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Irreversible Investments and Ambiguity

Aversion

Sebastian Jaimungal1

November 18, 2011

1Department of Statistics and Mathematical Finance Program, University of Toronto, 100 St.

George Street, Toronto, Ontario, M5S 3G3, Canada. Email:sebastian.jaimungal@utoronto.ca

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Abstract

Real-option valuation traditionally is concerned with investment under conditions of project-

value uncertainty, while assuming that the agent has perfect confidence in a specific model.

However, agents generally do not have perfect confidence in their models, and thisambiguity

affects their decisions. Moreover, real investments are not spanned by tradable assets andgenerate inherently incomplete markets. In this work, we account for an agents aversion to

model ambiguity and address market incompleteness through the notation ofrobust indiffer-

ence prices. We derive analytical results for the perpetual option to invest and the linear

complementarity problem that the finite time problem satisfies. We find that ambiguity aver-

sion has dual effects that are similar to, but distinct from, those of risk aversion. In particular,

agents are found to exercise options earlier or later than their ambiguity-neutral counterparts,

depending on whether the ambiguity stems from uncertainty in the investment or in a hedgingasset.

Key-words: Real Options; Ambiguity Aversion; Risk Aversion; Robust Optimal Control;

Indifference Pricing

JEL Codes: D81, G31, G11

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1 Introduction

Quantitative methods to analyze the option to invest in a project enjoy a long and distin-

guished history. The classical work of McDonald and Siegel (1986) (see also Dixit and Pindyck

(1994)) investigates the problem from the perspective of derivative pricing, and assigns the

following value to the option to invest irreversibly:

value = supT

erE [(P I)+] . (1)

In this relationship, the expected value is taken under an appropriate risk-adjusted measure,

Iis the cost of investing in the project,Ptis the value of the project at timet, and T denotes

the family of allowed stopping times in [0, T]. In the European case, the agent may invest in

the project only at maturity; in the Bermudan case, the agent may invest at a set of specific

times (e.g., monthly); and in the American case, the agent may invest at any time. As such,

the problem is generally a free-boundary problem, in which the optimal strategy is computed

simultaneously with the options value.

Traditionally, the project value is assumed to be a geometric Brownian motion (GBM)

and the investment amount is constant or deterministic, as in the pioneering work of Tourinho

(1979). Moreover, the bulk of the real-options literature assumes that the value attained by

investing is tradable (or at least completely spanned by a traded asset) and the agents are

risk-neutral. Clearly, these assumptions are violated in all but the most simplistic real-world

scenarios. For example, consider the case of a pharmaceutical company that is contemplating

whether to acquire rights to a new chemical process. With these rights, the firm can choose

when to invest according to the prevalent market conditions: they may develop the new pro-

cess now, or may delay and decide later whether to develop the process. Although the future

option may have significant value, this future value is not spanned by traded assets in the

financial market. Furthermore, the firm may be averse to the idiosyncratic risks embedded

in the additional cash-flows generated by implementing the process. The procedure of simply

applying a zero-risk premium to the unspanned risk (as is standard in the real-options litera-

ture) is undesirable, and such a choice will not reflect the firms decision process. Moreover,

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the firm may not be fully confident in any model of the cash-flows generated by making the

investment. In other words, the firm may be subject to model ambiguity.

Several recent works investigate the role of risk aversion in real-option valuation whenmarkets are incomplete. Henderson (2007) investigates how an agents risk aversion affects

the valuation of perpetual real options when the project value is only partially spanned by a

tradable asset. Grasselli (2011) applies the same approach, but considers instead the finite

horizon problem and develops a tree procedure for its valuation. Both works utilize the

concept of utility indifference for valuing the idiosyncratic risk and accounting for the agents

level of risk aversion. In particular, both studies find that increasing the agents level of risk

aversion induces the agent to invest earlier.Henderson (2007) finds that the parameter regime under which the agent invests early

is expanded relative to two benchmark cases: (i) when the market price of risk associated

with the unhedgable risk is zero, and (ii) when the project value is completely spanned by the

traded asset. Hugonnier and Morellec (2007) investigates a similar problem, incorporating the

effect of a control challenge from stake holders, if the managers policy erodes the firm value.

The authors conclude, similar to Henderson (2007) and Grasselli (2011), that increasing risk

aversion induces earlier investment. Miao and Wang (2007) study the perpetual investmentproblem from a novel angle. Rather than maximizing the agents utility of wealth, these

authors investigate the role of optimal consumption/savings and distinguish between lump-

sum and perpetual cash-flow streams. Their results agree with those of Henderson (2007) and

Grasselli (2011) when the investment provides a lump-sum payment, but their findings are

reversed when investing provides a cash-flow stream rather than a lump-sum payment.

In this work, we consider both the perpetual and finite-time American versions of the lump-

sum payment problem in an incomplete market setting. Moreover, our study also incorporates

ambiguity aversion, and distinguishes between ambiguities in a hedging asset and in the

project value. Our work is distinct from Nishimura and Ozaki (2007), Trojanowska and

Kort (2010), Roubaud, Lapied, and Kast (2010), and Miao and Wang (2011), in which the

resulting valuation effectively is reduced to considering the worst-case scenario out of all of the

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agents possible model choices. Furthermore, although these authors account for the agents

ambiguity aversion, they do not account for the agents risk aversion or distinguish between

ambiguities in the traded asset and in the project value.We develop an approach to consider both risk aversion and distinct forms of ambiguity

aversion, utilizing the concept ofrobust indifference pricing. This concept was first introduced

by Jaimungal and Sigloch (2010) in the context of credit markets. Our approach assumes that

the agent has a reference measure that is believed to be close to the truth, but the agent is

willing to consider a set of candidate measures as also possible. This consideration allows

the agent to probe other potential models, and the agent penalizes the measures according

to a scaled relative entropy. Our approach borrows ideas from the robust control approach(developed in Anderson, Hansen, and Sargent (1999), Uppal and Wang (2003), and Maenhout

(2004)), which is an alternative1 to the multiple priors approach introduced by Gilboa and

Schmeidler (1989) in a static setting, developed by Epstein and Wang (1994) in discrete time,

and axiomatized in Epstein and Schneider (2003). Here, however, we account for market

incompleteness.

