Discussion Papers In Economics
And Business

Real options with illiquidity of exercise opportunities
Michi NISHIHARA

Discussion Paper 19-01

March 2019

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANReal options with illiquidity of exercise opportunities∗
Michi NISHIHARA†,

Abstract
This paper presents a framework to study real options with illiquidity of option exercise
opportunities. I incorporate a constraint that the investment time is chosen from Poisson
arrival times in the standard real option value (ROV) model. I derive the closed-form
solution and show that illiquidity decreases the option value of waiting and the investment
threshold. I extend the results to a case with diﬀerent types of projects and show that aninferior project can be undertaken in the presence of illiquidity. I prove that the solutionof the illiquid model converges to that of the ROV model for higher liquidity and converges
to that of the net present value (NPV) model for lower liquidity. I also show that thesolution agrees with the limit of the corresponding regime-switching model. The results
fill the gaps in the NPV, ROV, and regime-switching models and reveal the eﬀects of
illiquidity on investment decisions.
JEL Classifications Code: D82; G13; G33.
Keywords: real option; net present value; regime switching; liquidity; search theory.

∗

This version was written on 4 March, 2019. The author thank Junichi Imai, Takashi Shibata, and Kazutoshi

Yamazaki for helpful comments. The paper was presented the conferences in Crete, Taipei, and Tokyo in 2018.
The author thanks the participants for helpful feedbacks. This work was supported by the JSPS KAKENHI(Grant number JP17K01254, JP17H02547).
†
Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka,
Osaka 560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Phone: 81-6-6850-5242, Fax: 81-6-6850-5277

1Introduction

In project valuation involving high uncertainty and managerial flexibility, the real option
value (ROV) method adds value to the basic net present value (NPV) method (cf. Dixit
and Pindyck (1994)). A number of papers have recently investigated corporate financial
issues, such as financing and investment, mergers and acquisitions, and bankruptcy andliquidation in the ROV framework. Most of these studies presume that an option holder
can exercise his/her real option at an arbitrary time, analogous to American call and put
options.
In the real world, however, the assumption of arbitrary exercise timing does not always
hold. For example, consider a firm that will expand its business by acquiring (a certain
section of) another firm. If a hostile acquisition is too costly, the firm begins by searchingfor a target that may potentially agree to be acquired. Only if the firm finds such a target
and the negotiation and/or bidding process succeeds, the firm can acquire the target
assets. Otherwise, the firm begins by searching for another target. In this case, the firm
can exercise the acquisition option not at an arbitrary time but only when it finds a target
satisfying certain conditions. Similarly, in an asset liquidation problem, a firm can sell
illiquid assets at fair prices not at an arbitrary time but only when it meets a counterparty.
Another example is a financially constrained firm’s investment. A firm can invest in a
project not at an arbitrary time but when it is able to raise funds for the project. Some
investments might be feasible only when the investment programs qualify for government
grants and subsidies. Boyle and Guthrie (2003) investigate investment timing decisions,assuming that investment is feasible only when internal funds are suﬃcient.
Real options frequently concern transactions of illiquid assets, such as real estate,technology, patents, and businesses. Capital market friction may also restrict real option
exercise opportunities. Nevertheless, in the real options context, few papers have investigated the eﬀects of illiquidity of exercise opportunities. On the other hand, a number of
papers focus on market illiquidity of labor, real estate, and financial assets such as debts
and derivatives, etc. For example, Duﬃe, Garleanu, and Pedersen (2005) reveal how intermediation and asset prices in over-the-counter markets are aﬀected by illiquidity. He
and Xiong (2012) and He and Milbradt (2014) show the eﬀects of debt market illiquidity
on a firm’s bankruptcy timing. These studies are based on search theory, which was first
proposed by Diamond (1982) to investigate labor market illiquidity.
This paper develops a new framework to study real option problems with illiquidity ofoption exercise opportunities. To do so, I incorporate a constraint that the investment time2

is chosen from Poisson arrival times in the investment timing model (hereafter, the ROV
model) of McDonald and Siegel (1986).1 The Poisson arrival process, as in search-theoretic
models, stands for illiquidity of option exercise opportunities. Most notably, I derive
the closed-form solution in the proposed illiquid model. The obtained solution exhibitsintermediate characteristics between the NPV and ROV solutions. Indeed, the option
value of waiting decreases due to illiquidity, and hence, the investment threshold decreases.
This implies that in the presence of illiquidity, the firm exercises an investment option
more eagerly than in the ROV model. Although Boyle and Guthrie (2003) document
similar results from capital market friction, they do not derive any analytical solution due
to the model complexity.
Furthermore, I also examined a case in which two types of projects arrive randomly.
Although I cannot derive a closed-form solution in this case, I obtain the analytical solution. When the profitability gap between the two types, liquidity, and the probabilityof the good-type appearance increase, the firm is more likely to wait for a good-type
project and forgo a bad-type project. When these factors are low, the firm accepts only
a good-type project for intermediate levels of the state variable and accepts even a badtype project for suﬃciently high levels of the state variable. The impact of illiquidity
on the bad-type investment threshold is much greater than on the good-type investment
threshold. This result is also numerically verified for a case with a continuum of projecttypes.
I prove that the solution of the illiquid model converges to that of the ROV modelfor higher liquidity and that of the NPV model for lower liquidity. I also reveal the relationship of the illiquid model to the regime-switching model of Hackbarth, Miao, andMorellec (2006); indeed, the limiting solution of the regime-switching model concurs withthe solution of the illiquid model. Thus, in terms of illiquidity of real option exercise
opportunities, this paper fills the gaps in the NPV, ROV, and regime-switching models.
The proposed models and solutions can potentially play a significant role as a new framework to study real options with illiquidity of option exercise opportunities. They can beapplied to various problems, including but not limited to problems of merger and acquisition timing with illiquidity of searching and matching, illiquid asset sales and liquidation
timing, and investment timing with illiquidity of fundraising.
The remainder of this paper is organized as follows. Section 2 introduces the model1

Although I describe the results of the investment timing model in full details, the same analysis can beapplied to asset sales and liquidation models (cf. Section 4.3).

3

setup. Then, Sections 3.1 and 3.2 explain the NPV and ROV models. Sections 3.3 and3.4 show the main results of the models with illiquidity of option exercise opportunities.
Section 4 extends these models in several ways, and Section 5 concludes the paper.

2

Setup

Consider a firm that has an option to invest in a project only when an investment opportunity arrives. Assume that the project arrival process follows a Poisson process with thearrival rate λ(> 0). If the firm rejects a project, the firm has to wait until another project
arrives. The firm cannot invest in more than one project.2
Project i requires the initial investment cost Ii and perpetually generates cash flowai X(t). The random shock X(t) follows a geometric Brownian motiondX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,where B(t) denotes the standard Brownian motion defined in a filtered probability space(Ω, F, P, {Ft }) and µ, σ(> 0) and x(> 0) are constants. Throughout the paper, for themodel tractability, I assume that B(t) is independent of the Poisson process. A positive
constant r denotes the discount rate, and for convergence I assume that r > µ.
I consider two types of projects, i.e., i = G (Good) and i = B (Bad), where 0 < aB <aG and 0 < IG ≤ IB . Define the expected payoﬀ of project i by πi (x) = ai x/(r − µ) − Ii .
In terms of profitability, a project G dominates a project B because πG (x) > πB (x) holds
for all x > 0.3 Section 3.3 examines a case in which the same type of projects arrive.
Section 3.4 examines a case in which a project type follows an independent draw with
i = G at probability q ∈ (0, 1) and i = B at probability 1 − q.
The model incorporates illiquidity of investment opportunities in the ROV model of
McDonald and Siegel (1986). As in search-theoretic models (e.g., Diamond (1982) andDuﬃe, Garleanu, and Pedersen (2005)), illiquidity is captured by the Poisson arrival
process. The illiquid model is relevant to real options involving transactions of illiquid
assets, such as real estate, technology, patents, and businesses. It can also represent
illiquidity of fundraising. Several papers (e.g., Baldwin and Meyer (1979) and Baldwin(1982)), prior to development of the real options literature, investigate investment timing
2
3

Investment projects are mutually exclusive or the firm chooses one project under financial constraints.
Unlike this paper, Décamps, Mariotti, and Villeneuve (2006) consider two mutually exclusive projects with

diﬀerent scales, i.e., 0 < aB < aG and 0 < IB < IG . They solved the optimal investment timing and sizing
problem under the standard real options assumption that investment is possible at an arbitrary time.

4

under the assumption that investment opportunities arrive according to a Poisson process,but these models do not include a continuous-time state process, i.e., X(t). Section 4.2
shows the relationship of the illiquid model to the regime-switching model of Hackbarth,Miao, and Morellec (2006). I can apply the same analysis used in this paper to put-type
real options (cf. Section 4.3).

