Discussion Papers In Economics
And Business

Evaluating the occurrence and disappearance of real
options
Michi NISHIHARA

Discussion Paper 10-19

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEvaluating the occurrence and disappearance of real
options
Michi NISHIHARA

Discussion Paper 10-19

July 2010

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEvaluating the occurrence and disappearance of real
options∗
Michi NISHIHARA†

Abstract
This paper investigates the decision-making of a ﬁrm that has an option to invest
in a single project among multiple alternatives. This type of option is called a maxoption, and the nature of a max-option has been investigated in several papers. I extend
the previous analysis to a model that allows the random occurrence and disappearance ofalternative projects in which to invest. The occurrence and disappearance of opportunities
in which to invest will be caused by changes in regulation, the exit and entry of rival ﬁrms,
technological innovation, political risk, catastrophes, etc. For example, the model applies
to land development with an alternative land use choice under uncertainty about changes
in zoning and development regulations. I show the properties of the exercise region andthe value function for the option. Speciﬁcally, I demonstrate that the prospective future
occurrence of an alternative has the signiﬁcant eﬀect of increasing the option value anddeferring the investment decision.
JEL Classiﬁcations Code: C61, G13, G31.
Keywords: real option, max-option, exercise region, regulatory risk, Poisson arrival.

∗†

This work was supported by KAKENHI 22710142. The ﬁrst version: June 30, 2010.
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 real options approach, in which option pricing theory is applied to capital budgeting decisions, better enables us to ﬁnd the optimal investment strategy and undertakeproject valuation under uncertainty than is possible under more classical methods. The
early literature has investigated a real option that has a rather simple payoﬀ structure,assuming that the dynamics of project value follow a one-dimensional stochastic process(see (Dixit and Pindyck 1994)). Naturally, the studies have developed into a more complicated analysis on the basis of a multidimensional process (e.g., (Geltner, Riddiough, andStojanovic 1996, Loubergé, Villeneuve, and Chesney 2002, Martzoukos 2009, BastianPinto, Brandão, and Alves 2010, Nishihara 2010)). For example, (Geltner, Riddiough,and Stojanovic 1996) investigates land development timing with an alternative land use
choice. In (Geltner, Riddiough, and Stojanovic 1996), the option’s payoﬀ depends on the
maximum of several underlying asset prices. This type of option is called a max-option,
which has been also investigated in (Stulz 1982, Johnson 1987, Broadie and Detemple
1997, Villeneuve 1999, Detemple 2006).1
This paper extends the previous max-option literature to a model that allows the
Poisson arrival or death of an alternative project in which to invest. The uncertain
occurrence or disappearance of the investment opportunity will be caused by changes
in regulation, the exit and entry of rival ﬁrms2 , technological innovation, political risk,
catastrophes, etc. The model captures these risks in addition to uncertainty about the
future market values of projects. For example, the model applies to land development withan alternative land use choice under uncertainty about changes in zoning and development
regulations. Especially in an emerging country, regulatory and political changes frequently
happen, and hence, a ﬁrm is required to evaluate the option value and optimize theinvestment strategy taking account of these risks.
Conventionally, numerous studies on ﬁnancial derivatives have modeled the catastrophic risks as a stock price following a discontinuous stochastic process with jumps.
Most of the real options studies have followed this convention. For example, (Martzoukos
and Trigeorgis 2002) investigates both a growth option and an extension option involving
various types of rare events, assuming the underlying asset value follows a jump diﬀusion
process. (Boyarchenko 2004) presents the option values and the optimal investment strategies for both one-shot investment with ﬁxed costs and incremental capacity expansion,
assuming the underlying asset value follows a geometric Lévy process.
A distinction between this paper and the previous works is that this paper directlymodels the possibility that an opportunity in which to invest occurs or disappears. This
1
2

(Nishihara 2010) reveals the nature of a combination of a max-option and a spread option.
An alternative approach is the game-theoretic approach. Strategic interactions among several ﬁrms areinvestigated in (Grenadier 1996, Grenadier 2002, Weeds 2002, Lambrecht and Perraudin 2003) among others.

1direct approach can provide a simpler and more appropriate valuation of the real options
than the previous approach. The model directly captures the eﬀects of the uncertain
disappearance or occurrence of an alternative. To my knowledge, this is the ﬁrst paper thatreveals the interactions among the random disappearance or occurrence of an alternative
project, investment timing, and project choice. Technically, this study links the standard
option to the max-option via the Poisson arrival or death. In other words, I investigate
an option that may change to the max-option by the Poisson arrival and a max-optionthat may change to the standard option by the Poisson death.
In the model, I reveal how the possibility of the occurrence or disappearance inﬂuences
the optimal exercise policy and the option value. Naturally, a higher intensity of theoccurrence (disappearance) plays the role in increasing (decreasing) the value of the option
to defer the investment timing and discouraging (encouraging) investment. In addition,
the properties, such as the monotonicity and convexity shown by the max-option literature(e.g., (Geltner, Riddiough, and Stojanovic 1996, Broadie and Detemple 1997, Detemple
2006)), hold true even if the occurrence or disappearance of the alternative are taken into
consideration. The result ensures the robustness of these properties.
Furthermore, the numerical analysis reveals the following characteristics. The possibility of uncertain changes inﬂuences the option value and the optimal investment policy
for the option that may change to the max-option greater than the max-option that may
change to the standard option. Indeed, for the rational parameter values, the prospective future occurrence of an alternative has the potential to enhance the option value byalmost 50%. The eﬀect becomes larger, especially for a weaker or negative correlation
between the project values, because a weaker or negative correlation increases the valueof the max-option that may appear in future.
This paper also entails real-world implications. For example, the results oﬀer rational
explanations for the behavior of an owner of farmland which has not been cultivated inmany years. Recently, an increase in idled farmland has been a serious issue in Japan.
Typically, idled farmland is restricted within the agricultural use because the zoning
and development ordinances prohibit nonagricultural uses. However, prospective future
regulatory and environmental changes may enable an owner to develop land for residential
or commercial uses. That is, an owner of idled farmland has the option that may change
to the max-option. In addition to the relatively high value of residential or commercial
land, the weak correlation between the alternative use and the agricultural use increases
the option value and deters an owner from cultivating farmland.

