GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.88

Hybrid or Electric Vehicles? A Real Options Perspective

Michi NISHIHARA

September 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Hybrid or Electric Vehicles? A Real Options Perspective∗
Michi NISHIHARA†

Abstract
This paper investigates the decision of an automaker concerning the alternative promotion of a hybrid vehicle (HV) and a full electric vehicle (EV). We evaluate the HV projectby considering the option to change promotion from the HV to the EV in the future. The
results not only extend previous ﬁndings concerning American options on multiple assets,
but also include several new implications. One notable observation is that the increased
market demand for EVs can accelerate the promotion of the HV because of the embedded
option.

JEL Classiﬁcations Code: C61, G13, G31, O32.
Keywords: real options, American options on multiple assets, exercise region, alternative
projects, hybrid and electric vehicles.

∗

The author would like to thank Atsuyuki Ohyama for useful comments and suggestions. This work was

supported by KAKENHI 20710116.
†
Corresponding Author, Address: Graduate School of Economics, Osaka University, 1-7 Machikaneyama-cho,
Toyonaka, Osaka 560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Tel: 81-6-6850-5277

1Introduction

The global ﬁnancial crisis beginning in 2007 has increased uncertainty about future market
demand in many industries throughout the world. As a result, it has becoming increasingly
important for ﬁrm project management to take into account uncertainty and ﬂexibility
in the future. The real options approach, in which option pricing theory is applied tocapital budgeting decisions, better enables us to ﬁnd an optimal investment strategy andundertake project valuation in this environment than is possible under more classicalmethods.
Using a real options approach, this paper investigates an automaker’s decisions concerning investment timing and project choice. Recently, increased concerns about theenvironmental impact of gasoline fueled cars, along with increasing fuel prices and theGreen New Deal, have promoted interest in hybrid and electric vehicles.1 This is because
both types of vehicles are perceived to be more environmentally friendly and cheaper tomaintain and run than comparable gasoline fueled vehicles. For example, the share of HVsales in total monthly car sales in Japan reached 12% in May, 2009. However, althoughinterest in HVs is now overwhelming, most experts predict that they are only a temporary
development that will ultimately be replaced by EVs (see [4]). Under these circumstances,an automaker faces an urgent problem: namely, what type of vehicle, a HV or an EV,should it seek to promote, and when should it undertake the large-scale market promotionof the vehicle chosen? For example, Toyota is currently undertaking the substantial sales
promotion of its HV, the Prius, while Nissan has announced that its EV, the Lief, will be
for sale from 2010.
This paper models the investment timing and project choice decision of an automakeras follows. The automaker has an option to promote two alternatives2 , namely the HVand the EV, with sunk costs at some arbitrary timing. If and only if the automakerchooses the HV does it have the option to change its promotion from the HV to the EV
with sunk costs. That is, the HV project is evaluated by taking into consideration thevalue of the option to replace the HV with the EV in the future. For brevity, we assume
the cash ﬂows from the HV and EV projects follow a bidimensional geometric Brownian
motion (GBM). Then, the investment problem becomes a problem combining a max-call1

In this paper, EV denotes a full electric vehicle, while HV denotes a hybrid vehicle that employs bothan engine and a battery. In turn, EVs are classiﬁed into several types depending on the type of battery.
Accordingly, the analysis considers not only fuel cells (which arguably have far to go before commercializationin vehicles) but also lithium–ion batteries, currently one of the most promising battery technologies owing to
its lightweight and power. See [4] for details.
2
It is assumed that the ﬁrm cannot promote both projects at the same time. Although some automakers
invest in both HV and EV technologies in the real world, most give emphasis to either HVs or EVs. For example,
Toyota emphasizes HVs while Nissan emphasizes EVs.