3

Model Solutions

3.1

The NPV model

As a benchmark, this subsection briefly explains the NPV framework. Suppose that a
project of type i, which is fixed at G or B, is feasible at the initial time.4 Consider thefirm that follows the NPV rule, i.e., the firm undertakes a project if and only if πi (x) ≥ 0
at the initial time.
The firm value with a project i, denoted by N P Vi (x), is given by
N P V i (x) = max{πi (x), 0} (i = G, B).

(1)

The investment threshold in the NPV model is given by
PV
xN
=
i

(r − µ)Ii
ai(i = G, B).

(2)

P V . Otherwise, it never invests in a project. The
The firm invests in a project i if x ≥ xNi

NPV rule is optimal if the firm has no option to defer the investment timing.

3.2

The ROV model

As another benchmark, this subsection explains the ROV model, which was first studiedin McDonald and Siegel (1986). Suppose that a project of type i, which is fixed G or
B, is feasible at any time.5 Consider the firm that follows the ROV rule, i.e., the firm
undertakes a project at the optimal time.
For future uses, I define, for y > 0,√(µ
)2 2(r + y)µβy = 0.5 − 2 +−0.5+(> 1),
2
σ
σ2
√ σ
(µ
)2 2(r + y)
µ
+
−
0.5
(< 0).
γy = 0.5 − 2 −
σ
σ2
σ2
4
5

(3)
(4)

The firm focuses only on project G when both types of projects are feasible at the initial time.
The firm focuses only on project G when both types of projects are feasible at an arbitrary time.

5For notational simplicity, I denote β = β(0) and γ = γ(0). In the standard manner (e.g.,
Dixit and Pindyck (1994)), I can derive the firm value with a project i as

)β
(
x

 πi (xROV )(x < xROV
),
i
i
ROV
x
ROVi (x) =(i = G, B),
i


ROV
πi (x)(x ≥ xi),(5)where the investment threshold is given by
xROV
=
i

PV
βxN
PV
i
(< xN
) (i = G, B).
i
β−1

(6)

For x ≥ xROV
, the firm invests in a project i at the initial time. In this case, ROVi (x) =i
N P Vi (x) holds, which means the ROV method adds no value to the NPV method. Forx < xROV, the firm invests in a project i as soon as the state variable X(t) hits xROV.
ii
The upper equation in (5) stands for the value of deferring investment. Because of theoption value of waiting, ROVi (x) > N P Vi (x) holds for x < xROV
.
i

3.3

The same type of projects arriving randomlyThis subsection examines the baseline model in which the same type of projects arrive
randomly. To be more precise, I assume that projects of type i, which is fixed at G or
B, arrive according to a Poisson process with the arrival rate λ. I denote by Vi (x) the
firm value when a project i is not feasible and by Ṽi (x) the firm value when a projecti arrives. I have Ṽi (x) = max{πi (x), Vi (x)} because the firm value Ṽi (x) becomes Vi (x)when a feasible project i is forgone.
I can show that the value function Ṽi (x) is continuous and that the optimal policy isa threshold policy. For the proof, see Appendix A. Then, the stopping region Si∗ = {x ∈R+ | Ṽi (x) = πi (x)} is expressed asSi∗ = {x ∈ R+ | x ≥ x∗i },(7)

where x∗i = min{x ∈ R+ | Ṽi (x) = πi (x)}, and Ṽi (x) is expressed as Vi (x) (x < x∗ ),iṼi (x) = π (x) (x ≥ x∗ ).
i

(8)

i

Because of N P Vi (x) ≤ Ṽi (x) ≤ ROVi (x), the investment threshold x∗i exists between
P V and xROV . Because of Theorem 4.4.9 in Karatzas and Shreve (1998), for thexNiithreshold policy, the expectation Vi (x) is a piecewise C 2 function and satisfies ordinary
diﬀerential equations (ODEs):µxVi′ (x) + 0.5σ 2 x2 Vi′′ (x) = rVi (x)(0 < x < x∗i ),µxVi′ (x) + 0.5σ 2 x2 Vi′′ (x) + λ(πi (x) − Vi (x)) = rVi (x) (x > x∗i ),

6

(9)
(10)

where λ(πi (x)−Vi (x)) corresponds to the fact that Vi (x) changes to πi (x) with probability
λdt in infinitesimal time interval dt. The piecewise C 2 property means that Vi (x) is
continuously diﬀerentiable at x∗i (see Theorem 4.4.9 in Karatzas and Shreve (1998)). By
solving (9) and (10) with the boundary conditions which will be specified in the proof, I
have the following proposition.
Proposition 1 A1,i xβ
Vi (x) =(x < x∗i ),

λIi
λai x
−
+ A2,i xγλ(r − µ)(r + λ − µ) r + λ(x ≥ x∗i ),(11)

where the investment threshold x∗i and coeﬃcients A1,i , A2,i arePV(r + λ − µ)((β − γλ )r + βλ)xNi,(r + λ)((β − γλ )(r − µ) + (β − 1)λ)πi (x∗i )A1,i =
,
x∗β
i
(
)
1
ai x∗irIi
A2,i = ∗γλ−.
r+λ−µ r+λ
xi

x∗i =

(12)
(13)
(14)

Proof. I have the boundary conditions
lim Vi (x) = 0

(15)

lim

Vi (x)<∞
x→∞ πi (x)(16)

Vi (x∗ ) = πi (x∗ )

(17)

x→0

along with the continuous diﬀerentiability of Vi (x) at x∗i . Conditions (15) and (16) are
trivial, while condition (17) follows from (8) and the continuity of Ṽi (x). The continuous
diﬀerentiability of Vi (x) at x∗i follows from Theorem 4.4.9 in Karatzas and Shreve (1998)).6
A general solution to ODE (9) with (15) is expressed as A1,i xβ , where A1,i is an
unknown coeﬃcient. Then, by (16), I have the expression (13). A general solution to
ODE (10) with (16) is expressed as
λai x
λIi
−
+ A2,i xγλ ,(r − µ)(r + λ − µ) r + λ
6This is not the smooth pasting (i.e., optimality) condition in Dixit and Pindyck (1994), but it is the same

condition as the piecewise C 2 property in the regime-switching models in Guo, Miao, and Morellec (2005) andHackbarth, Miao, and Morellec (2006). I do not have to impose the smooth pasting condition and verify the
optimality of the solution. Instead, I can directly calculate the expectation Vi (x) by condition (17) which stems
from the optimality of x∗i .

7where A2,i is an unknown coeﬃcient. Then, by (17), I have the expression (14). By thecontinuous diﬀerentiability of Vi (x) at x∗i , I haveβA1 x∗β−1
=
i

λai
λ −1
+ γλ A2 x∗γ.
i(r − µ)(r + λ − µ)(18)

By substituting (13) and (14) into (18), I have()
ai x∗irIi
λai x∗i+ γλ−(r − µ)(r + λ − µ)r+λ−µ r+λPV
(r + λ − µ)((β − γλ )r + βλ)xN
i
x∗i =
.
(r + λ)((β − γλ )(r − µ) + (β − 1)λ)βπi (x∗i ) =The proof is complete. □The firm’s optimal policy is to invest in a project i only if it arrives at time t satisfying
X(t) ≥ x∗i . Unlike in the standard real options model, the investment time is later than
the first hitting time to the investment threshold x∗i . Indeed, the firm cannot invest at
the first hitting time when a project is not feasible.7 Lower liquidity λ increases the gap
between the first hitting time and the investment time.
The upper equation in (11) stands for the value of waiting when the option value ishigher than the investment payoﬀ, i.e, Vi (x) > πi (x). The lower equation in (11) stands
for the value of waiting when the option value is lower than the investment payoﬀ, i.e,
Vi (x) ≤ πi (x). Then, the firm wishes to immediately exercise the investment option, but
it cannot do so because of illiquidity of exercise opportunities. To be more precise, thelower equation in (11) can be decomposed as follows. The first and second terms of thelower equation stand for the expected payoﬀ of investing as soon as a project i arrives.
The last term A2,i xγλ stands for the value of the option to forgo a project i when it arrives
at time t satisfying X(t) < x∗i .
Notably, I have the closed-form solution in Proposition 1. Indeed, the investment
threshold x∗i is derived in the closed-form expression (12). In a more generalized model(i.e., X(t) following a geometric Lévy process with one-sided jumps), Perez and Yamazaki(2018) show an analytical solution, but they do not derive any closed-form solution. Similarly, Hackbarth, Miao, and Morellec (2006) show an analytical solution in the generalregime-switching model, but they do not derive any closed-form solution.8 Sections 4.1–4.3
also present closed-form solutions in several extended models. By virtue of tractability,
the closed-form solutions obtained in this paper will be a framework to study real options
with illiquidity of option exercise opportunities.
7
8

Mathematically, the probability of the Poisson arrival at the first hitting time is equal to zero.
In Section 4.2, I derive the closed-form solution of the regime-switching model where investment is feasible

in one of the two regimes and reveal the relationship of the illiquid model with the regime-switching model.

