥ x1 +x2 −I1,2⇒x∈/ S1,D ∪ S2,D ∪ S1,2,D . xi + Vj (xj ) − Ii = x1 + x2 − I1,2 ≥ xj + Vi (xi ) − Ij (j 6= i)⇒x∈/ S1,D ∪ S2,D ∪ S1,2,D .
Proof For simplicity, we denote the payoﬀ function of problem (20) by4
f (x) = max{x1 + V2 (x2 ) − I1 , x2 + V1 (x1 ) − I2 , x1 + x2 − I1,2 , 0}.
(Convexity of the value function) By the convexity of Vi (xi ), the payoﬀ function
f (x) is also convex. Because of the convexity of the payoﬀ function the value function
VD (x) is convex (by Proposition A.6 in (Broadie and Detemple 1997), or equivalently,
Proposition 88 in (Detemple 2006)), when X(t) follows a GBM. Consider X(t) following
a BM with a drift. Similar to the case of a GBM, we can show the convexity of the valuefunction as follows. For any λ ∈ (0, 1), x ∈ R, and y ∈ R, we have
VD (λx + (1 − λ)y) = sup E(0,0) [e−rτ f (λ(x + X(τ )) + (1 − λ)(y + X(τ )))]
τ ∈T

≤ sup E(0,0) [e−rτ λf (x + X(τ )) + e−rτ (1 − λ)f (y + X(τ ))]
τ ∈T

(21)

≤ λ sup E(0,0) [e−rτ f (x + X(τ ))] + (1 − λ) sup E(0,0) [e−rτ f (y + X(τ ))]
τ ∈T

τ ∈T

= λVD (x) + (1 − λ)VD (y),where we use the convexity of f (x) in (21).
(Convexity of the simultaneous exercise region) Take any λ ∈ (0, 1), x ∈ S1,2,D ,and y ∈ S1,2,D . By the convexity of VD (x), we have
VD (λx + (1 − λ)y) ≤ λVD (x) + (1 − λ)VD (y)= λ(x1 + x2 − I1,2 ) + (1 − λ)(y1 + y2 − I1,2 )= λx1 + (1 − λ)y1 + λx2 + (1 − λ)y2 − I1,2 ,4For technical reasons we deﬁne f (x) as the nonnegative function. This does not matter because a ﬁrm never

exercises the option which yields a negative payoﬀ.

10where the last inequality implies λx + (1 − λ)y ∈ S1,2,D , and, hence, we have the convexityof the stopping region S1,2,D .
(Monotonicity of the simultaneous exercise region) First, assume that X(t) follows
a GBM. Take any x ∈ S1,2,D , x01 ≥ x1 , and x02 ≥ x2 .
VD (x0 ) = sup E(1,1) [e−rτ max{x01 X1 (τ ) + V2 (x02 X2 (τ )) − I1 , x02 X2 (τ ) + V1 (x01 X1 (τ )) − I2 ,τ ∈T
x01 X1 (τ ) + x02 X2 (τ ) − I1,2 }]≤ sup E(1,1) [e−rτ max{(x01 − x1 )X1 (τ ) + (x02 − x2 )X2 (τ ) + x1 X1 (τ ) + V2 (x2 X2 (τ )) − I1 ,τ ∈T(x01 − x1 )X1 (τ ) + (x02 − x2 )X2 (τ ) + x2 X2 (τ ) + V1 (x1 X1 (τ )) − I2 ,(x01 − x1 )X1 (τ ) + (x02 − x2 )X2 (τ ) + x1 X1 (τ ) + x2 X2 (τ ) − I1,2 }](22)≤ sup E(1,1) [e−rτ (x01 − x1 )X1 (τ )] + sup E(1,1) [e−rτ (x02 − x2 )X2 (τ )]
τ ∈T

τ ∈T

+ sup E

(1,1)

τ ∈T

[e

−rτ

max{x1 X1 (τ ) + V2 (x2 X2 (τ )) − I1 , x2 X2 (τ ) + V1 (x1 X1 (τ )) − I2 ,x1 X1 (τ ) + x2 X2 (τ ) − I1,2 }]= x01 − x1 + x02 − x2 +(23)