1

option (studied by [7, 3, 12]) with a spread option (studied by [3, 12, 8]).3 Indeed, thealternative project choice corresponds to the max-call option, while the replacement of
HV with EV corresponds to the spread option.
For the combined problem, we show the analytical properties of the project value
and the investment region. Along with a theoretical contribution to the literature on
American options on multiple assets, the results lead to several implications concerning
the automaker’s strategy. One notable ﬁnding is that the investment region for the HV
is not necessarily monotonic with respect to the market demand for EVs. This resultsfrom an interaction between the max-call option and the spread option embedded in the
problem. In terms of the max-call option, increased market demand for EVs discourages
the promotion of the HV because the EV becomes more favorable. Conversely, in termsof the spread option, increased market demand for EVs encourages the promotion of the
HV because the value of the spread option to replace the HV with the EV increases.
This tradeoﬀ determines the investment region for the HV. In particular, we ﬁnd that thelatter eﬀect (encouragement of the HV) can dominate the former eﬀect (discouragementof the HV) if the market demand for EVs is small and the correlation between the EV
and the HV is low or negative. This is consistent with the promotion strategies used byautomakers such as Toyota and Honda that emphasize the promotion of HVs despite theimminent dominance of EVs.
Although this paper is intended to better understand the decision by automakers on
investment timing and project choice, the use of this model is not restricted to a particularindustry. For example, the model also applies, say, to a developer’s decisions concerning
the renovation and rebuilding of condominiums. That is, the developer accounts for not
only the value of the renovated condominium but also the value of the option to rebuild
it in the near future.
The paper is organized as follows. Section 2 introduces the setup and the preliminary
results in two cases; the alternative choice between the HV and the EV (Section 2.1) andthe replacement of the HV with the EV (Section 2.2). Section 3.1 shows several propertiesof the value function and the stopping region for the combined problem. Section 3.2
presents numerical results and implications concerning the investment timing and projectchoice decision of an automaker. Section 4 concludes the paper.

3

Numerous studies propose new methods of computing the prices of American options on multiple assets,

including max-call and spread options. However, other than [7, 8], there are few studies concerning theirapplication to real options.

22Preliminaries

Consider an automaker that has an option to invest in a project i. Consider two kindsof projects i = 1 (HV) and i = 2 (EV). When the ﬁrm conducts project i at time t withsunk cost Ii (> 0), it receives a cash ﬂow Xi (t) for the ongoing project i. For analytical
purposes, this analysis builds on the continuous time model with exogenous cash ﬂows
X(t), which follows a bidimensional GBM
dXi (t) = µi Xi (t)dt + σi Xi (t)dBi (t),(1)

where (B1 (t), B2 (t)) is a two-dimensional Brownian Motion (BM) with correlation coeﬃcient ρ satisfying |ρ| < 1. The drift µi and the volatility σi (> 0) represent the mean growthrate and the volatility of the cash ﬂow from project i. As usual, we assume max(µ1 , µ2 ) < r
for convergence, where r(> 0) denotes the constant discount rate. Mathematically, themodel is built on the ﬁltered probability space (Ω, F, P ; Ft ) generated by (B1 (t), B2 (t)).
The set Ft is the available information set in time t, and a ﬁrm optimizes its investment
strategy under this information. Let T (> 0) denote the maturity of the option to choose
HV or EV. Although we can theoretically take T = ∞ (a perpetual option), we assume
recent circumstances oblige the automaker to make a short-term decision. Therefore, wetake a ﬁnite T , set to 1 year in the numerical example. In preparation for the main resultsin Section 3, we present two earlier results in this section.

2.1

Choice among HV and EV

As a benchmark, we consider the case of no replacement of the HV with the EV. The
automaker has an option to invest between the HV and the EV at an arbitrary time
before maturity T . However, the ﬁrm cannot execute both projects. The option value attime t(< T ) with X(t) = x ∈ R2++ is equal to the value function of the optimal stopping
problem as follows:
VA (x, t) = sup Ext [e−r(τ −t) maxτ ∈Tt,Tµ¶X2 (τ )
X1 (τ )
− I1 ,− I2 , 0 ],r − µ1r − µ2(2)where Tt,T denotes the set of all stopping times τ satisfying τ ∈ [t, T ] and Etx [·] is the
expectation conditional on X(t) = x. Throughout the paper, the subscript A denotes“Alternative choice.” Problem (2) has been essentially investigated in [7, 3] (also refer to
Section 6 in [6]).4 The optimal stopping time τA (t) in problem (2) becomes:τA (t) = inf{s ≥ t | X(s) ∈ SA (s) = SA,1 (s) ∪ SA,2 (s)},4(3)