8Next, I examine the eﬀects of liquidity λ on the investment threshold and firm values.
I will show the following results in the limiting cases.
Proposition 2 Vi (x), Ṽi (x), and x∗i monotonically increase in λ.
PVlim Vi (x) = 0, lim Ṽi (x) = N P V i (x), lim x∗i = xN
,
i

λ→0

λ→0

λ→0

.
lim Vi (x) = lim Ṽi (x) = ROVi (x), lim x∗i = xROV
i

λ→∞

λ→∞

λ→∞

Proof. By definition of the problem, Vi (x) and Ṽi (x) monotonically increase in the arrival
rate λ. By the monotonicity of Ṽi (x) and x∗i = min{x ∈ R+ | Ṽi (x) = πi (x)}, I have themonotonicity of x∗i .
By λ → 0 in (12), I havePV
lim x∗i = xN
.
i

λ→0

(19)

By substituting (19) into (13) and (14), I have limλ→0 A1,i = limλ→0 A2,i = 0. Then,limλ→0 Vi (x) = 0 holds. By (8), I also have limλ→0 Ṽi (x) = N P Vi (x).
Now I consider the limiting case of λ → ∞. Note that limλ→∞ γλ /λ = 0. By thisproperty and (12), I have
lim x∗i = xROV
.
i

λ→∞

(20)

By substituting (20) into (13), (14), and (11), I have limλ→∞ Vi (x) = ROVi (x). By (8), I
also have limλ→∞ Ṽi (x) = ROVi (x). The proof is completed. □Lower liquidity of investment opportunities decreases the value of waiting and theinvestment threshold x∗i below xROV
. In other words, the firm accepts an investmentiopportunity more eagerly than in the ROV model because the firm has to wait for anotherproject arrival if it forgoes an investment opportunity. This is similar to Boyle and Guthrie(2003)’s result that cash shortfall risk decreases the value of waiting and the hurdle rate
for investment, although they do not derive any analytical solution due to the model
complexity.
Proposition 2 shows that the NPV and ROV solutions are obtained in the limiting casesof λ → 0 and λ → ∞, respectively.9 Indeed, as λ decreases (increases), x∗i , Vi (x), and Ṽi (x)approach the NPV (ROV) solutions. The baseline model results fill the gaps between thetwo extreme cases (i.e., the NPV and ROV models) by showing the intermediate solution in
the closed form. Indeed, it is more general and plausible to assume that the firm can delay
9

Although Perez and Yamazaki (2018) numerically check the limiting cases (for a geometric Lévy processwith one-sided jumps X(t)), they do not analytically prove the limiting cases.

9the investment time but cannot choose an arbitrary investment time. A Bermudan option,
where the holder can exercise the option only on certain predefined dates, also captures anintermediate case between the NPV and ROV models. In this case, however, the solution
cannot be analytically derived but can be numerically computed in the corresponding
discrete-time model. In contrast, by virtue of randomizing exercise dates, I can derive theclosed-form solution in the baseline model.
Lastly, I check the impacts of illiquidity on the investment thresholds and firm values
in numerical examples. The baseline parameter values are set in Table 1. We now consider
the case of i = G. Figure 1 shows the firm value function VG (x) along with the benchmark
values ROVG (x) and N P VG (x) for x close to the investment threshold x∗G = 0.8088. Notethat N P VG (x) = πG (x) holds in the depicted region. As shown in Proposition 1, VG (x)crosses πG (x) at the investment threshold x∗G = 0.8088. As shown in Proposition 1, theP V = 0.2 and xROV = 0.9531.
investment threshold x∗G = 0.8088 lies between xN
GG

Figure 2 shows VG (x), ROVG (x), x∗G , and xROV
along with the liquidity cost LC =G(ROVG (x) − VG (x))/ROVG (x) and the threshold decrease T D = (xROV− x∗G )/xROV
G
G
for varying levels of 1/λ. Note that 1/λ is equal to the expected time interval (years)
between two projects. The firm values are computed for the initial value x = 0.2, where
N P VG (x) = 0 holds. As shown in Proposition 2, both VG (x) and x∗G monotonicallydecrease in 1/λ. More interestingly, I can see from the figure that the impact of illiquidity
is greater on the investment threshold than on the firm value. For instance, I have
LC = 0.005 and T D = 0.1515 for λ = 1. Note that LC does not depends on the initialvalue x for x ≤ x∗G and that LC increases to (r − µ)/(r + λ − µ) for x → ∞. As expectedfrom the expression (r − µ)/(r + λ − µ), LC can be significantly high for plausible levelsof 1/λ when r − µ is quite large (say, negative µ10 ). By Proposition 2, I can easily showthat limλ→0 LC = 1, limλ→0 T D = 1/β, and limλ→∞ LC = limλ→∞ T D = 0.
Figure 3 shows VG (x), ROVG (x), x∗G , xROV
, LC, and T D for varying levels of σ. The
Gparameter values other than σ are set in Table 1. It is well known that ROVG (x) andxROV
monotonically increase in σ because a higher σ increases the value of deferringG
investment (e.g., Dixit and Pindyck (1994)). In the NPV model, volatility has no eﬀect
on the firm value and investment threshold. Taking into account that the baseline modelis intermediate between the ROV and NPV models, I can conjecture that the volatility
eﬀects in the baseline model are weaker than in the ROV model. Figure 3 verifies thisconjecture. Indeed, both VG (x) and x∗G monotonically increase in σ, but the increases
10

If I set µ = −0.05 in Table 1, I have LC = 0.1428 and T D = 0.1079.

10

are moderate compared to ROVG (x) and xROV
(see the top panels). As a result, theG
impacts of illiquidity, i.e., LC and T D, monotonically increase in σ (see the bottom
P V (r + λ − µ)/(r + λ) for µ > 0
panels). By (12), I can easily show that limσ→0 x∗i = xNi
P V for µ ≤ 0. I also have lim∗
N P V (r + λ − µ)/(r − µ),and limσ→0 x∗i = xN
σ→∞ xi = xiiwhich is sharply contrasted with the unbounded threshold limσ→∞ xROV
= ∞ in thei
ROV model. Then, I can show that limσ→0 LC = 1 − ((r + λ)/(r + λ − µ))r/µ λ/(r +λ), limσ→0 T D = µ/(r + λ) for µ > 0 and limσ→0 T D = 0 for µ ≤ 0. I can also show that
limσ→∞ LC = (r − µ)/(r + λ − µ), limσ→∞ T D = 1.

3.4

Two types of projects arriving randomly

Throughout this paper, I investigate illiquidity of investment opportunities. Although I
focus on exclusive investment projects, projects might be diﬀerent from each others. For
instance of a merger and acquisition option with illiquidity of searching and matching,
the acquiring firm might meet diﬀerent targets, where the synergies in the merger andacquisition vary over targets. It is important to see how the baseline results in Section 3.3
change with multiple project types. To do so, this section extends the baseline model to acase in which two types of projects arrive randomly.11 To be more precise, I assume thatprojects arrive following a Poisson process with the arrival rate λ and that project typei equals G at probability q and equals B at probability 1 − q by an independent draw.
The firm can choose between acceptance and rejection after it observes project type i.
I denote by W (x) the firm value when no project is feasible and by W̃i (x) (i = G, B)
the firm values when a project i arrives. I have W̃i (x) = max{πi (x), W (x)} (i = G, B)
because the firm value W̃i (x) becomes W (x) when a feasible project i is forgone.
As in Section 3.3, I can show that W̃i (x) (i = G, B) are continuous and the optimal
policy is a threshold policy. For the proof, refer to Appendix B. To be more precise, thestopping regions Si∗∗ = {x ∈ R+ | W̃i (x) = πi (x)} (i = G, B) are expressed asSi∗∗ = {x ∈ R+ | x ≥ x∗∗i }(i = G, B)(21)

Because of N P VG (x) ≤ W˜G (x) ≤ ROVG (x), the good-type investment threshold x∗∗
G
P V and xROV . Because of W
˜B (x) ̸≤ ROVB (x), the bad-type thresholdexists between xNGG∗∗ N P V )) can be infinite, i.e., S ∗∗ can be empty. I will later clarify thex∗∗
B (≥ max(xG , xB
B
11

I can similarly derive the solution in the illiquid model with a finite number of project types. Section 4.4

examines a continuum of project types. In Baldwin and Meyer (1979) and Baldwin (1982), project types has acontinuous distribution, although the models do not include the state process, i.e., X(t).

11∗∗ is empty. The value functions W̃ (x) (i = G, B) are expressed
condition under which SBias W (x) (x < x∗∗ ),i
W̃i (x) = π (x) (x ≥ x∗∗ )i(i = G, B).