22Preliminaries

Consider a ﬁrm that has an option to invest in a project. There are two exclusive projects
i = 1 and 2. The risk-adjusted values of the projects, X(t) = (X1 (t), X2 (t)), are random
and follow GBMs (Geometric Brownian Motion)dXi (t) = µi Xi (t)dt + σi Xi (t)dBi (t),(1)

where B1 (t), B2 (t) are Brownian Motions (BM) with correlation coeﬃcient ρ satisfying|ρ| < 1. Constants µi and σi (> 0) denote the risk-adjusted growth rate and volatilityof the project value, respectively. Investing in project i requires an irreversible capital
expenditure of Ii (> 0). 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 at time t, and the ﬁrm optimizes the investment policy under this information.
The risk-free rate is a constant r(> 0). For convergence, I assume that r > µ.3 The
maturity of the options is T (> 0).

2.1

Standard option

As a benchmark, I consider a ﬁrm that has an option to invest in a given project i. A
ﬁrm cannot invest in project j(6= i). For Xi (t) = xi , the option value is equal to the valuefunction of the optimal stopping problem as follows:4
Vi (xi , t) := sup Ext i [1{τ ≤T } e−r(τ −t) (Xi (τ ) − Ii )],
τ ∈Tt

(2)

where Tt denotes the set of all stopping times τ ≥ t and Etxi [·] denotes the expectation
conditional on Xi (t) = xi . The subscript i denotes the option to invest in project i.
Note that problem (2) is analogous to an American call option written on a stock with adividend. The optimal stopping time for problem (2) becomes τi (t) := inf{s ≥ t | X(s) ∈Sii (s)}, where the stopping region Sii (s) is deﬁned bySii (s) := {x ∈ R2++ | Vi (xi , s) = xi − Ii }.

(3)

The superscript i means the immediate exercise region for project i. The optimal policy
is that a ﬁrm makes investment in project i as soon as X(s) hits Sii (s). The following
properties are well known (e.g., (Detemple 2006)):(Convexity of the value function) Vi (xi , t) is convex with respect to xi .
(Monotonicity of the stopping region) x ∈ Sii (t) ⇒ x0 ∈ Sii (t) (∀x0i ≥ xi , ∀x0j ∈ R++ ),where i 6= j.
3
4

Refer to (Dixit and Pindyck 1994) for the economic rationale for this assumption.
When the maturity is inﬁnite, I have only to replace 1{τ ≤T } with 1{τ <∞} .

3

The monotonicity implies that Sii (t) can be expressed as Sii (t) = {x ∈ R2++ | xi ≥ x∗i (t)},where x∗i (t) denotes the threshold. This type of optimal policy is called the thresholdpolicy.
Next, consider a case in which the investment opportunity for project i is killed at an
instantaneous rate λdt, where a positive constant λ denotes the intensity of the Poisson
death. I assume that the disappearance is independent of X(t). The disappearance ofan opportunity in which to invest will be caused by the enforcement of new regulations,
preemption by rival ﬁrms, political changes, natural disasters, etc. For Xi (t) = xi , theoption value prior to the death is equal to the value function of the optimal stopping
problem as follows:

Z ∞Vi→∅ (xi , t) := sup Ext i [τ ∈Tt01{τ ≤T } 1{τ <t+y} e−r(τ −t) (Xi (τ ) − Ii )λe−λy dy],(4)

where the subscript i → ∅ means that project i may be killed. I have
Z ∞
xi
Et [
1{τ ≤T } 1{τ <t+y} e−r(τ −t) (Xi (τ ) − Ii )λe−λy dy]0Z ∞xiλe−λy dy]
= Et [1{τ ≤T } e−r(τ −t) (Xi (τ ) − Ii )
τ −t
Z ∞
xi
−(r+λ)(τ −t)= Et [1{τ ≤T } e(Xi (τ ) − Ii )λe−λy dy]0=

Ext i [1{τ ≤T } e−(r+λ)(τ −t) (Xi (τ ) − Ii )].

(5)Then, problem (4) can be rewritten as follows:Vi→∅ (xi , t) = sup Ext i [1{τ ≤T } e−(r+λ)(τ −t) (Xi (τ ) − Ii )].
τ ∈Tt

(6)

That is, the problem with the random disappearance is equivalent to the standard problem(2) with the augmented discount rate r + λ. This has also been shown in (Weeds 2002).
Then, problem (6) satisﬁes the same properties, such as the convexity of the value function
and the monotonicity of the stopping region, as problem (2). Until the death the option
value follows the continuous process Vi→∅ (Xi (t), t), and it jumps downward to zero at theirandom death. In addition, because of Vi→∅ (xi , t) ≤ Vi (xi , t), the stopping region Si→∅(t)ifor problem (6) is larger than Sii (t). Note that Vi→∅ (xi , t) ↓ max{xi − Ii , 0}, Si→∅(t) ↑i(t) ↓ Sii (t) (λ ↓ 0).
{x ∈ R2++ | xi > Ii } (λ ↑ ∞) and Vi→∅ (xi , t) ↑ Vi (xi , t), Si→∅