In relation to problem (2), [5] investigated investment of diﬀerent scales under a one-dimensional statevariable. As an alternative, [11, 10] examined preemptive competition where two ﬁrms strategically preempt a
project between two alternative projects using one-dimensional and bidimensional models, respectively.

3

where the stopping region SA,i (s) is deﬁned by:
SA,i (s) = {x ∈ R2++ | VA (x, s) =xi
− Ii }.
r − µi(4)

If X(t) ∈ SA,i (t), the ﬁrm makes immediate investment in project i. Both the valuefunction VA (t) and the stopping region SA (t) cannot be derived in a closed form for thisbidimensional problem. However, for t < T , the following properties concerning VA (x, t)
and SA (t) hold (see Section 6 in [6]).
(Convexity of the value function) VA (x, t) is convex with respect to x ∈ R2++ .
(Convexity of each stopping region) SA,i (t) is a convex set.
(Monotonicity of each stopping region) x ∈ SA,1 (t) ⇒ x0 ∈ SA,1 (t) (∀x01 ≥ x1 , ∀x02 ∈(0, x2 ]). x ∈ SA,2 (t) ⇒ x0 ∈ SA,2 (t) (∀x01 ∈ (0, x1 ], ∀x02 ≥ x2 ).
(Behavior on the indiﬀerence line) x1 /(r − µ1 ) − I1 = x2 /(r − µ2 ) − I2 ⇒ x ∈/ SA (t).
The monotonicity of each stopping region ensures the intuition that a higher market
demand for HVs (resp. EVs) and a lower demand for EVs (resp. HVs) encourages thepromotion of the HV (resp. EV). The last property means that, in the situation wherethe value of the HV project is the same as that of the EV, the automaker waits and sees
which project is more promising. For other detailed properties, refer to [12, 6].

2.2

The replacement of the HV with the EV

Now, we consider that the automaker is promoting the HV and has an option to replace theHV with the EV. Assume that the replacement requires sunk costs IR . For consistency, we
also assume that IR > max(I2 −I1 , 0). Throughout the subscript R denotes “Replacementof the HV with the EV.” As mentioned earlier, maturity T for the option to initiate the HVor the EV project is reasonably considered as a short-term decision, while the replacementof the HV with the EV may take place in the longer term. We then consider the inﬁnite
maturity for the replacement option.
The option value at time t with X(t) = x ∈ R2++ is equal to the value function of thetime-homogeneous optimal stopping problem5 as follows:µ¶
X2 (τ )
X1 (τ )x −r(τ −t)
VR (x) = sup Et [e−− IR ].
r − µ2 r − µ1
τ ∈Tt,∞(5)

The project value including the ongoing HV project at time t becomes:
X1 (t)r−µ| {z 1}+ VR (X(t)) .
| {z }(6)

option value

perpetual HV

Problem (5) is essentially the same as the value function of an American spread option(refer to [9, 6]). In terms of real options, [8] used a similar problem to ﬁnd the optimal
5

Assume that in (5) the payoﬀ is zero in the case of τ = ∞.

4

timing in changing the method of nuclear waste disposal. According to [8], the optimal
stopping time τR (t) in problem (5) allows a closed form:τR (t) = inf{s ≥ t | SR = {X2 (s) ≥ c1 X1 (s) + c2 }},(7)

where c1 (> 0), c2 (> IR ) are constants derived in a closed form (for details, see [8]).
Although the value function VR (x) cannot be derived in a closed form, the following
properties are known. For other detailed properties, refer to [12, 6].
(Convexity of the value function) VR (x) is convex with respect to x ∈ R2++ .
Furthermore, we provide the following lemmas concerning VR (x) for the next section.
Lemma 1VR (x0 ) − VR (x) ≤x1 − x01 x02 − x2+r − µ1r − µ2(∀x01 ∈ (0, x1 ], ∀x02 ≥ x2 ).