(22)

i

∗∗ is empty, I can derive W (x) in the same manner as in Proposition 1. Suppose
If SB

that x∗∗
B is finite. As in Section 3.3, for the threshold policy, the expectation W (x) is apiecewise C 2 function and satisfies ODEs:µxW ′ (x) + 0.5σ 2 x2 W ′′ (x) = rW (x) (0 < x < x∗∗G ),∗∗µxW ′ (x) + 0.5σ 2 x2 W ′′ (x) + λq(πG (x) − W (x)) = rW (x) (x∗∗G < x < xB ),µxW ′ (x) + 0.5σ 2 x2 W ′′ (x) + λ(π̄(x) − W (x)) = rW (x) (x > x∗∗
B ),

(23)
(24)
(25)

where I define π̄(x) = qπG (x) + (1 − q)πB (x). In (24), λq(πG (x) − W (x)) corresponds tothe fact that W (x) changes to πG (x) with probability λqdt in infinitesimal time interval
dt, and in (25), λ(π̄(x) − W (x)) corresponds to the fact that W (x) changes to π̄(x) withprobability λdt in dt. As in Vi (x) in Section 3.3, W (x) is continuously diﬀerentiable atthresholds x∗∗i (i = G, B) because of Theorem 4.4.9 in Karatzas and Shreve (1998). By
solving (23)–(25) with the boundary conditions which will be specified in the proof, I
have the following proposition. In the proposition, I define ā = qaG + (1 − q)aB andI¯ = qIG + (1 − q)IB , and I denote (8), (11), and (12) by V˜G (x; λ), VG (x; λ), and x∗G (λ),
respectively, to indicate argument λ.
Proposition 3 Suppose that
r−µ
aG
.
+1≤
λq
aB

(26)

The firm has no possibility of investing in a project B. The firm value W (x) is given by
W (x) = VG (x; λq).
The investment thresholds x∗∗i (i = G, B) are given by∗∗∗x∗∗
G = xG (λq), xB = ∞.

Suppose that (26) does not hold. Assume that there exists a unique solution (xG , xB )P V ≤ x < x andsatisfying xN
GBG

λqaG xGβγ+ βλq B2 xGλq + γλq B3 xGλq = βπG (xG ),(r − µ)(r + λq − µ)λāxBλqaG xBβγ+ βλq B2 xBλq + γλq B3 xBλq =+ γ λ C3 ,(r − µ)(r + λq − µ)(r − µ)(r + λ − µ)

12

(27)
(28)

where Bi (i = 1, 2, . . . , 4) and Ci (i = 1, 2, 3) are defined by
B1 =πG (xG )
xβG

,

γ

B2 =

γ

xGλq C2 − xBλq C1
β

γ

β

β

B3 =

γ

xBλq xGλq − xGλq xBλq,

β

xBλq C1 − xGλq C2

,
β
γ
β
γ
xBλq xGλq − xGλq xBλq
C3B4 = γλ ,
xB
aG xG
rIG
C1 =
−
,
r + λq − µ r + λq
λqaG xB
λqIG
C2 = πB (xB ) −+,(r − µ)(r + λq − µ) r + λq
λāxB
λI¯
C3 = πB (xB ) −+.
(r − µ)(r + λ − µ) r + λ
The investment thresholds x∗∗i (i = G, B) are the above solution, i.e.,∗∗x∗∗
G = xG , x B = xB .

The firm value W (x) is given by



B1 xβ




λqIG
λqaG x−+ B2 xβλq + B3 xγλq
W (x) =
(r
−
µ)(r
+
λq
−
µ)
r
+
λq



λāx
λI¯


−
+ B4 xγλ(r − µ)(r + λ − µ) r + λ(x < x∗∗G ),∗∗(x ∈ [x∗∗G , xB )),(29)(x ≥ x∗∗B ).

Proof. Assume that (26) holds. In this case, I can show that VG (x; λq) > πB (x) (x ∈ R+ )as follows. For x < x∗G (λq), I have
VG (x; λq) = V˜G (x; λq) > πG (x) ≥ πB (x).

(30)

By A2,i > 0 in (11), for x ≥ x∗G (λq), I haveλqaG xλqIG−(r − µ)(r + λq − µ) r + λq
aB x
λqIG
≥
−
r − µ r + λq

VG (x; λq) >> πB (x),

(31)
(32)

where (31) follows from (26). By (30) and (32), I have VG (x; λq) > πB (x) (x ∈ R+ ).
Then, I haveW˜B (x) ≥ W (x) ≥ VG (x; λq) > πB (x) (x ∈ R+ ),13

∗∗ is empty. The firm has no possibility of investing in a project B,
which implies that SB

and hence, the problem is reduced to the problem when projects G arrive according to a
Poisson process with the arrival rate λq. By Proposition 1, I have W (x) = VG (x; λq) and∗x∗∗
G = xG (λq).

Next, assume that (26) does not hold. I can readily see that VG (x′ ; λq) < πB (x′ ) for a∗∗ is not empty. Indeed, if S ∗∗ is empty, I have
suﬃciently large x′ . This implies that SB
BW̃B (x′ ) = max{πB (x′ ), W (x′ )} = W (x′ ) = VG (x′ ; λq),
which contradicts the inequality VG (x′ ; λq) < πB (x′ ). Then, two finite thresholds x∗∗i (i =G, B) exist, and hence, I have ODEs (23)–(25). I have the boundary conditions
lim W (x) = 0

(33)

lim

W (x)
<∞
x→∞ πB (x)

(34)

∗∗
W (x∗∗i ) = πi (xi ) (i = G, B),

(35)

x→0

along with the continuous diﬀerentiability of the continuous diﬀerentiability of W (x) atx∗∗
i (i = G, B). Conditions (33) and (34) are trivial, and condition (35) follows from(22) and the continuity of W̃i (x) (i = G, B). The continuous diﬀerentiability of W (x) atx∗∗
i (i = G, B) follows from Theorem 4.4.9 in Karatzas and Shreve (1998). ODE (23) withboundary conditions (33) and (35), ODE (24) with boundary condition (35), and ODE(25) with boundary conditions (34) and (35) lead to the first, second, and third rows in(29), respectively. Because of the continuous diﬀerentiability of W (x) at x∗∗i (i = G, B),
the investment thresholds x∗∗i (i = G, B) satisfy (27) and (28). The proof is complete. □The solution varies according to whether (26) holds or not. In the former case, thefirm always ignores a project B, and hence, the solution is the same as in Proposition 1
with i = G and the arrival rate λq. I can see from (26) that higher aG /aB , λ, q, and lowerr − µ are likely to lead this case.
When (26) does not hold, the firm takes the following policy. The firm forgoes any
project when it arrives at time t satisfying X(t) < x∗∗
G . The firm undertakes a project
G if it arrives at time t satisfying X(t) ≥ x∗∗
G . Notably, the firm accepts only a project∗∗
G for X(t) ∈ [x∗∗G , xB ). The firm undertakes even a project B if it arrives at time tsatisfying X(t) ≥ x∗∗
B . As explained after Proposition 1, the investment time is later
than the first hitting time to the threshold. The paths of X(t) and the Poisson process
determine whether the firm invests in a project G or B. This unpredictability is a key
diﬀerence from the ROV model in which the firm invests in a project G at the first hitting

14

time. Due to illiquidity of investment opportunities, unlike in the ROV model, the firm
can potentially invest in a project B.
∗∗
Although I cannot derive closed-form expressions of x∗∗
G and xB , I have the analyticalexpression of W (x) in Proposition 3. In (29), the first row stands for the value of waiting
when the option value is higher than the good-type investment payoﬀ, i.e, W (x) > πG (x).
The second row stands for the value of waiting when the option value is between thebad- and good-type investment payoﬀs, i.e, πB (x) < W (x) ≤ πG (x). Then, the firm
wishes to immediately invest in a project G, but it cannot do so because of illiquidity ofinvestment opportunities. I can decompose the second row in (29) as follows. The firstand second terms represent the expected payoﬀ of investing whenever a project G arrives.
B2 xβλq + B3 xγλq represents the value of the option to revise this investment policy when∗∗
X(t) passes either x∗∗
G or xB before the arrival of a project G. The last row in (29) stands

for the value of waiting when the option value is lower than the bad-type investment
payoﬀ, i.e, W (x) ≤ πB (x). The first and second terms of this equation represent the
expected payoﬀ of investing whenever a project arrives regardless of its type. The lastterm B4 xγλ represents the value of the option to revise this investment policy when X(t)passes x∗∗
B before the arrival of a project.
Next, I examine the eﬀects of liquidity λ on the investment thresholds and firm values.
I will show the following results in the limiting cases under the assumptions in Proposition
3.
Proposition 4 W (x), W̃i (x), and x∗∗
i monotonically increase in λ.
NP V
lim W (x) = 0, lim W̃i (x) = N P Vi (x), lim x∗∗(i = G, B).
i = xi

λ→0

λ→0

λ→0

When λ → ∞, (26) is satisfied, andROV
lim W (x) = lim W˜G (x) = lim W˜B (x) = ROVG (x), lim x∗∗, lim x∗∗
G = xG
B = ∞.