Now, consider a case in which the investment opportunity for project i occurs at an
instantaneous rate λdt, where a positive constant λ denotes the intensity of the Poisson
arrival. I assume that the occurrence is independent of X(t). The occurrence of an
opportunity in which to invest will be caused by deregulation, the exit of rival ﬁrms,
technological innovation, etc. For Xi (t) = xi , the option value prior to its arrival is equal
to the following expectation:

Z ∞V∅→i (xi , t) := Ext i [e−ry Vi (Xi (t + y), t + y)λe−λy dy],

0

4

(7)

where the subscript ∅ → i means that project i may be available in future. A ﬁrm waits forthe arrival of the option, and after the arrival it adopts the optimal policy for problem (2).
At the time of arrival, the option value jumps upward from V∅→i (Xi (t), t) to Vi (Xi (t), t).
Note that V∅→i (xi , t) ↑ Vi (xi , t) (λ ↑ ∞) and V∅→i (xi , t) ↓ 0 (λ ↓ 0).

2.2

Max-option

As a benchmark, this subsection considers an option to invest in a single project between
projects 1 and 2. The model applies not only to a case in which two projects are exclusive (e.g., alternative land use in (Geltner, Riddiough, and Stojanovic 1996)) but also
to a case where a ﬁrm faces a budget constraint. This type of option is identiﬁed as an
American max-option. European max-options have been studied in (Stulz 1982, Johnson
1987), while American max-options have been studied in (Geltner, Riddiough, and Stojanovic 1996, Broadie and Detemple 1997, Villeneuve 1999, Detemple 2006). Although atypical max-option has a multidimensional state variable, (Décamps, Mariotti, and Villeneuve 2006) studies a max-option based on a one-dimensional state variable in order toinvestigate investment timing with an alternative scale choice.
For X(t) = x, the option value is equal to the value function of the optimal stopping
problem as follows:
V1,2 (x, t) := sup Ext [1{τ ≤T } e−r(τ −t) max{Xi (τ ) − Ii }],
i=1,2
τ ∈Tt
|
{z
}

(8)

project choice

where the subscript 1, 2 represents the option to invest between projects 1 and 2.
The optimal stopping time for problem (2) becomes τ1,2 (t) := inf{s ≥ t | X(s) ∈1 (s) ∪ S 2 (s)}, where the stopping region S i (s) is deﬁned by
S1,2 (s) = S1,2
1,2
1,2
i
S1,2
(s) := {x ∈ R2++ | V1,2 (x, s) = xi − Ii }(9)for i = 1 and 2. The max-option literature (e.g., (Geltner, Riddiough, and Stojanovic
1996, Broadie and Detemple 1997)) shows the following properties:(Convexity of the value function) V1,2 (x, t) is convex with respect to x.
i (t) is a convex set.
(Convexity of each stopping region) S1,2i (t) ⇒ x0 ∈ S i (t) (∀x0 ≥ x , ∀x0 ≤(Monotonicity of each stopping region) x ∈ S1,2
i
1,2
i
j

xj ), where i 6= j.
(Behavior on the indiﬀerence line) x1 − I1 = x2 − I2 ⇒ x ∈/ S1,2 (t).
i (t) implies that an increase (decrease) in the value of project i
The monotonicity of S1,2

(j 6= i) facilitates investment in project i. The behavior on the indiﬀerence line meansthat a ﬁrm waits and sees which project is better when the values of two projects equal.

5

3

Main Results

This section links the standard option in Section 2.1 and the max-option in Section 2.2
via a random variable distributed exponentially. Section 3.1 investigates a max-optionthat may change to the standard option, while Section 3.2 investigates an option that
may change to the max-option. I show several properties of the option values and exercise
regions. The model applies to the decision-making process about land development withan alternative land use choice under regulatory risks.

3.1

Max-option that will change to the standard option

As in Section 2.2, consider an option to invest between projects 1 and 2. Assume thatthe investment opportunity for project 2 is killed at an instantaneous rate λdt which is
independent of X(t).5 For X(t) = x, the option value is equal to the value function of
the optimal stopping problem as follows:6
Z ∞x
V1,2→1 (x, t) := sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) max{Xi (τ ) − Ii }
τ ∈Tt

i=1,2

0

+1{τ ≥t+y} e−ry

V1 (X1 (t + y), t + y)}λe−λy dy],

(10)

where y denotes the term until the death. The subscript 1, 2 → 1 represents the maxoption that may change to the standard option to invest in project 1.
Using the similar calculation to (5), I can rewrite problem (10) as
V1,2→1 (x, t) = sup Ext [1{τ ≤T } e−(r+λ)(τ −t) max{Xi (τ ) − Ii − V∅→1 (X1 (τ ), τ )}] + V∅→1 (x1 , t),
i=1,2

τ ∈Tt

(11)
where V∅→1 (x1 , t) is deﬁned by (7). The optimal stopping time for problem (11) becomes
12
τ1,2→1 (t) := inf{s ≥ t | X(s) ∈ S1,2→1 (s) = S1,2→1(s) ∪ S1,2→1(s)}, where the stoppingi
region S1,2→1(s) is deﬁned byi
S1,2→1(s) := {x ∈ R2++ | V1,2→1 (x, s) = xi − Ii }(12)

for i = 1 and 2.
Proposition 1(Convexity of the value function) V1,2→1 (x, t) is convex with respect to x.
i(Convexity of each stopping region) S1,2→1(t) is a convex set.
ii(Monotonicity of each stopping region) x ∈ S1,2→1(t) ⇒ x0 ∈ S1,2→1(t) (∀x0i ≥xi , ∀x0j ≤ xj ), where i 6= j.
5(Nishihara 2009) models rival preemption for the max-option endogenously. Indeed, (Nishihara 2009) derives

the equilibrium of a preemption game in which two ﬁrms compete for two alternatives.
6
This problem is the same as the max-option problem (8) replaced X2 (t) with the killed process.