Proof First, note that for any x ∈ R2++∂VR−1≤(x) < 0r − µ1
∂x1

(8)

and
1
∂VR
(x) ≤
,
(9)
∂x2
r − µ2where the equalities in (8) and (9) hold if and only if x ∈ SR (t). These are readily proved0<by the diﬀerentiability (refer to [3]) and convexity of VR (x) and VR (x) = x2 /(r − µ2 ) −x1 /(r − µ1 ) − I1 (x ∈ SR (t)). By (8), (9), and the convexity of VR (x, t), we have for any
x01 ≤ x1 and x02 ≥ x2 :
∂VR 0
∂VR 0
(x )(x1 − x01 ) +(x )(x2 − x02 )∂x1∂x2
x1 − x01 x02 − x2−,
≥ VR (x0 ) −r − µ1r − µ2VR (x) ≥ VR (x0 ) +¤where the last inequality completes the proof.

Lemma 2 Fix any x ∈/ SR (t). There exist constants a1 ∈ (0, 1/(r − µ1 )), a2 ∈ (0, 1/(r −µ2 )), and a3 ≥ I1 (which may depend on x) such thatx01
+ VR (x0 ) − I1 ≥ a1 x01 + a2 x02 − a3r − µ1(∀x0 ∈ R++ ),where the equality holds when x0 = x.
Proof By the convexity of VR (x), we have for any x0 ∈ R2++ :x01+ VR (x0 ) − I1
r − µ1
x01
∂VR
∂VR
≥
+ VR (x) +(x)(x01 − x1 ) +(x)(x02 − x2 ) − I1
r − µ1
∂x1
∂x2
µ
¶
1
∂VR
∂VR
∂VR
∂VR
=
+
(x) x01 +(x) x02 + VR (x) −(x)x1 −
(x)x2 − I1 .
r − µ1
∂x1
∂x2
∂x1
∂x2
| {z }
|
{z
}
|
{z
}
a1

−a3

a2

5

By (8), (9), and x ∈/ SR (t), we have a1 ∈ (0, 1/(r − µ1 )) and a2 ∈ (0, 1/(r − µ1 )).
Considering the limit x01 ↓ 0, x02 ↓ 0, we have a3 ≥ I1 .

¤Note that in Lemma 2 the right-hand side is the ﬁrst order Taylor approximation to
the left-hand function for x0 near a ﬁxed point x.

3

Main Results

This section combines Sections 2.1 and 2.2. The automaker initiates the promotion ofeither the HV or the EV before maturity T . If and only if the automaker chooses the HVdoes it have an option to change its promotions from the HV to the EV in future. Before
the initiation of any promotion, the option value at time t(< T ) with X(t) = x ∈ R2++ isequal to the value function of the optimal stopping problem: X (τ )
X2 (τ )
 1
+ VR (X(τ )) −I1 ,− I2 , 0].
VAR (x, t) = sup Ext [e−r(τ −t) max 
r
−
µ
r
−
µ


1
2
τ ∈Tt,T
|
{z
}

(10)

equation (6)Throughout the paper, the subscript AR denotes “Alternative choice including the replacement option.” The optimal stopping time τAR (t) in problem (10) then becomes:τAR (t) = inf{s ≥ t | X(s) ∈ SAR (s) = SAR,1 (s) ∪ SAR,2 (s)},(11)

where each stopping region SAR,i (s) is deﬁned by:
SAR,1 (s) = {x ∈ R2++ | VAR (x, s) =x1
+ VR (x) − I1 }.
r − µ1(12)x2− I2 }.
r − µ2(13)

andSAR,2 (s) = {x ∈ R2++ | VAR (x, s) =Problem (10) diﬀers from problem (2) in that the value of the HV includes the spread
option value VR (X(τ )). This paper focuses on the eﬀects of the spread option to replace
the HV with the EV.