λ→∞

λ→∞

λ→∞

λ→∞

λ→∞

Proof. By definition of the problem, W (x) and W̃i (x) monotonically increase in the∗∗arrival rate λ. Then, the monotonicity of x∗∗i follows from xi = min{x ∈ R+ | W̃i (x) =πi (x)}. First, consider the limiting case of λ → 0. (26) is not satisfied as λ → 0. In thelimiting case, I have∗∗γ∗∗
B2 x∗∗β
G + B3 xG = πG (xG )

(36)

∗∗
by W (x∗∗G ) = πG (xG ) (see (29)) and∗∗γ∗∗
βB2 x∗∗β
G + γB3 xG = βπG (xG )

15

(37)

∗∗
N P V > 0, I haveby (27). By (36) and (37), I have (β − γ)B3 x∗∗γ
G = 0. Because of xG ≥ xG

B3 = 0. Similarly, I have∗∗γ∗∗
B2 x∗∗β
B + B3 xB = πB (xB )

(38)

∗∗
by W (x∗∗B ) = πB (xB ) (see (29)) and∗∗γ∗∗
βB2 x∗∗β
B + γB3 xB = γπB (xB )

(39)

∗∗
∗∗
by (28). By (38) and (39), I have (β − γ)B2 x∗∗β
B = 0. Because of xB > xG > 0, I have∗∗
B2 = 0. By B2 = B3 = 0, (36), and (38), I have πG (x∗∗G ) = πB (xB ) = 0. Then, I have
N P V (i = G, B) and B = B = 0.
x∗∗
14i = xi

Next, consider the limiting case of λ → ∞. Because of aG /aB > 1 and limλ→∞ (r −µ)/λq = 0, (26) is satisfied in the limiting case. Then, by Proposition 3, I have x∗∗
G =x∗G (x; λq), W (x) = W˜B (x) = VG (x; λq), and W˜G (x) = V˜G (x; λq). The limiting results
follow from Proposition 2. The proof is complete. □Proposition 4, as well as Proposition 2, shows that the illiquidity solution fills the gaps
between the NPV and ROV solutions. Indeed, as λ decreases (increases), x∗∗i , W (x), andW̃i (x) approach the NPV (ROV) solutions. When λ is suﬃciently high, (26) holds, andhence the firm always forgoes a project B and invests only in a project G. This result is
in line with that of the ROV model, in which the firm always invests in a project G whenboth projects G and B are feasible. In contrast, for a lower λ, the firm can invest in a
project B if it is feasible.
Lastly, I examine the impacts of illiquidity on the investment thresholds and firm∗∗
ROV alongvalues in numerical examples. Figure 4 shows W (x), ROVG (x), x∗∗
G , xB , and xG

with the liquidity cost LC = (ROVG (x) − W (x))/ROVG (x) and the threshold decrease
ROV for varying levels of 1/λ. The parameter values other than λT D = (xROV− x∗∗
GG )/xG

are set in Table 1. For 1/λ ≤ 0.5, (26) is satisfied, and hence x∗∗
B = ∞ holds. As shown∗∗in Proposition 4, W (x), x∗∗
G , and xB monotonically decrease in 1/λ. As in Figure 2, I

find that the impact of illiquidity is greater on the investment thresholds than on the firm
value. Indeed, I have LC = 0.0112, T D = 0.2142, and x∗∗
B = 1.6409 for λ = 1. Notably,∗∗
I can see from Figure 4 that x∗∗
B decreases much more sharply than xG for 1/λ ≈ 0.5∗∗
because x∗∗
B , unlike xG , becomes infinity for 1/λ ≤ 0.5. In the absence of illiquidity ofinvestment opportunities, the firm can always choose the best investment project, butwith illiquidity, the firm might give up waiting for the best project and invest in theinferior project. Thus, the investment threshold for the inferior project depends greatly
on the degree of illiquidity. I will verify this result in a case of continuous project types
in Section 4.4.

16

LC and T D in Figure 4 are higher than those in Figure 2 because a project is B
at probability 0.5 in this example. Note that LC does not depends on the initial value
x for x ≤ x∗∗
G and that LC increases to (r − µ)/(r + λq − µ) for 1/λ ≤ 0.5 and (r +λ − µ − λā/aG )/(r + λ − µ) for 1/λ > 0.5 as x → ∞. LC can be significantly high forplausible levels of 1/λ when r − µ is quite large.12 By Proposition 4, I can also show that
limλ→0 LC = 1, limλ→0 T D = 1/β, and limλ→∞ LC = limλ→∞ T D = 0.
∗∗
ROV for varying levels of σ. The paFigure 5 shows W (x), ROVG (x), x∗∗
G , xB , and xG

rameter values other than σ are set in Table 1. Note that (26) does not hold, and hence,∗∗
I have finite x∗∗
B . The impacts of σ on W (x) and xG are weaker than those in the ROV

model because the illiquid model is intermediate between the ROV and NPV models.
Then, LC and T D in Figure 5 monotonically increase in σ. These findings are similar
to those of the baseline model (cf. Figure 3). In conclusion, I argue that the eﬀects
of illiquidity of investment opportunities increase with a higher volatility of the project
payoﬀ.
Although I cannot derive any closed-form expression of limσ→0 x∗∗
G for µ > 0, it follows
N P V for µ ≤ 0 and lim∗∗
N P V (r + λ − µ)/(r + λ −from (29) that limσ→0 x∗∗
σ→∞ xi = xi
G = xGµ − λā/ai ) (i = G, B). Then, I can show that limσ→0 T D = 0 for µ < 0, limσ→∞ LC =(r + λ − µ − λā/aG )/(r + λ − µ), and limσ→∞ T D = 1.

4

Extensions

4.1

Search cost

In this subsection, I show that the same analysis as in Sections 3.3 and 3.4 can be applied
to the illiquid model with cost flows of searching for investment opportunities. In mergers
and acquisitions, the acquiring firm might continue costly searches and negotiations until
it succeeds. A financially constrained firm might continue to apply for government grants
until it receives suﬃcient funds. To capture these situations, in addition to the baselinemodel, I assume that the firm incurs cost flows, denoted by c, when searching for a project.
The firm can freely suspend the search and switch between search and suspension. I use
subscript c to stand for the case with search cost c, instead of omitting project type i.
As in Section 3.3, I can show that a threshold policy is optimal, and I have two ODEs.
ODE (9) does not change because the firm does not search a project for X(t) in the12

If I set µ = −0.05 in Table 1, I have LC = 0.2143 and T D = 0.124.

17

rejection region. ODE (10) changes toµxVc′ (x) + 0.5σ 2 x2 Vc′′ (x) + λ(π(x) − Vc (x)) − c = rVc (x)(x > x∗c )(40)

because the firm has to pay cost flows c in the acceptance region. As in the proof of
Proposition 1, by solving ODEs (9) and (40) with the continuous diﬀerentiability condition at x∗c , as well as the boundary conditions limx→0 Vc (x) = 0, limx→∞ Vc (x)/π(x) <∞, Vc (x∗c ) = π(x∗c ), I can show the following proposition.
Proposition 5 A1c xβ

Vc (x) =(x < x∗c ),λI + cλax−+ A2c xγλ(r − µ)(r + λ − µ)r+λ(x ≥ x∗c ),(41)

where the investment threshold x∗c and coeﬃcients A1c , A2c are(r − µ)(r + λ − µ)(((β − γλ )r + βλ)I − γλ c),a(r + λ)((β − γλ )(r − µ) + (β − 1)λ)π(x∗ )
A1c = ∗βc ,xc)(
1rI + cax∗cA2c = ∗γλ
−
.
r+λ−µ
r+λ
xc
x∗c =

(42)

I omit the proof of Proposition 5 because it can be proved in the same fashion as in the
proof of Proposition 1. As in Proposition 1, the firm invests in a project only when it
arrives at time t satisfying X(t) ≥ x∗c . Note that −γλ c > 0 of the denominator in (42).
Because of cost flows c, the investment threshold x∗c is higher than that of the baselinemodel. The upper equation in (41) stands for the value of waiting when the option value
is higher than the investment payoﬀ, i.e, Vc (x) > π(x). The lower equation in (41) stands
for the value of waiting when the option value is lower than the investment payoﬀ, i.e,Vc (x) ≤ π(x). The first and second terms of the lower equation stand for the expected
payoﬀ of continuing search and investing whenever a project is found. The last term
A2c xγλ stands for the value of the option to suspend search when X(t) passes x∗c before
the arrival of a project. I can show similar results to Propositions 2–4, although I do not
redundantly present them. To summarize, the main results in Sections 3.3 and 3.4 remainunchanged with the presence of search cost.