VD (x)| {z }=x1 +x2 −I1,2=

x01 + x02 − I1,2 ,where we use Lemma 1 in (22), and the last inequality implies x0 ∈ S1,2,D . Hence, wehave x ∈ S1,2,D ⇒ x0 ∈ S1,2,D (∀x01 ≥ x1 , ∀x02 ≥ x2 ) in the case of a GBM.
Similarly, we can show the monotonicity in the case of a BM with a drift as follows.
For any x ∈ S1,2,D , x01 ≥ x1 , and x02 ≥ x2 , we have
VD (x0 ) = sup E(0,0) [e−rτ max{x01 + X1 (τ ) + V2 (x02 + X2 (τ )) − I1 ,τ ∈Tx02 + X2 (τ ) + V1 (x01 + X1 (τ )) − I2 , x01 + X1 (τ ) + x02 + X2 (τ ) − I1,2 }]≤ sup E(0,0) [e−rτ max{x01 − x1 + x02 − x2 + x1 + X1 (τ ) + V2 (x2 + X2 (τ )) − I1 ,τ ∈Tx01 − x1 + x02 − x2 + x2 + X2 (τ ) + V1 (x1 + X1 (τ )) − I2 ,x01 − x1 + x02 − x2 + x1 + X1 (τ ) + x2 + X2 (τ ) − I1,2 }](24)≤ sup E(0,0) [e−rτ (x01 − x1 )] + sup E(0,0) [e−rτ (x02 − x2 )]τ ∈T+ sup E
τ ∈T

τ ∈T

(0,0)

[e

−rτ

max{x1 + X1 (τ ) + V2 (x2 + X2 (τ )) − I1 ,x2 + X2 (τ ) + V1 (x1 + X1 (τ )) − I2 , x1 + X1 (τ ) + x2 + X2 (τ ) − I1,2 }]= x01 − x1 + x02 − x2 +(25)

VD (x)| {z }=x1 +x2 −I1,2=

x01 + x02 − I1,2 ,where we use Lemma 1 in (24), and the last inequality implies x0 ∈ S1,2,D .
(Semi-monotonicity of the individual exercise regions) First, consider the case ofa GBM. Take any x ∈ S1,D , x01 ≥ x1 , and x02 = x2 . In the same manner as the proof of11the monotonicity of the simultaneous exercise region, we have
VD (x0 ) ≤ (23)(26)

= x01 − x1 +=VD (x)| {z }=x1 +V2 (x2 )−I10
x1 + V (x2 ) − I1 ,where the last inequality implies x0 ∈ S1,D . Hence, we have x ∈ S1,D ⇒ x0 ∈ S1,D (∀x01 ≥x1 , x02 = x2 ) in the case of a GBM. By the symmetry, we have the semi-monotonicity of
S2,D .
Next, assume that X(t) follows a BM with a drift. Take any x ∈ S1,D , x01 ≥ x1 ,and x02 = x2 . In the same manner as the proof of the monotonicity of the simultaneous
exercise region, we have
VD (x0 ) ≤ (25)
= x01 − x1 +=VD (x)| {z }=x1 +V2 (x2 )−I10
x1 + V (x2 ) − I1 ,where the last inequality implies x0 ∈ S1,D .

By the symmetry, we have the semi-

monotonicity of S2,D .
(Behavior on the indiﬀerence lines) Assume that x1 +V2 (x2 )−I1 = x2 +V1 (x1 )−I2 ≥x1 + x2 − I1,2 . Note that x ∈/ S1 ∪ S2 because of I1,2 < I1 + I2 . By Proposition 3, we havex∈
/ S1,D ∪ S2,D , which implies
VD (x) > x1 + V2 (x2 ) − I1 = x2 + V1 (x1 ) − I2 ≥ x1 + x2 − I1,2 .
Thus, we have x ∈/ S1,D ∪ S2,D ∪ S1,2,D .
Assume that x1 + V2 (x2 ) − I1 = x1 + x2 − I1,2 ≥ x2 + V1 (x1 ) − I2 . Note that x ∈/ S2
because of I1,2 < I1 + I2 . First, consider the case of a GBM. By the convexity of V2 (x02 ),we have
V2 (x02 ) ≥ c1 x02 + c2
whereβ2
dV (x2 )
=
c1 =
dx2
x2

µ

x2
x∗2

(x02 ∈ R++ )

¶β2

and
dV (x2 )+ V (x2 ) = −(β2 − 1)c2 = −x2dx2(27)(x∗2 − I2 ) ∈ (0, 1)
µ

x2
x∗2

¶β2

(x∗2 − I2 ) ∈ (−I2 , 0).