3.1

Theoretical results

The following proposition provides the properties of the value function VAR (x, t) and thestopping region SAR (t) in problem (10) for t < T .
Proposition 1(Convexity of the value function) VAR (x, t) is convex with respect to x ∈ R2++ .
(Convexity of the stopping region for EV) SAR,2 (t) is a convex set.
(Monotonicity of the stopping region for EV) x ∈ SAR,2 (t) ⇒ x0 ∈ SAR,2 (t) (∀x01 ∈6

(0, x1 ], ∀x02 ≥ x2 ).
(Behavior on the indiﬀerence curve) x1 /(r − µ1 ) + VR (x) − I1 = x2 /(r − µ2 ) − I2⇒x∈
/ SAR (t).
(Comparison with the case of no replacement) VAR (x, t) ≥ VA (x, t), SAR,1 (t) ⊃SA,1 (t), and SAR,2 (t) ⊂ SA,2 (t).

Proof For simplicity, we denote the payoﬀ function of the combined option by
µ
¶
x1
x2
f (x) := max+ VR (x) − I1 ,− I2 , 0 .
r − µ1r − µ2(Convexity of the value function) By the convexity of VR (x), the payoﬀ function
f (x) is also convex. Because the payoﬀ function is convex, the value function VAR (x, t) isconvex with respect to x ∈ R2++ (by Proposition A.6 in [3], or equivalently, Proposition
88 in [6]).
(Convexity of the stopping region for EV) Take any λ ∈ (0, 1), x ∈ SAR,2 (t), andy ∈ SAR,2 (t). By the convexity of VAR (x, t) with respect to x ∈ R2++ , we have
VAR (λx + (1 − λ)y, t) ≤ λVAR (x, t) + (1 − λ)VAR (y, t)
¶
µ
¶
µ
y2
x2
− I2 + (1 − λ)− I2
= λr − µ2r − µ2λx2 + (1 − λ)y2
− I2 ,=
r − µ2where the last inequality implies λx + (1 − λ)y ∈ SAR,2 (t), i.e., the convexity of the
stopping region SAR,2 (t).
(Monotonicity of the stopping region for EV) Take any x ∈ R2++ , x01 ∈ (0, x1 ], andx02 ≥ x2 . By Lemma 1, we have:x01
x1
x0 − x2+ VR (x0 ) − I1 ≤+ VR (x) − I1 + 2.
r − µ1r − µ1r − µ2(14)

By (14) we have:¶x01
x2
x02 − x2
0
f (x ) = max+ VR (x ) − I1 ,− I2 +,0r − µ1r − µ2r − µ2µ¶x1x2x0 − x2≤ max+ VR (x) − I1 ,− I2 , 0 + 2
r − µ1r − µ2r − µ2|{z}0µ=f (x)

x0 − x2= f (x) + 2.
r − µ2(15)

7

Then, for x ∈ SAR,2 (t), we have:
VAR (x0 , t) =
≤
≤

(1,1)

sup Etτ ∈Tt,Tsupτ ∈Tt,T[e−r(τ −t) f (x01 X1 (τ ), x02 X2 (τ ))](1,1)
Et [e−r(τ −t)(1,1)

sup Etτ ∈Tt,T= VAR (x, t) +=¶(x02 − x2 )X2 (τ )]f (x1 X1 (τ ), x2 X2 (τ )) +r − µ2[e−r(τ −t) f (x1 X1 (τ ), x2 X2 (τ ))] + sup Et