6

(Behavior on the indiﬀerence line) x1 − I1 = x2 − I2 ⇒ x ∈/ S1,2→1 (t).
1 (t) ⊂ S 1(Comparison) max{V1 (x1 , t), V2→∅ (x2 , t)} ≤ V1,2→1 (x, t) ≤ V1,2 (x, t), S1,2
1,2→1 (t) ⊂2 (t) ⊂ S 2
2
S11 (t), and S1,2
1,2→1 (t) ⊂ S2→∅ (t).

Proof(Convexity of the value function) For any x, x0 ∈ R++ and α ∈ [0, 1], I have
V1,2→1 (αx + (1 − α)x0 , t)
Z ∞
(1,1)
= sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) max{(αxi + (1 − α)x0i )Xi (τ ) − Ii }
τ ∈Tt

0
−ry

i=1,2

V1 ((αx1 + (1 − α)x01 )X1 (t + y), t + y)}λe−λy dy]+1{τ ≥t+y} e
Z ∞
(1,1)
≤ sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) (α max{xi Xi (τ ) − Ii } + (1 − α) max{x0i Xi (τ ) − Ii })
τ ∈Tt

0
−ry

i=1,2

0
−ry

i=1,2

i=1,2

(αV1 (x1 X1 (t + y), t + y) + (1 − α)V1 (x01 X1 (t + y), t + y))}λe−λy dy]
Z ∞
(1,1)
≤ α sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) max{xi Xi (τ ) − Ii }+1{τ ≥t+y} eτ ∈Tt

V1 (x1 X1 (t + y), t + y)}λe−λy dy]
Z ∞
(1,1)
+(1 − α) sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) max{x0i Xi (τ ) − Ii }+1{τ ≥t+y} e

τ ∈Tt
0
−ry
0
+1{τ ≥t+y} e V1 (x1 X1 (t + y), t + y)}λe−λy dy]i=1,2= αV1,2→1 (x, t) + (1 − α)V1,2→1 (x0 , t),where (13) results from the convexity of V1 (x, t) and maxi=1,2 {xi − Ii }.
i(Convexity of each stopping region) For any x, x0 ∈ S1,2→1(t) and α ∈ [0, 1], I have

V1,2→1 (αx + (1 − α)x0 , t) ≤ αV1,2→1 (x, t) + (1 − α)V1,2→1 (x0 , t)(14)

= α(xi − Ii ) + (1 − α)(x0i − Ii )= αxi + (1 − α)x0i − Ii ,where (14) results from the convexity of V1,2→1 (x, t), and the last inequality implies αx +i(1 − α)x0 ∈ S1,2→1(t).

(Monotonicity of each stopping region) It follows from the convexity of V1 (x1 , t)with respect to x and ∂V1 (x1 , t)/∂x1 = 1 for x1 > x∗1 (t) that
V1 (x01 , t) ≤ V1 (x1 , t) + x01 − x1

7(x01 ≥ x1 )

(15)

(13)

1
(t) and x0 satisfying x01 ≥ x1 and x02 ≤ x2 . I have
Take any x ∈ S1,2→1

V1,2→1 (x0 , t)
Z ∞
(1,1)
= sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) max{x0i Xi (τ ) − Ii }
τ ∈Tt

0
−ry

i=1,2

V1 (x01 X1 (t + y), t + y)}λe−λy dy]+1{τ ≥t+y} e
Z ∞
(1,1)
≤ sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) (max{xi Xi (τ ) − Ii } + (x01 − x1 )X1 (τ ))
τ ∈Tt

0
−ry

i=1,2

0
−ry

i=1,2

+1{τ ≥t+y} e (V1 (x1 X1 (t + y), t + y) + (x01 − x1 )X1 (t + y))}λe−λy dy]
Z ∞
(1,1)
≤ sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) max{xi Xi (τ ) − Ii }
τ ∈Tt

(16)

+1{τ ≥t+y} e (V1 (x1 X1 (t + y), t + y)}λe−λy dy]
Z ∞1+ sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) (x01 − x1 )X1 (τ )
τ ∈Tt

0
−ry

+1{τ ≥t+y} e(x01 − x1 )X1 (t + y)}λe−λy dy]≤ V1,2→1 (x, t) + sup E1t [1{τ ≤T } e−r(τ −t) (x01 − x1 )X1 (τ )]τ ∈Tt= x1 − I1 + x01 − x1
= x01 − I1 ,(17)

1
where (16) follows from (15). The last inequality (17) implies x0 ∈ S1,2→1(t), and hence,1
2
I have the monotonicity of S1,2→1(t). Similarly, I can show the monotonicity of S1,2→1(t).

(Behavior on the indiﬀerence line) Take any x satisfying x1 − I1 = x2 − I2 . First,
note that
V1,2→∅ (x, t) :=
=
>

Z ∞
x
sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) max{Xi (τ ) − Ii }λe−λy dy]
i=1,2
τ ∈Tt
0
x
−(r+λ)(τ −t)
sup Et [1{τ ≤T } e
max{Xi (τ ) − Ii }]
(18)
i=1,2
τ ∈Tt
x1 − I1 ,(19)

where I can show (18) by the similar calculation to (5). The last inequality (19) follows
from the behavior on the indiﬀerence line for the max-option in Section 2.2. Then, I have
V1,2→1 (x, t) ≥ V1,2→∅ (x, t)> x1 − I1 ,1
2
where the last inequality implies that x ∈/ S1,2→1(t). Similarly, I have x ∈/ S1,2→1(t).