Note that the right-hand side of (27) is the ﬁrst order Taylor approximation to V2 (x02 )around the point x2 . In (27), the equality holds if and only if x02 is equal to x2 . For any

12t ≥ 0, by (27) we have
VD (x) ≥ Ex [e−rt max{X1 (τ ) + c1 X2 (t) + c2 − I1 , X1 (t) + X2 (t) − I1,2 , 0}]≥ e−rt Ex [X1 (τ ) + c1 X2 (t) + c2 − I1 ]|{z}
→x1 +V2 (x2 )−I1 (t↓0)

−rt x

+e
|

E [max{(1 − c1 )X2 (t) − I1,2 − c2 + I1 , 0}] .
{z
}

(28)

↓0 (t↓0)

In (28), the ﬁrst term → x1 +c1 x2 +x2 −I1 = x1 +V2 (x2 )−I1 (t ↓ 0) at a ﬁnite rate while thesecond term ↓ 0 (t ↓ 0) at a rate that increases to inﬁnity in the limit. Note that the second
term corresponds to the value of an at-the-money European call option with maturity t.
Therefore, there exists some t > 0 such that (28) is strictly larger than x1 + V2 (x2 ) − I1 .
This implies that VD (x) > x1 + V2 (x2 ) − I1 = x1 + x2 − I1,2 , i.e., x ∈/ S1,D ∪ S2,D ∪ S1,2,D .
By the symmetry, we have x2 + V1 (x1 ) − I2 = x1 + x2 − I1,2 ≥ x1 + V2 (x2 ) − I1 ⇒x∈
/ S1,D ∪ S2,D ∪ S1,2,D . Similarly, using the ﬁrst order Taylor approximation, we canshow that the statement holds for a BM with a drift.

¤

Proposition 4 extends previous ﬁndings by (Geltner, et al 1996, Broadie and Detemple
1997, Bobtcheﬀ and Villeneuve 2005) to a case allowing a convex function Vi (x). Technically, the proof of the behavior on the indiﬀerence lines has been accomplished in thesame manner as (Nishihara 2010). The monotonicity of S1,2,D leads to the straightforward prediction that simultaneously increased values of two projects encourage simultaneous investment. Although we prove neither the convexity of Si,D nor a property thatx ∈ Si,D ⇒ x0 ∈ Si,D (∀x0i = xi , x0j ≤ xj (j 6= i)), these properties will be veriﬁed numerically in Section 4. Then, the increased value of either project tends to provide the
incentive for a ﬁrm to undertake the better project individually. The behavior on theindiﬀerence lines means that a ﬁrm must delay the decision on the investment type when
the two types of investment yield the same value. This is in line with the previous ﬁndingsin the max-options analysis (e.g., (Geltner, et al 1996, Detemple 2006)). By Proposition
2, we can also show the following comparative statics with respect to the correlationcoeﬃcient ρ.
Proposition 5 Assume that X(t) follows a BM with a drift.
(Monotonicity of the value function) VD (x) monotonically increases with ρ.
(Monotonicity of the exercise regions) Si,D and S1,2,D monotonically decrease withρ.
Proof The monotonicity of VD (x) immediately follows from Proposition 2. We attachρsuperscript ρ to VD (x) and S1,D to avoid confusing. For any x ∈ S1,D
and ρ0 < ρ, we have
0

VDρ (x) ≤ VDρ (x) = x1 + V2 (x2 ) − I1 ,0ρ
. Similarly, we can show the monotonicity of S2,D and S1,2,D .
which implies x ∈ S1,D