(1,1)

|

=

µ

{z

}

τ ∈Tt,T[e−r(τ −t)(16)(x02 − x2 )X2 (τ )]r − µ2=VAR (x,t)
0(x2 − x2 )r − µ2(x02 − x2 )x2+
r − µ2r − µ20x2,r − µ2(17)(18)

where (16) and (17) follow from (15) and x ∈ SAR,2 (t), respectively. The last inequality(18) implies x0 ∈ SAR,2 (t), and hence we have x ∈ SAR,2 (t) ⇒ x0 ∈ SAR,2 (t) (∀x01 ∈(0, x1 ], ∀x02 ≥ x2 ).6(Behavior on the indiﬀerence curve) Take any x ∈ R2++ satisfying x1 /(r − µ1 ) +VR (x) − I1 = x2 /(r − µ2 ) − I2 . Note that x ∈/ SR (t) because of the assumption IR >max(I2 − I1 , 0). Using the constants a1 , a2 , and a3 in Lemma 2, we have for any s ∈ (t, T ]:VAR (x, t)¶
X2 (τ )
a1 X1 (τ ) + a2 X2 (τ ) + a3 ,− I2 , 0 ]≥ sup
r − µ2τ ∈Tt,T¶µX2 (s)−r(s−t) x− I2 ]≥ e
Et [max a1 X1 (s) + a2 X2 (s) + a3 ,r − µ2X2 (s)≥ e−r(s−t) Ext [− I2 ]
r − µ2|{z}
Ext [e−r(τ −t) maxµ↑x2 /(r−µ2 )−I2 (s↓t)+ e−r(s−t) Ext [max|µµa1 X1 (s) −¶¶1− a2 X2 (s) + a3 − I2 , 0 ] .
r − µ2{z}(19)↓0 (s↓t)In the right-hand side of (19), the ﬁrst term ↑ x2 /(r − µ2 ) − I2 (s ↓ t) at a ﬁnite rate while
the second term7 ↓ 0 (s ↓ t) at a rate that increases to inﬁnity in the limit (refer to Lemma
B.1 in [3], or equivalently, Lemma 91 in [6]). Therefore, there exists some s ∈ (t, T ] suchthat the right-hand side of (19) is strictly larger than x2 /(r − µ2 ) − I2 . This implies that
VAR (x, t) > x2 /(r − µ2 ) − I2 = x1 /(r − µ1 ) + VR (x) − I1 , i.e., x ∈/ SAR (x, t).
(Comparison with the case of no replacement) The inequality VAR (x, t) ≥ VA (x, t)
6
7

We used a similar method of the proof of Proposition A.3 in [3], or equivalently, Proposition 85 in [6].
This is equal to the value of the European spread option with maturity s.

8

is clear. We have for any x ∈ SA,1 (t):VAR (x, t)¶¶
X2 (τ )
X1 (τ )≤ sup
VR (X(τ )) + max− I1 ,− I2 , 0 ]r − µ1r − µ2τ ∈Tt,Tµ¶
X1 (τ )
X2 (τ )x −r(τ −t)x −r(τ −t)≤ sup Et [eVR (X(τ ))] + sup Et [emax− I1 ,− I2 , 0 ]r − µ1r − µ2τ ∈Tt,Tτ ∈Tt,T|{z}
Ext [e−r(τ −t)µµ=VA (x,t)= VR (x) + VA (x, t)
x1= VR (x) +− I1 ,
r − µ1
where the last inequality implies x ∈ SAR,1 (t). We have for any x ∈ SAR,2 (t):VA (x, t) ≤ VAR (x, t)
x2=− I2 ,
r − µ2where the last inequality implies x ∈ SA,2 (t).

¤Proposition 1 extends previous ﬁndings by [7, 3, 1] allowing only a linear function toa case allowing a nonlinear function VR (x). The diﬀerence from problem (2) in Section
2.1 is that the investment region for the HV, SAR,1 (t), does not necessarily satisfy either
convexity or monotonicity. The monotonicity of SAR,2 (t) brings about the straightforward
fact that an increased demand for EVs and a decreased demand for HVs accelerates theautomaker’s investment in the EV. However, this monotonicity does not necessarily holdfor the HV because of the spread option to replace the HV with the EV. Indeed, as will
be shown in the numerical example, increased demand for the EV may encourage thepromotion of the HV because the value of the spread option increases. Note that SAR,1 (t)includes SA,1 (t), which has both monotonicity and convexity (see the ﬁnal property in
Proposition 1). On the other hand, the behavior on the indiﬀerence curve is inheritedfrom the option without replacement. The automaker delays the decision on project choice
when the market demand X(t) lies on the curve where both project values are equal. The
property of SAR,1 (t) ⊃ SA,1 (t) and SAR,2 (t) ⊂ SA,2 (t) supports the natural intuition thatthe potential replacement of the HV with the EV encourages (resp. discourages) the HV(resp. EV).