(Comparison) The inequalities with respect to the value functions are clear. For any
i (t), I havex ∈ S1,2

V1,2→1 (x, t) ≤ V1,2 (x, t)= xi − Ii ,8i(t). Similarly, by the inequalities with respect
where the last inequality implies x ∈ S1,2→1
1
2
2
to the value functions, I can show that S1,2→1(t) ⊂ S11 (t) and S1,2→2(t) ⊂ S2→∅(t).

¤

Proposition 1 extends previous ﬁndings by (Geltner, Riddiough, and Stojanovic 1996,
Broadie and Detemple 1997, Detemple 2006) to a case in which the investment opportunity may be killed. The properties for the max-option in Section 2.2 remain true forithe generalized case. The monotonicity of S1,2→1(t) implies that an increase (decrease)in the value of project i (j 6= i) encourages investment in project i. The behavior on
the indiﬀerence line means that a ﬁrm will wait and see which project is better when theivalues of both projects are equal. Clearly, V1,2→1 (x, t) (S1,2→1(t)) monotonically decreases(increases) with the intensity λ. This means that an increased possibility of the disappearance reduces the value of waiting and encourages investment. Note that V1,2→1 (x, t) ↓1
2
max{x2 −I2 , V1 (x1 , t)}, S1,2→1(t) ↑ S11 (t)\{x ∈ R2++ | x2 −I2 ≥ V1 (x1 , t)}, S1,2→1(t) ↑ {x ∈ii (t) (λ ↓ 0).
R2++ | x2 − I2 > V1 (x1 , t)} (λ ↑ ∞) and V1,2→1 (x, t) ↑ V1,2 (x, t), S1,2→1(t) ↓ S1,2

A higher volatility σi increases V1,2 (x, t) and V1 (x1 , t), and hence, it increases (decreases) V1,2→1 (X(t), t) (S1,2→1 (t)). On the other hand, an increase in the correlation coeﬃcient ρ tends to decrease (increase) V1,2→1 (X(t), t) (S1,2→1 (t)). This is because V1,2 (x, t)
tends to increase with ρ (see (Geltner, Riddiough, and Stojanovic 1996, Detemple 2006)).
Note that the option value jumps downward from V1,2→1 (X(t), t) to V1 (X1 (t), t) at thetime of the Poisson death (see Figure 1). The jump size is endogenously determined
as V1,2→1 (X(t), t) − V1 (X1 (t), t).7 The jump size decreases with λ and ρ. Typically,
V1 (X1 (t), t) is more volatile than V1,2→1 (X(t), t) in which X1 (t) and X2 (t) diversify therisk. Accordingly, the volatility of the option value jumps upward on the same timing.
This may account for empirical observations (e.g., (Black 1976, Christie 1982)) that thevolatility of a stock price increases when the stock price decreases.

3.2

Option that will change to the max-option

This subsection considers an option that may change to the max-option. This option
contrasts with the option studied in Section 3.1. Assume that the investment opportunity
for project 2 is created at an instantaneous rate λdt which is independent of X(t). For
X(t) = x, the option value prior to the Poisson arrival is equal to the value function of
the optimal stopping problem as follows:Z ∞x
V1→1,2 (x, t) := sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) (X1 (τ ) − I1 )
τ ∈Tt

0

+1{τ ≥t+y} e−ry V1,2 (X(t + y), t + y)}λe−λy dy](20)

where y denotes the term until the arrival. The subscript 1 → 1, 2 denotes the optionthat may change to the max-option. Note that V1→1,2 (x, t), unlike V1 (x1 , t), depends not
7

In contrast, in a model based on the discontinuous stochastic process the jump size must be exogenously

presumed.

9

ValueJump size0

PoissondeathtFigure 1: The downward jump caused by the disappearance of the alternative.
only x1 but also x2 because of the potential arrival of project 2.
Using the similar calculation to (5), I can rewrite problem (20) as
V1→1,2 (x, t) = sup Ext [1{τ ≤T } e−(r+λ)(τ −t) (X1 (τ ) − I1 − V∅→1,2 (X(τ ), τ ))] + V∅→1,2 (x, t),
τ ∈Tt

(21)
where V∅→1,2 (x, t) is deﬁned by
Z ∞V∅→1,2 (x, t) = Ext [e−ry V1,2 (X(t + y), t + y)λe−λy dy].

(22)

0

The optimal stopping time for problem (21) becomes τ1→1,2 (t) := inf{s ≥ t | X(s) ∈1
1
S1→1,2(s)}, where the stopping region S1→1,2(s) is deﬁned by
1
S1→1,2(s) := {x ∈ R2++ | V1→1,2 (x, s) = x1 − I1 }.

(23)

1
Note that, the exercise region S1→1,2(s) exists only for project 1 although it also depends

on the value of project 2.
Proposition 2(Convexity of the value function) V1→1,2 (x, t) is convex with respect to x.
1(Convexity of the stopping region) S1→1,2(t) is a convex set.
1
1
(Monotonicity of the stopping region) x ∈ S1→1,2(t) ⇒ x0 ∈ S1→1,2(t) (∀x01 ≥x1 , ∀x02 ≤ x2 ).
1 (t) ⊂ S 1
1(Comparison) V1 (x1 , t) ≤ V1→1,2 (x, t) ≤ V1,2 (x, t) and S1,2
1→1,2 (t) ⊂ S1 (t).