13¤

Note that a decrease in S1,2,D with ρ does not mean a decrease in the possibility
that a ﬁrm chooses simultaneous investment. An increase in VD (x) with ρ results froman increase in V1,2 (x). Then, as in static management (cf. Proposition 2), a ﬁrm tends
to make simultaneous investment in strongly correlated markets. We also make a briefcomment regarding the one-dimensional model by (Décamps, et al 2006). Problem (20) is
similar to that of (Décamps, et al 2006), when X1 (t) and X2 (t) follow the same dynamics(which means that ρ = 1) with diﬀerent initial values. In this case, as in (Décamps, etal 2006), we can derive VM (x) and VD (x) in closed forms, and it can readily demonstratethat there is no gap between VM (x) and VD (x) for a suﬃciently small x. Accordingly, we
recognize that an imperfect correlation is a source of the gap between static and dynamic
management. This eﬀect can be intuitively explained as follows. As the correlation
becomes weaker, a static choice of the investment type is more likely to result in anincorrect choice ex post. Then, the gap between static and dynamic management increases
with a weaker correlation. This result implies that managerial ﬂexibility can be a key tosuccess in market entry into several countries and regions in Asia with wide diversity.
This view will be examined numerically in Section 4.

3.2

Extensions and limitations

This section investigates the robustness of the results in Section 3.1 with respect to changes
in the model assumptions. First, we consider the eﬀects of strategic interactions. Separate
investment may entail a higher risk of rival preemption than simultaneous investment.
This is because the ﬁrst investment induces potential rivals to invest in the remaining
business opportunity. We can incorporate this into our model as follows.5 Assume thatthe second investment opportunity for project i is killed at an instantaneous rate λi dt,
where a positive constant λi denotes the intensity. Let Ṽi (x) denotes the value function(2) for the killed process of Xi (t). For X(0) = x, ﬁrm value under dynamic management
is evaluated by
VD (x) = sup Ex [e−rτ max{X1 (τ ) + V˜2 (X2 (τ )) − I1 , X2 (τ ) + V˜1 (X1 (τ )) − I2 ,τ ∈T

X1 (τ ) + X2 (τ ) − I1,2 }].

(29)

Clearly, Proposition 3 holds for the killed problem (29). It is also clear that Ṽi (x)monotonically decreases with λi . Then, similar to Proposition 5, we can show a monotonic
decrease in VD (x) and monotonic increases in Si,D and S1,2,D with respect to λi . We caneasily derive Ṽi (x) in a closed form when X(t) follows either a GBM or a BM with adrift. In this case, Propositions 4 and 5 hold for the killed problem (29). Accordingly,
the results in Section 3.2 are relatively robust with respect to consideration of strategic
5

Another approach to strategic interactions is the game-theoretic approach (e.g., (Grenadier 1996, Huisman

2001)). For example, (Nishihara 2009) investigates a duopoly real options game concerning two projects.

14interactions. In addition, for X(t) following a nonnegative process, including a GBM, thevalue function (29) approaches to V1,2 (x) as λi → ∞ (i = 1, 2). This means that thethreat of rival preemption provides the incentive for a ﬁrm to undertake both projects
simultaneously.
So far, we assume that I1,2 < I1 + I2 to capture the positive synergy of simultaneous
investment. However, synergy may change not only the costs but also the proﬁts. Whenthe value of simultaneous investment can be expressed as a linear combination of X1 (t)and X2 (t) with positive coeﬃcients, few diﬃculties arise from the technical viewpoint.
Propositions 4 and 5 hold for the case. When there is nonlinear synergy of simultaneous
investment, it is mathematically diﬃcult to show the properties of the value function andthe stopping regions. In such cases, the results depend on parameter values, and we must
calibrate the model carefully.
The eﬀect of learning is another important issue that should be addressed. When aﬁrm undertakes projects sequentially, it may beneﬁt in learning from the ﬁrst investment.
From the ﬁrst investment, a ﬁrm may acquire skill, know-how, reputation, etc. If thisis the case, a ﬁrm will make the second investment more eﬃciently. We can capture theeﬀect by assuming that the second investment requires the sunk costs I˜j , which is lower
than Ij .6 As I˜j decreases, the possibility of individual investment increases. In particular,when Ii + I˜j (i 6= j) decreases below I1,2 , the positive synergy of simultaneous investment
is oﬀset by the positive eﬀect of learning in separate investment. In this case, a ﬁrm
always chooses individual investment with the beneﬁt from project ﬂexibility.
In this paper, we consider two projects. One of its natural extensions is to take into
consideration more than two projects. Because the number of combinations of projects
which are undertaken simultaneously increases exponentially with the number of projects,
the formulation and computation of the optimal exercise policy become much more diﬃcult. However, the theoretical results in Propositions 3–5 remain essentially unchanged.