4.2

The relationship with the regime-switching modelThis subsection reveals the relationship between the baseline model in Section 3.3 and theregime-switching model of Hackbarth, Miao, and Morellec (2006). I consider the following

18

regime-switching model with two regimes. In one of the regimes (say, boom), the firm
can invest in a project at an arbitrary time, whereas in the other regime (say, bust), thefirm has no chance to invest in a project. For notational simplicity, I skip the subscript
i for the project type. The project payoﬀ function π(x) is the same as in Section 3. A
bust regime switches to a boom regime following a Poisson process with the arrival rateλ(> 0), whereas a boom regime switches to a bust regime following a Poisson processwith the arrival rate η(> 0). I assume that the Poisson processes are independent of
the state process X(t). I denote by Vr (x) and V˜r (x) the firm values in bust and boom
regimes, respectively. The subscript r stands for the regime-switching model. The regimeswitching model can be regarded as a simplified version of Hackbarth, Miao, and Morellec(2006)’s model in which cash flows from investment depend on the regimes.
As in Section 3.3 (see Hackbarth, Miao, and Morellec (2006) and Bensoussan, Yan,and Yin (2012) for more general cases), the following ODEs hold:µxVr′ (x) + 0.5σ 2 x2 Vr′′ (x) + λ(V˜r (x) − Vr (x)) = rVr (x),(x < x∗r ),(43)µxVr′ (x) + 0.5σ 2 x2 Vr′′ (x) + λ(π(x) − Vr (x)) = rVr (x),(x > x∗r ),

(44)

′
′′
µxV˜r (x) + 0.5σ 2 x2 V˜r (x) + η(Vr (x) − V˜r (x)) = rV˜r (x) (x < x∗r ),(45)

where x∗r is the investment threshold in a boom regime. The terms λ(V˜r (x) − Vr (x)) in(43) and λ(π(x) − Vr (x)) in (44) mean that Vr (x) changes to V˜r (x) and π(x) when a bust
regime switches to a boom regime. The term η(Vr (x) − V˜r (x)) in (45) means that V˜r (x)changes to Vr (x) when a boom regime switches to a bust regime. The boundary conditions
are limx→0 Vr (x) = 0, limx→∞ Vr (x)/π(x) < ∞, limx→0 V˜r (x) = 0, V˜r (x) = π(x) (x ≥ x∗r ),and the the continuous diﬀerentiability of Vr (x) and V˜r (x) at x = x∗r .13
As in the proof of Proposition 1, by solving ODEs (43)–(45) with the conditions above,
I can show the following proposition.

13

The continuous diﬀerentiability of V˜r (x) at x = x∗r corresponds to the smooth pasting condition in Dixit

and Pindyck (1994), although the continuous diﬀerentiability of Vr (x) at x = x∗r follows from Theorem 4.4.9
in Karatzas and Shreve (1998). See also Hackbarth, Miao, and Morellec (2006) and Bensoussan, Yan, and Yin(2012).

19

Proposition 6λ A1r xβ − A2r xβλ+ηη
Vr (x) =
λax
λI


−
+ A3r xγλ(r − µ)(r + λ − µ) r + λ A1r xβ + A2r xβλ+η (x < x∗ ),rV˜r (x) = π(x)(x ≥ x∗ ),(x < x∗r ),(x ≥ x∗r ),

(46)

(47)

r

where the investment threshold x∗r and coeﬃcients A1r , A2r , A3r are(r + λ − µ)(β(r + λ)(βλ+η η + λ(βλ+η − γλ )) − γλ η(βλ+η r + βλ))xN P V,(r + λ)(λ(β − 1)(βλ+η − γλ )(r + λ + η − µ) + η(β − γλ )(βλ+η − 1)(r − µ))()(βλ+η − 1)ax∗r
1− βλ+η I ,
A1r =
r−µ
(βλ+η − β)x∗βr)(
1(β − 1)ax∗r
A2r =
βI −
,
∗β
r−µ
(βλ+η − β)xr λ+η
)
(
1
λax∗r
λI
λ
∗βλ+η
∗β
A3r = ∗γλ A1r xr − A2r xr−+.
η(r − µ)(r + λ − µ) r + λ
xr
x∗r =

I omit the proof of Proposition 6 because the calculation is lengthy but can be done in
the same manner as in the proof of Proposition 1. Notably, Proposition 6 shows theclosed-form solution, although Hackbarth, Miao, and Morellec (2006) do not derive any
closed-form solution. In a boom regime, the firm invests in the project at time t satisfying
X(t) ≥ x∗r . The investment time is either the first hitting time to x∗r or a boom arrival
time t satisfying X(t) ≥ x∗r .
The firm value in a bust regime, Vr (x) in (46), is similar to Vi (x) in (11) because thefirm cannot invest in the project in a bust regime. The upper equation in (46) stands
for the value of waiting when the option value is higher than the investment payoﬀ, i.e,Vr (x) > π(x). Because of the transition probability from a bust to a boom, unlike in (11),it has an extra term A2r xβλ+η λ/η. The lower equation in (46) stands for the value of
waiting when the option value is lower than the investment payoﬀ, i.e, Vr (x) ≤ π(x). The
first and second terms of the lower equation stand for the expected payoﬀ of investing
whenever a boom regime arrives. The last term A3,r xγλ stands for the value of the option
to change the policy when X(t) passes x∗r .
On the other hand, the firm value in a boom regime, V˜r (x) in (47) is similar to ROVi (x)in (5) because the firm can invest in the project at an arbitrary time in a boom regime.
The upper equation in (47) stands for the value of waiting when the option value is higher
than the investment payoﬀ, i.e, Vr (x) > π(x). Because of the transition probability from20

a boom to a bust, unlike in (5), it has an extra term A2r xβλ+η . The lower equation in(47) represents the investment payoﬀ.
By virtue of the closed-form expression of x∗r , I can directly show that limη→0 x∗r =xROV and limη→∞ x∗r = x∗ . Then, I can easily show the following proposition.
Proposition 7 Vr (x) and V˜r (x), and x∗r monotonically decrease in η.
lim V˜r (x) = ROV (x), lim x∗r = xROV

η→0

η→0

lim Vr (x) = V (x), lim V˜r (x) = Ṽ (x), lim x∗r = x∗ .

η→∞

η→∞

η→∞

Proposition 7 shows that the regime-switching model is intermediate between the ROV
model and the baseline model in Section 3.3. Indeed, the solution of the regime-switchingmodel converges to that of the ROV model for lower transition probability from a boom
to a bust, η, and converges to that of the baseline model for a higher η. Intuitively, witha higher η, a boom ends sooner, and hence, the value of deferring investment within aboom regime decreases. In the limiting case of η → ∞, the firm has to decide whether
or not to invest as soon as a boom regime arrives, which leads to the same solution as
the baseline solution in Section 3.3. Thus, the regime-switching model is regarded as anextended model of the illiquid model in Section 3.3.14

4.3

Put-type option

Although this paper has examined call-type real options so far, I can also derive the solutions for put-type real options, such as asset sales, liquidation, and bankruptcy options, inthe same fashion as in Sections 3.3 and 3.4. This subsection examines the stylized model
below. Suppose that a firm has an illiquid and indivisible asset that generates continuous streams of cash flows aX(t), where a is a positive constant. The firm occasionallymeets acquirers who want to buy the asset at a constant price I(> 0) and optimizes
the asset sales policy.15 Assume that the acquirers’ arrival process is a Poisson processwith the arrival rate λ. If the firm rejects an acquirer’s oﬀer, the firm has to wait until
another acquirer arrives. The Poisson arrival process stands for illiquidity of asset sales
opportunities.
14

Similarly, the regime-switching model with three regimes can be regarded as an extension of the illiquid

model with two project types in Section 3.4.
15
I can similarly derive the closed-form solution when the sales price is expressed as a linear function of X(t).

21As in Section 3.3, I can show that a threshold policy is optimal and that the firm’s
asset value Vp (x) satisfiesµxVp′ (x) + 0.5σ 2 x2 Vp′′ (x) + λ(I − Vp (x)) + ax = rV (x) (x < x∗p ),(48)µxVp′ (x) + 0.5σ 2 x2 Vp′′ (x) + ax = rV (x) (x > x∗p ),(49)

where x∗p denotes the asset sales threshold below which the firm wishes to sell the asset.
The subscript p stands for the put-type option. Note that ax in (48) and (49) corresponds
to cash flows from the asset, while λ(I − Vp (x)) corresponds to the fact that Vp (x) changes
to I with probability λdt in infinitesimal time interval dt.
As in the proof of Proposition 1, by solving ODEs (48) and (49) with the continuous diﬀerentiability condition at x∗p along with the boundary conditions limx→0 Vp (x) <∞, limx→∞ Vp (x)/ax < ∞, Vp (x∗ ) = I, I can show the following proposition.
Proposition 8