Proof

10

(Convexity of the value function) Using the convexity of V1,2 (x, t), I can construct
the same proof as the convexity of the value function in Proposition 1.
(Convexity of the stopping region) Using the convexity of V1→1,2 (x, t), I can construct the same proof as the convexity of the stopping region in Proposition 1.
(Monotonicity of the stopping region) Using
V1,2 (x0 , t) ≤ V1,2 (x, t) + x01 − x1(∀x01 ≥ x1 , ∀x02 ≤ x2 ),I can construct the same proof as the monotonicity of the stopping region in Proposition
1.
(Comparison) The relationships among the value functions are clear. Then, I can easily
show the relationships among the stopping regions.

¤

To my knowledge, there has been no studies that investigates the option that willchange to the max-option. Proposition 2 presents the ﬁrst result for this type of option.
Propositions 1 and 2 bridge the gap between the standard option and the max-option
1
from both sides. The exercise region S1→1,2(t) has the same convexity and monotonicity
1 (t) for the max-option. The monotonicity of S 1
as S1,2
1→1,2 (t) implies that an increase (de-crease) in the value of project 1 (2) accelerates investment in project 1. In the presence ofthe potentially available project 2, a high value of project 2 increases the option value and1delay investment in project 1. It follows from expression (21) that S1→1,2(t) is containedin {x ∈ R++ | x1 ≥ I1 + V∅→1,2 (x, t)}. In particular, byZ ∞xV∅→1,2 (x, t) ≥ Et [e−ry (X2 (t + y) − I2 )λe−λy dy]
0

=

λI2
λx2
−
,
r+λ−µ r+λ1
I can show that x1 ≥ I1 + λx2 /(r + λ − µ) − λI2 /(r + λ) for x ∈ S1→1,2(t). This indicates
1that for a high x2 , S1→1,2(t) is much smaller than S11 (t) = {x ∈ R2++ | x1 ≥ x∗1 (t)}.

Clearly, V1→1,2 (x, t) (S1→1,2 (t)) monotonically increases (decreases) with the intensity λ.
This means that an increased possibility of the arrival enhances the value of waiting and
1
1 (t) (λ ↑ ∞) anddiscourages investment. I have V1→1,2 (x, t) ↑ V1,2 (x, t), S1→1,2(t) ↓ S1,2
1
V1→1,2 (x, t) ↓ V1 (x1 , t), S1→1,2(t) ↑ S11 (t) (λ ↓ 0).

As in Section 3.1, V1→1,2 (X(t), t) (S1→1,2 (t)) increases (decreases) with the volatility σi and decreases (increases) with the correlation coeﬃcient ρ. Note that the option value jumps upward from from V1→1,2 (X(t), t) to V1,2 (X(t), t) at the time of the
Poisson arrival (see Figure 2). The jump size, which is endogenously determined asV1,2 (X(t), t) − V1→1,2 (X(t), t), decreases with λ.

3.3

Extensions and limitations

This section explains several extensions of the results. First, consider the max option thathas n investment opportunities. Assume that projects n, n − 1, . . . , m + 1 will be killed

11ValueJump size0tPoissonarrivalFigure 2: The upward jump caused by the occurrence of the alternative.
sequentially with the intensity λ. For X(t) = x, the option value prior to the Poisson
death is equal to the value function of the optimal stopping problem as follows:
V1,...,n→1,...,n−1→···→1,...,m (x, t)
Z ∞x:= sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) max {Xi (τ ) − Ii }τ ∈Tt

i=1,...,n0+1{τ ≥t+y} e−ry V1,...,n−1→1,...,n−2→···→1,...,m (X(t + y), t + y)}λe−λy dy](n > m),which is deﬁned backward from
V1,...,m→1,...,m (x, t):= V1,...,m (x, t):=

sup Ext [1{τ ≤T } e−r(τ −t) max {Xi (τ ) − Ii }].
i=1,...,mτ ∈TtThis is a generalized version of problem (10). Note that V1,...,m (x, t) is convex with respect
to x. Then, using backward induction, I can show the same properties as Proposition 1.
Similarly, a generalized version of problem (20) is expressed as
V1,...,m→1,...,m+1→···→1,...,n (x, t)
Z ∞x:= sup Et [1{τ ≤T } {1{τ <t+y} e−r(τ −t) max {Xi (τ ) − Ii }τ ∈Tt

i=1,...,m0−ry+1{τ ≥t+y} e

V1,...,m+1→···→1,...,n (X(t + y), t + y)}λe−λy dy].

I can show the same properties as Proposition 2 by the backward induction. Of course, itdoes not matter if the intensities vary over projects. Furthermore, I can show the same
properties even if the order of the disappearances or occurrences is not presumable.

12For the problem in Section 3.2, a new value X2 (t) may be unobservable until the arrival.
For example, value of an alternative that proceeds from some technical innovation may
not be evaluated correctly prior to the innovation. This partial information problem isessentially diﬀerent from problem (20) in that the option value and the investment strategy
depend only on X1 (t).

4

Numerical Examples

This section provides numerical examples of the options studied in the previous section.
I use base parameter values as follows8 :r = 8%, µ1 = µ2 = 0%, σ1 = σ2 = 20%, ρ = 0%.

(24)

For expositional purposes, I set the investment cost I1 = I2 = 100 and the option values
are computed at the money, i.e., x = (100, 100). The maturity of the option is set at
T − t = 3 years. For the intensity λ, the probability that an opportunity in which toinvest disappears or occurs within 3 years is expressed asZ 3
λe−λt dt = 1 − e−3λ .
0I set this probability at 25%, 50%, and 75%. These correspond to λ = 0.096, 0.231, and0.462. In the numerical procedure, I make a discretization with 200 time steps per 1 year,
and use a bivariate version of the lattice binomial method (e.g., (Boyle 1988)). Technically,
I compute the lattice model for maturity T = 4 years, and draw the investment regions
for t = 1 year.
First, I set the intensity λ = 0.231. Figure 3 illustrates the exercise regions S1,2→1 (t) =12
1
S1,2→1(t) ∪ S1,2→1(t) and S1→1,2(t) for the base parameter values (24). For comparison,
1 (t)∪S 2 (t) and S 1 (t). I can checkthe ﬁgure also presents the exercise regions S1,2 (t) = S1,2
1,2
1
i
1
the convexity and monotonicity of S1,2→1 (t) and S1→1,2 (t), as well as the relationship that
1 (t) ⊂ S 1
1
2
2
1
1
1
S1,2
1,2→1 (t) ⊂ S1 (t), S1,2 (t) ⊂ S1,2→1 (t), and S1,2 (t) ⊂ S1→1,2 (t) ⊂ S1 (t).