4

Numerical examples

This section reveals further properties of static and dynamic management in numericalexamples. Assume that X(t) follows a GBM. We use base parameter values as follows7 :r = 8%, µ1 = µ2 = 0%, σ1 = σ2 = 20%.

6

(30)

An alternative modeling for learning is the ﬁltering approach (e.g., (Bernardo and Chowdhry 2002, Décamps,

et al 2005, Shibata 2008)). Extending our model to a ﬁltering model will be a diﬃcult but important challenge
in future work.
7
These parameter values are similar to (Geltner, et al 1996, Detemple 2006). We carried out a lot ofcomputations with varying parameter values and distilled robust results into this section.

15

For expositional purposes, we set I1 = I2 = 10, I1,2 = 15. The positive synergy ofsimultaneous investment is (20 − 15)/20 = 25%. We can derive Vi (x) in a closed form,
but we must rely on numerical methods to compute V1,2 (x), VM (x), and VD (x). We makea bivariate lattice model8 that approximates to a GBM, and we execute a value function
iteration algorithm to compute V1,2 (x), VM (x), and VD (x).
First, we explore the nature of static management. Table 1 shows the option values
Vi (x), V1,2 (x), VM (x) = max{V1 (x) + V2 (x), VM (x)}, VD (x) and the investment thresholdx∗i with varying levels of the correlation coeﬃcient ρ. We set x = (10, 10), which is thesame as the sunk cost I1 = I2 = 10 for individual investment. Note that Vi = 16.404
and x∗i = 1.802 do not depend on ρ. For ρ = −0.5, x = (10, 10) lies in the stopping
region S1,2,D , and, hence, V1,2 (x) = VM (x) = VD (x) = x1 + x2 − I1,2 = 5 holds. For
other levels of ρ, x = (10, 10) lies in S1,2,M \ (S1,D ∩ S2,D ∩ S1,2,D ), which means that
V1,2 (x) = VM (x) < VD (x). In Table 1, the value of the option to invest simultaneously,V1,2 (x), monotonically increases with ρ. This is because a higher ρ makes X1 (t) + X2 (t)more volatile and increases the option value. Figure 1 illustrates the stopping region S1,2
for the basket option. As mentioned in Section 2.2, S1,2 satisﬁes the convexity and themonotonicity. We see a monotonic decrease in S1,2 with respect to ρ. This is analogous
to the monotonic increase in V1,2 (x) with respect to ρ. This monotonicity of V1,2 (x) andS1,2 is the same as the case of a BM with a drift (cf. Proposition 1).
Figure 4 illustrates the simultaneous investment region under static management,
S1,2,M , with varying levels of ρ. The region S1,2,M monotonically decreases with ρ becauseof the monotonicity of V1,2 (x). Similar to the case of a BM with a drift (cf. Proposition 2),
a ﬁrm is more likely to make simultaneous investment in strongly correlated markets. This
result is consistent with empirical observations. For a large xi , as mentioned in Section
2.3, the boundary of S1,2,M coincides with a line xj = xˆj = 5.2729 (j 6= i), where xˆj is aunique solution to Vj (xj ) = xj − I1,2 + Ii . For a large ρ, S1,2,M shows neither monotonicity
nor convexity. As ρ approaches 100%, S1,2,M approaches the region a1 ≤ x1 /x2 ≤ a2 fora small x, where ai is a constant. This is because the proportion of x1 to x2 mattersto a GBM. In contrast, the diﬀerence between x1 and x2 matters to a BM with a drift.
Then, when ρ is very large, S1,2,M is like the region a1 ≤ x1 − x2 ≤ a2 for a small x. The
patterns of static management are revealed ﬁrst by our analysis.
Next, we explore the nature of dynamic management. Figure 3 illustrates the stopping
regions Si,D and S1,2,D for ρ = 0%. For comparison, we plot S1,2,M under static management. We can check all of the properties in Propositions 3 and 4. The convexity andmonotonicity of Si,D can be also veriﬁed numerically. The continuation region becomesmuch smaller as xi increases. Indeed, we see that the continuation region approaches to a8

We make a discretization with 100 time steps per 1 year following a bivariate version of the lattice binomial

method (see (Boyle 1988)).