λI
ax
+
+ A2p xβλ
r
+
λ
−
µ
r
+
λ
Vp (x) =ax+ A1p xγ
r−µ(x ≤ x∗p ),(x > x∗p ),(50)

where the asset sales threshold x∗ and coeﬃcients A1p , A2p are(r − µ)(r + λ − µ)((γ − βλ )r + γλ)I,a(r + λ)((γ − βλ )(r − µ) + (γ − 1)λ)
)
(
ax∗p
1
A1p = ∗γ I −
,
r−µ
xp
)
(
ax∗p
rI
1
−
.
A2p = ∗βxp λ r + λ r + λ − µx∗p =The proof of Proposition 8 can be done in the same manner as in the proof of Proposition 1.
The firm liquidates the asset only when an acquirer arrives at time t satisfying X(t) ≤ x∗p .
I can easily show that the asset sales threshold is higher than that of the liquid model.
In other words, lower asset liquidity decreases the option value of waiting, and hence, thefirm more eagerly accepts an acquirer’s oﬀer.
The lower equation represents the value of waiting when the firm’s asset value is higher
than the sales proceeds, i.e, Vp (x) > I. More precisely, the first term represents the asset
value without the asset sales opportunity, and the second term represents the value of theasset sales option. The upper equation in (50) stands for the value of waiting when theasset value is lower than the sales proceeds, i.e, Vp (x) ≤ I. The first and second terms of
the upper equation stand for the expected payoﬀ when the firm sells the asset whenever22

an acquirer arrives, whereas the last term A2p xβλ stands for the value of the option tosuspend the asset sales at time t satisfying X(t) > x∗p . I can show results analogous to
Proposition 2-7, although I do not redundantly present them. Thus, for put-type real
options, the illiquid model also fills the gaps in the NPV, ROV, and regime-switchingmodels.

4.4

A continuum of project types

This subsection extends the illiquid model with two project types in Section 3.4 to acase with a continuum of project types. Suppose that projects with payoﬀ πk (x) =ak x/(r − µ) − Ik (k ∈ [0, 1]) may arrive, where I define ak = kaG + (1 − k)aB and
Ik = kIG + (1 − k)IB . Then, πk (x) increases in k from the the worst type k = 0 (i.e.,
project B in Section 3.4) to the best type k = 1 (i.e., project G). For simplicity, I assume
that project type k follows a uniform distribution on [0, 1], independently of X(t).
Although I cannot analytically derive the solution, I can numerically compute thesolution to ODEµxU ′ (x) + 0.5σ 2 x2 U ′′ (x) + λ(1 − k ∗ (x))(π0.5+0.5k∗ (x) (x) − U (x)) = rU (x),(51)

where U (x) denotes the firm value when no project is feasible, and I define k ∗ (x) =min{k ∈ [0, 1] | U (x) ≤ πk (x)}. If U (x) > π1 (x) holds, I artificially define k ∗ (x) = 1
to remove the term λ(1 − k ∗ (x)). The type k ∗ (x) stands for the lowest acceptable typefor x. In other words, the firm undertakes only a project of type k ≥ k ∗ (X(t)) andforgoes a project of type k < k ∗ (X(t)) when it arrives at time t. From the opposite
perspective, the investment threshold for type k becomes the inverse function k ∗−1 (k). In(51), λ(1 − k ∗ (x))(π0.5+0.5k∗ (x) (x) − U (x)) corresponds to the fact that the firm value U (x)changes to π0.5+0.5k∗ (x) (x) with probability λ(1 − k ∗ (x))dt in infinitesimal time interval
dt. Note that π0.5+0.5k∗ (x) (x) and λ(1 − k ∗ (x))dt represent the mean payoﬀ and the arrival
probability of the acceptable types, respectively.
Figure 6 shows the lowest acceptable type k ∗ (x) for 1/λ = 0.6, 0.8, and 1. I can see
from the figure that lower liquidity decreases k ∗ (x). Lower liquidity decreases the valueof waiting for a superior project, and the firm accepts a project more eagerly. This result
is consistent with Proposition 3 (cf. (26)). In the figure, I also find that the diﬀerencesof the three lines are larger for lower k ∗ (x). This means that the impact of illiquidity on
the investment threshold for type k, i.e., k ∗−1 (k), decreases in k. This result shows therobustness of Section 3.4’s result that the investment threshold for an inferior project is23

more aﬀected by the degree of illiquidity. Because the results of LC and T D and thecomparative statics results of σ are similar to those of Sections 3.3 and 3.4, I omit thefigures.

5

Conclusion

This paper investigated the eﬀects of illiquidity of investment opportunities on investment
decisions. I derived the firm value and investment threshold in the closed forms wheninvestment opportunities follow a Poisson process. Lower liquidity decreases the option
value of waiting and the hurdle rate for investment. For cash flows with a positive growthrate, the impact of illiquidity on the investment threshold is greater than on the firm
value. A higher cash flow volatility amplifies the impacts of illiquidity on the firm value
and investment threshold.
Furthermore, I extended the results to a case with two types of projects arriving
randomly. Although I could not derive a closed-form solution, I derived the analyticalsolution. When the profitability gap between the two types, liquidity, and the probabilityof the good-type appearance are higher, the firm is more likely to accept only a goodtype project and forgo a bad-type project. When these factors are low, the firm accepts
only a good-type project for intermediate levels of the state variable and accepts even a
bad-type project for suﬃciently high levels of the state variable. The impact of illiquidity
on the bad-type investment threshold is much greater than on the good-type investment
threshold.
I also proved that the solution of the illiquid model converges to those of the NPV andROV models for lower and higher liquidity, respectively. I also showed that the solutionagrees with the limit of the corresponding regime-switching model. Thus, this paper fills
the gaps in the NPV, ROV, and regime-switching models in terms of illiquidity of option
exercise opportunities. The proposed models and solutions can be a new framework to
study real options with illiquidity of option exercise opportunities. Potential applications
include but are not limited to mergers and acquisitions with illiquidity of searching andmatching, illiquid asset sales, and corporate investment with illiquidity of fundraising.

24

A

The continuity of Ṽi(x) and the optimality of athreshold policyFirst, we will show
0 ≤ Ṽi (x + ∆) − Ṽi (x) ≤

∆ai
r−µ

(x, ∆ ∈ R+ )(52)

as follows. We denote the set of Ft -stopping times by T . We also denote by TλP thesubset of T restricted to the Poisson arrival times with the arrival rate λ. Assume that
the Poisson process arrives at time t = 0. For x, ∆ ∈ R+ , I haveṼi (x + ∆))(µ−0.5σ 2 )τ +σB(τ )
a
(x
+
∆)e
i
−I ]
= sup E[e−rτr−µτ ∈TλP()()
2
2
ai xe(µ−0.5σ )τ +σB(τ )ai ∆e(µ−0.5σ )τ +σB(τ )−rτ−rτ≤ sup E[e− I ] + sup E[e
]
r−µ
r−µ
τ ∈TλPτ ∈TλP)(()(µ−0.5σ 2 )τ +σB(τ )(µ−0.5σ 2 )τ +σB(τ )
a
∆e
a
xe
i
i
− I ] + sup E[e−rτ≤ sup E[e−rτ
]
r−µ
r−µ
τ ∈T
τ ∈T P(

λ=Ṽi (x) +

ai ∆
.
r−µ

Similarly, I can easily show that Ṽi (x+∆) ≥ Ṽi (x) by e−rτ ai ∆e(µ−0.5σ )τ +σB(τ ) /(r−µ) ≥ 0.
2

Then, I have (52).
Note that (52) shows the continuity of the value function Ṽi (x). By the continuityof Ṽi (x) and πi (x), Si∗ = {x ∈ R+ | Ṽi (x) = πi (x)} is a closed set. Because Ṽi (x) ≤ROVi (x) = πi (x) holds for x ≥ xROV
, Si∗ includes {x ∈ R+ | x ≥ xROV}. Because ofii
P V }. Therefore, by the closeness of S ∗ ,Ṽi (x) ≥ 0, Si∗ is included by {x ∈ R+ | x ≥ xNii
P V and xROV . I denote this minimum by x∗ . I can showmin{x ∈ Si∗ } exists between xNiiithat the optimal policy is the threshold policy, i.e., Si∗ = {x ∈ R+ | x ≥ x∗i } as follows.
Indeed, by (52), for x > x∗i , I have(x∗i − x)air−µ∗
(x − x)ai= πi (x∗i ) + i
r−µṼi (x) ≤ Ṽi (x∗i ) += πi (x),which implies x ∈ Si∗ . The proof is complete. □Although this paper presented the simple proof in the standard manner (e.g., see
Detemple (2006)), Perez and Yamazaki (2018) proved the properties in a more generalcase in a diﬀerent manner.

25

B

The continuity of W̃i(x) and the optimality ofa threshold policy
In the same fashion as in Appendix A, we can show∆aG
0 ≤ W˜G (x + ∆) − W˜G (x) ≤r−µ(x, ∆ ∈ R+ )(53)

and by (53), I can show the continuity of W̃i (x), the optimality of a threshold policy, andN P V , xROV ].
the existence of the optimal threshold x∗∗
G ∈ [xG
G

Next, suppose that (26) does not hold. Then, I have
aB −

aG λq
> 0.
r + λq − µ(54)

I will show
0 ≤ W˜B (x + ∆) − W˜B (x) ≤

∆aB
r−µ

(x, ∆ ∈ R+ ).