In Figure 3, let us take a look at the max-option that will change to the standard
1
option. When X(t) hits the boundary of S1,2→1(t) (the lower-right curve), a ﬁrm invests
2
in project 1. On the contrary, when X(t) hits the boundary of S1,2→1(t) (the upper-leftcurve), the ﬁrm invests in project 2. The ﬁrm delays the decision on project choice for X(t)between the two curves, although project 2 may be killed in the waiting time. The ﬁgure
2
2 (t) is greater than the diﬀerenceindicates that the diﬀerence between S1,2→1(t) and S1,2
1
1 (t). The prospective disappearance of project 2 encourages
between S1,2→1(t) and S1,2

investment in project 2 prior to the disappearance, while it does not have a signiﬁcant
8

These parameter values are similar to (Geltner, Riddiough, and Stojanovic 1996, Detemple 2006). I carriedout a lot of computations with varying parameter values and distilled robust results into this section.

132(t) approaches theinﬂuence on the investment timing in project 1. The boundary of S1,2→1

threshold x∗2 (t) replaced r = 0.08 with r + λ = 0.08 + 0.231 = 0.311 when x1 → 0. Then,2
2 (t). On the other hand, both boundaries offor x1 ≈ 0, S1,2→1(t) is much larger than S1,2
1 (t) and S 1∗
S1,2
1,2→1 (t) converges to the threshold x1 (t) when x2 → 0. Then, for x2 ≈ 0,
1
1 (t).
S1,2→1(t) is almost the same as S1,2

In Figure 3, let us now turn to the option that may change to the max-option. When
1
X(t) hits the boundary of S1→1,2(t) (the lower-right curve), a ﬁrm invests in project

1. Otherwise, the ﬁrm delays investment, and in the waiting time project 2 may become
1available. I see that the gap between S1→1,2(t) and S11 (t) increases with x2 . This is because

a higher x2 increases an incentive for the ﬁrm to wait for the occurrence of project 2. On
1the other hand, for x2 ≈ 0 the boundary of S1→1,2(t) converges to the threshold x∗1 (t),and hence there is no gap between them.
Now, consider the comparative statics. First, I explore the eﬀects of the intensity λ.
1
2
1
Figures 4 and 5 draw the exercise regions S1,2→1 (t) = S1,2→1(t) ∪ S1,2→1(t) and S1→1,2(t)iwith varying levels of λ. As mentioned in Section 3.2, S1,2→1(t) increases with λ, while1
S1→1,2(t) decreases with λ. For the max-option that may change to the standard option

for project 1, a higher λ increases an incentive for a ﬁrm to invest in project 2 prior to thedisappearance. On the other hand, for the option that may change to the max-option, ahigher λ increases the value of waiting for project 2. Comparing Figures 4 and 5, I ﬁnd
that the impact of λ on the exercise policy is stronger for the option that will change to the
max-option. This suggests that a ﬁrm should take into careful consideration an alternative
which may be available in future rather than an alternative which may disappear in future.
Next, I explore the eﬀects of the correlation coeﬃcient ρ. Figures 6 and 7 illustrate12
1
the exercise regions S1,2→1 (t) = S1,2→1(t) ∪ S1,2→1(t) and S1→1,2(t) with varying levels ofi1
ρ. In both ﬁgures, the intensity λ is ﬁxed at λ = 0.231. I see that S1,2→1(t) and S1→1,2(t)monotonically enlarge with ρ. It is known (e.g., (Geltner, Riddiough, and Stojanovic
i (t) tends to increase with ρ because a higher ρ decreases
1996, Detemple 2006)) that S1,2

the value of the option to delay the decision concerning project choice. Figures 6 and7 demonstrate that the same result holds even if the disappearance or occurrence of analternative is taken into consideration. This means the robustness of the previous ﬁndings.
The eﬀects of ρ also appear in Tables 1 and 2. The tables present the option values
V1,2→1 (x, t) and V1→1,2 (x, t) for varying levels of λ and ρ. Note that V1,2→1 (x, t) andV1→1,2 (x, t) are computed at the money, i.e., x = (100, 100). The option values monotonically decrease with ρ. The impact of ρ on V1,2→1 (x, t) weakens with λ, while the impactof ρ on V1→1,2 (x, t) strengthens with λ. This can be explained as follows. When λ → ∞,V1,2→1 (x, t) approaches V1 (x, t) which is independent of ρ. On the other hand, V1→1,2 (x, t)
approaches V1 (x, t) when λ → 0. Then, a higher (lower) λ weakens the impact of ρ on
V1,2→1 (x, t)(V1→1,2 (x, t)). The impact of λ is signiﬁcant to V1→1,2 (x) especially for a low