1650
ρ=−0.5ρ=−0.25
ρ=0ρ=0.25
ρ=0.5

40

x2

30

20
S1,2
10

10

20

30

40

50

x1

Figure 1: The exercise region for the option to invest simultaneously. This ﬁgure plots S1,2
with varying levels of ρ. The parameter values are set at the base case (30).
50ρ=−0.5ρ=−0.25
ρ=0ρ=0.25
ρ=0.5

40

x2

30

20
S1,2,M

10

10

20

30

40

50

x1

Figure 2: The simultaneous investment region under static management. This ﬁgure plots
S1,2,M with varying levels of ρ. The parameter values are set at the base case (30).

17ρ\ Value

Table 1: The option values.
x∗i
Vi (xi ) V1,2 (x) VM (x) VD (x)−50%

16.404

1.802

5

5

5

−25%

16.404

1.802

5.023

5.023

5.054

0%

16.404

1.802

5.118

5.118

5.15

25%

16.404

1.802

5.239

5.239

5.259

50%

16.404

1.802

5.371

5.371

5.38

line xj = xˆj = 5.2729 (j 6= i) when xi → ∞. This means that for a large xi , a ﬁrm undertakes investment in a short time, whether individually or simultaneously, even if the two
types of investment have the same value. This ﬁnding contrasts with that of max-option
analysis (Geltner, et al 1996, Detemple 2006, Nishihara 2010), in which the waiting timeis rather long when two values are equivalent. Figure 4 shows the comparative statics of
Si,D and S1,2,D with respect to ρ. A monotonic decrease in S1,2,D can be seen clearly,
while Si,D is robust with respect to changes in ρ (cf. Proposition 5). This is becausechanges in ρ inﬂuence V1,2 (x) rather than Vi (x). Table 1 shows that VD (x), like VM (x),monotonically increases with ρ. This follows from a monotonic increase in V1,2 (x) withrespect to ρ.
We proceed with an analysis of a gap between dynamic and static management. Unlessx lies in the stopping regions Si,D and S1,2,D , VD (x) is strictly higher than VM (x). Thisgap measures the impact of the managerial ﬂexibility on ﬁrm value. Figures 5 and 6 plot
contour lines of VD (x)/VM (x). Figure 5 shows the comparative statics of VD (x)/VM (x)with respect to ρ. We see from each panel that VD (x)/VM (x) becomes large on theboundary of S1,2,M for a small x. This ﬁnding can be interpreted as follows. When theoption values of individual and simultaneous investment are similar, a ﬁrm must wait and
see which type is more eﬃcient. In this case, there is a remarkable advantage to dynamic
corporate management over static management. Otherwise, it does not matter whether aﬁrm chooses the investment type statically or dynamically.
From Figure 5, we recognize that VD (x)/VM (x) is rather robust with respect to changes
in ρ. To examine it more accurately, VD (x)/VM (x) tends to be higher when ρ approaches
0%. This is mainly because a weaker correlation increases the possibility that an initialdecision of whether to exercise options individually or simultaneously leads to ineﬃciency
ex post. This result predicts that the dynamic management capability will be a major
success determinant for a ﬁrm’s expansion into Asia’s emerging markets which involve avariety of risks depending on countries and regions. The gap VD (x)/VM (x) also depends
on r, µi , and σi . For example, Figure 5 shows the comparative statics with respect toσi . We see that VD (x)/VM (x) decreases with σi . This is because the positive synergy

18

of simultaneous investment becomes smaller relative to the option value, which greatly
increases with σi . Similarly, an increase in µ and a decrease in r enhance the option value
and then reduce VD (x)/VM (x).