3.2

Numerical results and implications

This subsection provides a numerical example with some implications. As reported in [4],it is diﬃcult to forecast the future market share of HVs and EVs. What appears to becertain is that EVs have more potential and more volatile than HVs in the future, but
at present the market demand for EVs is much lower. Considering this, we set the baseparameter values as: µ1 = 1%, σ1 = 20% for the HV, µ2 = 5%, σ2 = 40% for the EV, and9the discount rate r = 8%. The sunk costs Ii (i = 1, 2) are not essential because they are
adjustable by X(t) = x. We set I1 = 100 for the HV, I2 = 100 for the EV, and IR = 100
for the replacement. The maturity of the option is reasonably set as T − t = 1 year. In
the numerical procedure, we make a discretization with 200 time steps per 1 year, and use
a bivariate version of the lattice binomial method (see [2]). We compute the perpetualspread option VR (x) using a value iteration algorithm in the ﬁrst step, and using thecomputed VR (x) compute the max-call option VAR (x, t) backward from maturity T using
a dynamic programming algorithm.
Figure 18 illustrates the investment region SAR (t) along with SA (t) and SR (t). In
Figure 1, the investment region for the HV, SAR,1 (t), does not satisfy monotonicity whenthe market demand for EVs, x2 , is small (see X(t) = x ≈ (9, 2)). This ﬁnding leads to thefollowing implication. In the current circumstance, where the market demand for EVs ismuch lower than that of HVs, an increase in the market demand for EVs (or equivalently,
a technical innovation in the EV) can also accelerate the promotion of HVs because of theeﬀect of the spread option to replace the HV with the EV. This supports the promotion
strategies of automakers such as Toyota and Honda that emphasize the promotion of HVs
as a temporary measure before the introduction of EVs.9 In general, whether an increasein the demand for EVs encourages HVs or not is determined by the tradeoﬀ between
the max-call option (discouragement of HVs) and the spread option (encouragement of
HVs). According to computations using a wide range of parameter values, a larger gapµ2 − µ1 and lower replacement costs IR (> I2 − I1 ) make the eﬀect of the spread option(encouragement of HVs) dominant for a small x2 . In Figure 1, SAR,1 (t) does not satisfy
convexity. This results from the nonlinearity of the indiﬀerence curve.
Another important feature of SAR,1 (t) is the sensitivity with respect to the correlationcoeﬃcient ρ. Figure 2 depicts the investment regions SAR (t) with ρ = −0.5, 0, and0.5. It has been numerically veriﬁed in [7, 6] that SA,i (t) grows monotonically with ρ.
This is because a higher ρ decreases the value of the option to postpone the projectchoice and therefore hastens the investment in each project. In contrast, in Figure 2, theinvestment region SAR,1 (t) does not present monotonicity. In the combined problem, ahigher ρ decreases not only the value of the option to postpone the decision, but also thevalue of the spread option embedded in the HV project. The latter eﬀect of a higher ρdecreases the project value of the HV and then delays its promotion. Indeed, in the current
circumstance where X(t) = x ≈ (9, 2) in Figure 2, a lower ρ encourages the promotionof the HV. Given the prospect that HVs will be replaced with EVs (ρ is negative), an
8

Technically, we compute the lattice model with 400 time steps for maturity T = 2 year, and Figure 1 shows

the investment regions SAR (t), SA (t), and SR (t) for t = 1 year.
9
Of course, Nissan’s emphasis on EVs is not necessarily criticized if one considers strategic competition
against Toyota and Honda, carmakers that have taken a lead in HV technologies. This strategic interaction is
one of several important issues to be addressed in future work.