14

λ\ρ

Table 1: The option values V1,2→1 (x, t).
−75% −50% −25% 0%
25% 50%

0

22.8

22.1

21.28

20.32

19.2

17.82 15.98

0.096

21.07

20.46

19.75

18.93

17.97

16.8

0.231

19.36

18.83

18.23

17.55

16.75 15.79 14.53

0.462

17.5

17.07

16.59

16.04

15.42 14.68 13.73

λ\ρTable 2: The option values V1→1,2 (x, t).
−75% −50% −25% 0%
25% 50%

75%

0

11.7

11.7

11.7

11.7

11.7

11.7

0.096

14.42

14.24

14.02

13.77

13.47 13.12 12.65

0.231

17.16

16.81

16.39

15.9

15.33 14.62

13.7

0.462

19.93

19.41

18.79

18.07

17.22 16.18

14.8

11.7

75%
15.26

ρ. Indeed, for ρ = −50%, the Poisson arrival with the intensity λ = 0.096, 0.231, and0.462 enhances the option value by 21%, 44%, and 66%, respectively. For ρ = 0% the
Poisson arrival with the intensity λ = 0.096, 0.231, and 0.462 increases the option value
by 18%, 36%, and 54%, respectively.
Let us examine Tables 1 and 2 from a diﬀerent aspect. As mentioned in Section 3.1
and 3.2 (see Figures 1 and 2), the gap between V1,2→1 (x, t) and V1 (x, t) = 11.7 is equal
to the jump size at the time of the Poisson death, while the gap between V1→1,2 (x, t) andV1,2 (x, t) is equal to the jump size at the time of the Poisson arrival. Comparing Tables 1
and 2, I ﬁnd that the downward jump size tends to be larger than the upward jump size.
This ﬁnding may be related to empirical ﬁndings (e.g., (Skinner 1994)) that the stock
price response to bad news is larger than the stock price response to good news.

5

Conclusion

The paper has investigated the nature of two types of option; the max-option that willchange to the standard option by the Poisson death, and the option that will change to the
max-option by the Poisson arrival. These uncertain changes in real options can be causedby changes in regulatory, political, competitive, and technological environment. The
model, unlike the previous studies, directly captures these catastrophic risks in capitalbudgeting. In the model, I have revealed how the possibility of the disappearance oroccurrence inﬂuences the optimal exercise policy and the option value. The results are
summarized as follows.

15

500

∂S

(t)

1,2→ 11∂S(t)1→ 1,2400∂ S1,2(t)∂ S11(t)

x2

300
200
100

100

200

300

400

500

x1

Figure 3: The exercise regions for the four options. This ﬁgure plots the boundaries of the
1
2
1
1
2
exercise regions S1,2→1 (t) = S1,2→1(t) ∪ S1,2→1(t), S1→1,2(t), S1,2 (t) = S1,2(t) ∪ S1,2(t) and S11 (t).

The parameter values are set at the base case (24) with the intensity λ = 0.231.

A higher intensity of the disappearance (occurrence) decreases (increases) the option
value and encourages (discourages) investment. The properties, such as the monotonicity
and convexity shown by the max-option literature, remain true when the random change
is taken into consideration. The impact of the uncertain change is relatively greater forthe option that may change to the max-option. Notably, the impact is signiﬁcant fora weaker or negative correlation between the project values. The results oﬀer rational
explanations for the behavior of an owner of farmland which has not been cultivated inmany years. The results have the potential to account for the asymmetric market reaction
to good and bad news, although the model is not intended to investigate the stock pricereactions.

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Economic Review 94, 557–568.

16

Boyle, P., 1988, A lattice framework for option pricing with two state variables, Journalof Financial and Quantitative Analysis 23, 1–12.
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Detemple, J., 2006, American-Style Derivatives valuation and computation (Chapman &Hall/CRC: London).
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18

500

0
0.096
0.231
0.462

400
2
S1,2→
(t)
1

x2

300
200
100

S11,2→ 1(t)
100

200

300

400

500

x1

Figure 4: The exercise regions for the max-option that will change to the standard option.
1
2
This ﬁgure plots the boundaries of the exercise regions S1,2→1 (t) = S1,2→1(t) ∪ S1,2→1(t) for theintensity λ = 0, 0.096, 0.231, and 0.462. For λ = 0, the region is equal to the exercise region forthe max-option, S1,2 (t). The parameter values are set at the base case (24).

500

0
0.096
0.231
0.462

400

x2

300
200
S11→ 1,2(t)

100

100

200

300

400

500

x1

Figure 5: The exercise regions for the option that will change to the max-option. This ﬁgure
1(t) for the intensity λ = 0, 0.096, 0.231, andplots the boundaries of the exercise regions S1→1,2

0.462. For λ = 0, the region is equal to the exercise region for the standard option, S11 (t). The
parameter values are set at the base case (24).

19

500

−0.75
−0.5
−0.25
0
0.25
0.5
0.75

400
2
S1,2→
(t)
1

x2

300
200
100

S11,2→ 1(t)
100

200

300

400

500

x1

Figure 6: The exercise regions for the max-option that will change to the standard option.
1
2
This ﬁgure plots the boundaries of the exercise regions S1,2→1 (t) = S1,2→1(t) ∪ S1,2→1(t) for thecorrelation coeﬃcient ρ = −75%, −50%, −25%, 0%, 25%, 50%, and 75%. The intensity is set at
λ = 0.231. The other parameter values are set at the base case (24).

500

−0.75
−0.5
−0.25
0
0.25
0.5
0.75

400

x2

300
200
100

S11→ 1,2(t)
100

200

300

400

500

x1

Figure 7: The exercise regions for the option that will change to the max-option.

This1(t) for the correlation coeﬃcient
ﬁgure plots the boundaries of the exercise regions S1→1,2

ρ = −75%, −50%, −25%, 0%, 25%, 50%, and 75%. The intensity is set at λ = 0.231. The
other parameter values are set at the base case (24).

20