5Conclusion

The paper proposed a model for management of multiple real options to ﬁll a great gap
between the project portfolio and real options literature. In particular, we focused on the
problem of whether a ﬁrm should exercise two real options individually or simultaneously.
The model assumes that simultaneous exercise of both options has positive synergy, suchas economies of scale, scope, and networks. The analysis revealed the characteristics oftwo styles of management, namely static and dynamic management. A ﬁrm under static
management determines whether it exercises options individually or simultaneously ex
ante, while a ﬁrm under dynamic management makes the choice at the time of exercise.
We veriﬁed the natural intuition that a lower correlation among project values gives aﬁrm the incentive to invest individually rather than simultaneously. Further, and more
importantly, we emphasized the signiﬁcance of dynamic corporate management to a ﬁrm
entering weakly correlated markets. The model applies to a ﬁrm’s strategic decision on
business expansion into several countries and regions in Asia. Our results imply thatthe heterogeneous dynamics of Asia’s markets across countries and regions increase the
incentive for a ﬁrm to enter each market individually rather than the whole market simultaneously. Further, our results imply that the dynamic management capability will be amajor success determinant for a business in Asia.

References
Bernardo, A., and B. Chowdhry, 2002, Resources, real options, and corporate strategy,
Journal of Financial Economics 63, 211–234.
Bobtcheﬀ, C., and S. Villeneuve, 2005, Irreversible investment in competitive projects: A
new motive for waiting to invest, Working Paper, Université de Toulouse 1.
Boyle, P., 1988, A lattice framework for option pricing with two state variables, Journalof Financial and Quantitative Analysis 23, 1–12.
Broadie, M., and J. Detemple, 1997, The valuation of american options on multiple assets,
Mathematical Finance 7, 241–286.
Childs, P., S. Ott, and A. Triantis, 1998, Capital budgeting for interrelated projects: A
real options approach, Journal of Financial and Quantitative Analysis 33, 305–334.

19

50
staticdynamic
40

x2

30
S2,D
20
S1,2,D
S1,2,M

10

S1,D
10

20

30

40

50

x1

Figure 3: Static and dynamic management. This ﬁgure plots the stopping regions Si,D and
S1,2,D under dynamic management along with S1,2,M under static management. The parametervalues are set at the base case (30) with ρ = 0%.
50
ρ=−0.5ρ=−0.25
ρ=0ρ=0.25
ρ=0.5

40

30
x2

S2,D
20
S1,2,D
10
S1,D
10

20

30

40

50

x1

Figure 4: The exercise regions under dynamic management. This ﬁgure plots Si,D and S1,2,D
with varying levels of ρ. The parameter values are set at the base case (30).

20

50

50
ρ=−0.5 1.1240ρ=−0.25 1.12
40

1.1

20

1.06

30

1.08

20

1.06

x2

1.08

x2

30

1.1

1.04

10

1.04

10

1.02
10

20

30

40

1.02
50

10

20

x1

30

40

50

x1

50

50
ρ=0 1.1240

ρ=0.25 1.12
40

1.1

20

1.06

30

1.08

20

1.06

x2

1.08

x2

30

1.1

1.04

10

1.04

10

1.02
10

20

30

40

1.02
50

10

20

x1

30

40

50

x1

50

50
ρ=0.5 1.1240

ρ=0.75 1.12
40

1.1

20

1.06

30

1.08

20

1.06

x2

1.08

x2

30

1.1

1.04

10

1.04

10

1.02
10

20

30

40

1.02
50

10

x1

20

30

40

50

x1

Figure 5: The gap between static and dynamic management. These panels plot contour lines
of VD (x)/VM (x) for ρ = −50%, −25%, 0%, 25%, 50%, and 75%. The contour levels are set at
1.02, 1.04, 1.06, 1.08, 1.1, and 1.12. The parameter values are set at the base case (30).