10automaker then accelerates the investment in the HV. This also supports the rapid spreadof the promotion of HVs in present circumstances. Lastly, we remark that VAR (x, t) is
about 1.5 times higher than VA (x, t) for X(t) = x ≈ (9, 2) due to the spread option,
though the paper focuses on the investment strategy rather than the value.

4

Conclusion

From a real options perspective, this paper investigated an automaker’s decisions concerning investment timing and project choice between the HV and EV. We modelled the
problem as the max-call option including the spread option to replace the HV with the
EV. We showed the analytical properties of the project value and the investment region.
A notable diﬀerence from the case of no replacement option is that the increased market
demand for EVs may accelerate the promotion of not only the EV but also the HV because of the replacement option. Especially, the encouragement of the HV is predicted in
the current circumstance where the market demand for EVs is much smaller than that of
HVs and the correlation between the EV and the HV is low or negative. This supports
the promotion strategies used by automakers such as Toyota and Honda that emphasize
the promotion of HVs despite the imminent dominance of EVs.

References[1] C. Bobtcheﬀ and S. Villeneuve. Irreversible investment in competitive projects: A
new motive for waiting to invest. Working Paper, Université de Toulouse 1, 2005.
[2] P. Boyle. A lattice framework for option pricing with two state variables. Journal of
Financial and Quantitative Analysis, 23:1–12, 1988.
[3] M. Broadie and J. Detemple. The valuation of American options on multiple assets.
Mathematical Finance, 7:241–286, 1997.
[4] J. Chanaron and J. Teske. Hybrid vehcles: a temoporary step. International Journalof Automotive Technology and Management, 7:268–288, 2007.
[5] J. Décamps, T. Mariotti, and S. Villeneuve. Irreversible investment in alternative
projects. Economic Theory, 28:425–448, 2006.
[6] J. Detemple. American-Style Derivatives valuation and computation. Chapman &Hall/CRC, London, 2006.
[7] D. Geltner, T. Riddiough, and S. Stojanovic. Insights on the eﬀect of land use
choice: The perpetual option on the best of two underlying assets. Journal of Urban
Economics, 39:20–50, 1996.

11

∂ SR

∂ SA(t)∂ SAR(t)

50
45
40
35

SR
SAR,2(t)

x2

30
25

SA,2(t)20

SAR,1(t)15
10 Non−monotonic

SA,1(t)

5
5

10

15

20

25
x1

30

35

40

45

50

Figure 1: The boundaries of the investment regions SAR (t), SA (t), and SR . The parametervalues are set as µ1 = 1%, σ1 = 20%, µ2 = 5%, σ2 = 40%, r = 8%, I1 = I2 = IR = 100, ρ = 0%.
ρ=−50%

ρ=0%

ρ=50%

50
45
40
SAR,2(t)

35

x2

30
25
SAR,1(t)

20
15
10
5
5

10

15

20

25
x1

30

35

40

45

50

Figure 2: The boundaries of the investment regions of SAR (t) and SA (t) for correlation coefﬁcient ρ = −50%, 0%, and 50%. The other parameter values are the same as those in Figure1.
12[8] H. Loubergé, S. Villeneuve, and M. Chesneyc. Long-term risk management of nuclear
waste: a real options approach. Journal of Economic Dynamics & Control, 27:157–
180, 2002.
[9] W. Margrabe. The value of an option to exchange one asset for another. Journal of
Finance, 33:177–186, 1978.
[10] M. Nishihara. Preemptive investment game with alternative projects. Discussion
Papers in Economics and Business, Osaka University 09-16, 2009.
[11] M. Nishihara and A. Ohyama. R&D competition in alternative technologies: A real
options approach. Journal of the Operations Research Society of Japan, 51:55–80,
2008.
[12] S. Villeneuve. Exercise regions of American options on several assets. Finance andStochastics, 3:295–322, 1999.

13