(55)

as follows. Below, I assume that the Poisson process arrives at time t = 0 with i(0) = B,
where i(τ ) denotes the feasible project type at the Poisson arrival time τ .
W˜B (x + ∆)()(µ−0.5σ 2 )τ +σB(τ )
a
(x
+
∆)e
i(τ
)
= sup E[e−rτ−I ]r−µτ ∈TλP(())
22
ai(τ ) ∆e(µ−0.5σ )τ +σB(τ )ai(τ ) xe(µ−0.5σ )τ +σB(τ )−rτ−rτ≤ sup E[e− I ] + sup E[e
]
r−µ
r−µ
τ ∈TλPτ ∈TλP()()(µ−0.5σ 2 )τ +σB(τ )(µ−0.5σ 2 )T +σB(T )
a
∆e
a
∆e
B
G
≤W˜B (x) + sup E[1{τ <T } e−rτ
+ 1{τ ≥T } e−rT
]
r−µ
r−µ
τ ∈T

=W˜B (x) + sup E[e−(r+λq)τ
τ ∈T

(
˜
=WB (x) + aB −=W˜B (x) +

(
aB −

aG λq
r + λq − µ

)

aG λq
r + λq − µ

)(

)

2

(56)

∆e(µ−0.5σ )τ +σB(τ )aG λq∆]+r−µ(r − µ)(r + λq − µ)
(57)

aB ∆
aG λq∆+
r − µ (r − µ)(r + λq − µ)

∆aB
,
r−µ

where in (56), T denotes the first arrival time of the Poisson process with the arrival
rate λq (i.e., project G). The optimal stopping problem in (56) is the problem in which2

the firm can gain either the bad-type payoﬀ aB ∆e(µ−0.5σ )τ +σB(τ ) /(r − µ) at the op2

timal time τ or the good-type payoﬀ aG ∆e(µ−0.5σ )T +σB(T ) /(r − µ) at the first arrival
time of project G. Note that the firm has no incentive to defer the good-type payoﬀ

26

2

aG ∆e(µ−0.5σ )T +σB(T ) /(r − µ). In (57), the second term represents the option value ofgaining the bad-type payoﬀ before the first arrival time of project G, whereas the lastterm represents the value of the good-type payoﬀ at the first arrival time. Because of(54), the optimal stopping time in (57) is equal to 0, which implies that the firm has no
incentive to defer the bad-type payoﬀ. I can easily show that W˜B (x + ∆) ≥ W˜B (x) bye−rτ ai(τ ) ∆e(µ−0.5σ )τ +σB(τ ) /(r − µ) ≥ 0. Hence, I have (55).
2

By (55), I can show the continuity of W˜B (x), the optimality of a threshold policy, and∗∗ N P V ) in the same fashion as in
the existence of the optimal threshold x∗∗
B ≥ max(xG , xB

Appendix A. The proof is complete. □References
Baldwin, C., 1982, Optimal sequential investment when capital is not readily reversible,
Journal of Finance 37, 763–782.
Baldwin, C., and R. Meyer, 1979, Liquidity preference under uncertainty: A model ofdynamic investment in illiquid opportunities, Journal of Financial Economics 7, 347–
374.
Bensoussan, A., Z. Yan, and G. Yin, 2012, Threshold-type policies for real options using
regime-switching models, SIAM Journal on Financial Mathematics 3, 667–689.
Boyle, G., and G. Guthrie, 2003, Investment, uncertainty, and liquidity, Journal of Finance 58, 2143–2166.
Décamps, J., T. Mariotti, and S. Villeneuve, 2006, Irreversible investment in alternative
projects, Economic Theory 28, 425–448.
Detemple, J., 2006, American-Style Derivatives valuation and computation (Chapman &Hall/CRC: London).
Diamond, P., 1982, Aggregate demand management in search equilibrium, Journal of
Political Economy 90, 881–894.
Dixit, A., and R. Pindyck, 1994, Investment Under Uncertainty (Princeton University
Press: Princeton).
Duﬃe, D., N. Garleanu, and L. Pedersen, 2005, Over-the-counter markets, Econometrica
73, 1815–1847.
Guo, X., J. Miao, and E. Morellec, 2005, Irreversible investment with regime shifts, Journal of Economic Theory 122, 37–59.

27

Hackbarth, D., J. Miao, and E. Morellec, 2006, Capital structure, credit risk, and macroeconomic conditions, Journal of Financial Economics 82, 519–550.
He, Z., and K. Milbradt, 2014, Endogenous liquidity and defaultable bonds, Econometrica
82, 1443–1508.
He, Z., and W. Xiong, 2012, Rollover risk and credit risk, Journal of Finance 67, 391–430.
Karatzas, I., and S. Shreve, 1998, Brownian Motion and Stochastic Calculus (Springer:
New York).
McDonald, R., and D. Siegel, 1986, The value of waiting to invest, Quarterly Journal of
Economics 101, 707–727.
Perez, J., and K. Yamazaki, 2018, American options under periodic exercise opportunities,
Statistics and Probability Letters 135, 92–101.

28

Table 1: Baseline parameter values.
r

µ

σ

λ

x

aG

IG

0.07

0.05

0.2

1

0.5

1

10

aB

IB

q

1/1.02 10.2 0.5

31.2
V G(x)ROVG(x)

31

NPV G(x)

30.8

30.6

30.4

30.2

30

29.8
0.8

0.802

0.804

0.806

0.808
0.81
0.812
xG =0.8088

0.814

0.816

0.818

0.82

Figure 1: Firm value functions. The figure plots functions VG (x), ROVG (x), and N P VG (x).
The parameter values other than x are set in Table 1.

29

1
5.22
0.95

5.215

x∗G

5.21
0.9

xROV
G

5.205
5.2

VG(x)

5.195

ROVG(x)

5.19

0

0.2

0.85

0.4

0.6

0.8

0.8

1

0

0.2

0.4

1/λ

0.6

0.8

1

0.6

0.8

1

1/λ

−3

5

x 10

0.2

TD

LC
4

0.15

3
0.1
2
0.05

1
0

0

0.2

0.4

0.6

0.8

0

1

1/λ

Figure 2:

0

0.2

0.4

1/λ

Comparative statics with respect to illiquidity 1/λ.

The figure plots

VG (x), ROVG (x), x∗G , xROV
, LC, and T D. The other parameter values are set in Table 1.
G

30

6

1.2

5.5

VG (x)xG

ROV G (x)

xROV
G

1

5

0.8

4.5
0.1

0.15

0.25

0.6
0.1

0.3

0.15

0.2

0.25

0.3

0.2

0.25

0.3

10 -3

7

0.2

TD

LC

6

0.18

5

0.16

4

0.14

3

0.12

2
0.1

Figure 3:

0.2

0.1
0.15

0.2

0.25

0.3

0.1

0.15

Comparative statics with respect to volatility σ.

The figure plots

VG (x), ROVG (x), x∗G , xROV
, LC, and T D. The other parameter values are set in Table 1.
G

31

5.24

1

5.22
xG

0.9

xROV
G

5.2
0.8

W(x)
ROVG (x)

5.18
5.16

0.7
0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

0.6

0.8

1

1/

1/
0.012

0.25
TD

LC

0.01

0.2

0.008

0.15

0.006
0.1

0.004

0.05

0.002
0

0
0

0.2

0.4

0.6

0.8

1

0

1/

0.2

0.4

1/

10
xB

8
6
4
2
0
0

0.2

0.4

0.6

0.8

1

1/

Figure 4: Comparative statics with respect to λ in the case of two project types. The figure∗∗
ROVplots W (x), ROVG (x), x∗∗, LC, and T D. The parameter values are set in Table 1.
G , xB , xG

32

6

1.2
xG

W(x)
ROV G (x)

5.5

5

4.5
0.1

xROV
G

1

0.8

0.15

0.2

0.25

0.6
0.1

0.3

0.15

0.2

0.25

0.3

0.2

0.25

0.3

0.3

0.015
LC

TD

0.25
0.01
0.2

0.005
0.1

0.15
0.15

0.2

0.25

0.3

0.2

0.25

0.3

0.1

0.15

1.75
xB

1.7

1.65

1.6
0.1

0.15

Figure 5: Comparative statics with respect to σ in the illiquid model with two project types.
∗∗
ROV
The figure plots W (x), ROVG (x), x∗∗, LC, and T D. The other parameter values are
G , xB , xG

set in Table 1.

33

1
k (x) for 1/ =0.6
k (x) for 1/ =0.8
k (x) for 1/ =1

0.9
0.8
0.7

k

0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

Figure 6: The lowest acceptable type k ∗ (x) in the illiquid model with a continuum of projecttypes. The figure plots k ∗ (x) with respect to illiquidity 1/λ. The other parameter values are
set in Table 1.

34