21

50

50
σ=0.1 1.2

σ=0.2 1.2

40

40
1.15

1.15
x2

30

x2

30
1.1

20

1.05

10
10

20

30

40

1.1

20

1.05

10
50

10

20

x1

30

40

50

x1

50

50
σ=0.3 1.2

σ=0.4 1.2

40

40
1.15

1.15
x2

30

x2

30
1.1

20

1.05

10
10

20

30

40

1.1

20

1.05

10
50

10

x1

20

30

40

50

x1

Figure 6: The gap between static and dynamic management. These panels plot contour
lines of VD (x)/VM (x) for σ1 = σ2 = 10%, 20%, 30%, and 40%. The contour levels are set at
1.02, 1.04, 1.06, . . . , 1.2. The other parameter values are set at the base case (30) with ρ = 0%.

22

Cortazar, G., M. Gravet, and J. Urzua, 2008, The valuation of multidimensional american
real options using the lsm simulation method, Computers and Operations Research 35
35, 113–129.
Décamps, J., T. Mariotti, and S. Villeneuve, 2005, Investment timing under incomplete
information, Mathematics of Operations Research 30, 472–500.
Décamps, J., 2006, Irreversible investment in alternative projects, Economic Theory 28,
425–448.
Detemple, J., 2006, American-Style Derivatives valuation and computation (Chapman &Hall/CRC: London).
Dixit, A., and R. Pindyck, 1994, Investment Under Uncertainty (Princeton University
Press: Princeton).
Geltner, D., T. Riddiough, and S. Stojanovic, 1996, Insights on the eﬀect of land use
choice: The perpetual option on the best of two underlying assets, Journal of Urban
Economics 39, 20–50.
Grenadier, S., 1996, The strategic exercise of options: development cascades and overbuilding in real estate markets, Journal of Finance 51, 1653–1679.
Grenadier, S., 2002, Option exercise games: an application to the equilibrium investment
strategies of ﬁrms, Review of Financial Studies 15, 691–721.
Grenadier, S., and N. Wang, 2005, Investment timing, agency, and information, Journalof Financial Economics 75, 493–533.
Huisman, K., 2001, Technology Investment: A Game Theoretic Real Options Approach(Kluwer Academic Publishers: Boston).
Lambrecht, B., and W. Perraudin, 2003, Real options and preemption under incomplete
information, Journal of Economic Dynamics and Control 27, 619–643.
Loubergé, H., S. Villeneuve, and M. Chesney, 2002, Long-term risk management of nuclear
waste: a real options approach, Journal of Economic Dynamics and Control 27, 157–
180.
Luehrman, T., 2004, Strategy as a portfolio of real options, Real Options and Investment
Under Uncertainty (edited by E. Schwartz and L. Trigeorgis) pp. 385–404.
Martzoukos, S., 2009, Real R&D options and optimal activation of two-dimensional random controls, Journal of the Operations Research Society 60, 843–858.
Mauer, D., and S. Sarkar, 2005, Real options, agency conﬂicts, and optimal capital structure, Journal of Banking and Finance 29, 1405–1428.
Meier, H., N. Christoﬁdes, and G. Salkin, 2001, Capital budgeting under uncertainty- an integrated approach using contingent claims analysis and integer programming,
Operations Research 49, 196–206.

23

Nishihara, M., 2009, Preemptive investment game with alternative projects, Discussion
Papers in Economics and Business, Osaka University 09-16.
Nishihara, M., 2010, Hybrid or electric vehicles? a real options perspective, Operations
Research Letters 38, 87–93.
Nishihara, M., and M. Fukushima, 2008, Evaluation of ﬁrm’s loss due to incomplete
information in real investment decision, European Journal of Operational Research 188,
569–585.
Peskir, G., and A. Shiryaev, 2006, Optimal Stopping and Free-Boundary Problems(Birkhauser: Basel).
Shibata, T., 2008, The impacts of uncertainties in the real options model under incomplete
information, European Journal of Operational Research 187, 1368–1379.
Shibata, T., and M. Nishihara, 2010, Dynamic investment and capital structure undermanager-shareholder conﬂict, Journal of Economic Dynamics and Control 34, 158–
178.
Trigeorgis, L., 1993, The nature of option interactions and the valuations of investmentswith multiple real options, Journal of Financial and Quantitative Analysis 28, 1–20.
Wang, J., and W. Hwang, 2007, A fuzzy set approach for R&D portfolio selection using
a real options valuation model, Omega 35, 247–257.

24