Our results show that the effect of ambiguity aversion is similar to, but quite distinct

from, risk aversion, and plays a crucial role in determining exercise policies and the value ofthe option to invest. One of our key findings is that increasing ambiguity does not always

induce the agent to invest earlier-in contrast to the results of Nishimura and Ozaki (2007),

Trojanowska and Kort (2010), Roubaud, Lapied, and Kast (2010), and Miao and Wang

(2011), who find that increasing ambiguity always accelerates investment for the lump-sum

case. Instead, we find that, as the agent becomes more averse to ambiguity in the invesment

asset, (s)he will delay investment; in contrast, as the agent becomes more averse to ambiguity

in the project value, (s)he will invest earlier. The basic intuition for the first result is that,

as ambiguity in the traded asset increases, the agent loses the ability to hedge properly. The

additional uncertainty forces the agent to wait until (s)he can lock in a significant profit. The

1Hansen, Sargent, Turmuhambetova, and Williams (2001) shows that both formulations of ambiguity

aversion are related through a Legendre transform.

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second result carries the intuition that, as the project value becomes more uncertain, the

agent prefers to lock in a smaller profit earlier, in lieu of waiting for larger gains in the future,

because the probability of those gains is ambiguous.As is well known in standard-indifference valuation, the limiting case of a risk-neutral

agent values cash flows under the minimal entropy martingale measure (MEMM) (see, e.g.,

Rouge and El Karoui (2000)). When ambiguity aversion is included, we demonstrate that

this MEMM is distorted to account for the agents aversion to ambiguity. This distortion

is responsible for the economic results mentioned above. Moreover, in solving the robust

control problem, we show that the optimal measure under which the agent computes expected

utility contains a drift adjustment, so as to produce mean-reverting project values, despite thereference measure being a GBM. Surprisingly, however, when the agent computes expected

utility under this measure, the result is identical to a risk-averse but ambiguity-neutral agent,

assuming that the project value is a GBM, but with an ambiguity-adjusted drift for the

project-value.

2 Problem Setup

2.1 An Incomplete Market Model

In this section, we consider an agent who is faced with the option to invest (irreversibly)

in a project over a finite lifetime and who, upon investment, receives a random lump-sum

payment. For simplicity2, the projects value Pt is assumed to be a GBM:

dPt= Pt dt+ Pt dWt. (2)

2

It is not difficult to extend the model to include other diffusion processes, such as mean-reverting projectvalues; however, it will be more difficult to interpret the results, because the remaining model parameters

(e.g. the mean reversion rate and level), risk aversion, and ambiguity aversion will interact in a non-trivial

manner.

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The agent cannot, in general, trade this project value; however, we assume that the agent can

trade in a strongly correlated (hedging) asset denoted St, also modeled as a GBM:

dSt= St dt+ St dBt. (3)

In equations (2) and (3),Wtand Btdenote two correlatedP standard Brownian motions with

correlation . Here, P is considered to be a reference measure3, which reflects the agents

belief about the market and investment dynamics. This measure may or may not be the

historical one.

Although the agent is unable to replicate the option perfectly4, (s)he can partially hedge

away risk by trading in the correlated asset. As such, the real option to invest can be viewed asan American option on a nontraded asset. This unhedgeable risk represents the idiosyncratic

risk inherent in the project. Because the market is incomplete, the value of this idiosyncratic

risk is not unique. Rather, the agents utility function will play a key role in deciding how

the risk is priced and in setting the agents optimal strategy.

There are two types of risk that an agent may face: (i) known odds (risk) and (ii) ambigu-

ous odds (uncertainty), as first categorized by Knight (1921) and highlighted in the famous

Ellsberg (1961) paradox. The concept of known odds is what is typically referred to whendiscussing risk; it refers to the fact that, although the outcome is unknown, the probability

of the outcome is known. The concept of ambiguous odds pertains to the fact that an

agent may not be confident in the probabilities associated with the outcomes, or even in

the outcomes themselves. In this study, we will only deal with the case of ambiguity in the

probabilities. In the current real options context, an agent may be highly confident in his/her

model for the traded asset St, but not very confident in the model for the project value Pt.

This scenario is particularly true for projects embedded in irreversible investments.3As usual, we work on a completed filtered probability space (,F,P), where F = {(Ft)0tT}, Ft =

((Wt, Bt)0tT) is the sigma-algebra generated by the driving Brownian motions, and P is a given reference

measure. In practice, the reference measure would be attained by calibrating to market data.4If = 1, then the risk is spanned by the traded asset and perfect hedging is, in fact, possible. Our results

will cover this case as well, but it is most interesting to study the partially spanned case.

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An agents aversion to risk can be accounted for through a utility function via utility

indifference pricing, along the lines of Henderson (2007) (for the perpetual case) and Grasselli

(2011) (for the finite-horizon case). Although these works do not account for the agentsambiguity aversion, several recent approaches do (e.g., Nishimura and Ozaki (2007), Tro-

janowska and Kort (2010), Roubaud, Lapied, and Kast (2010), and Miao and Wang (2011)).

These latter approaches all involve choosing, from among all of the choices that the agent is

willing to consider, the drift of the underlying source of uncertainty that is the worst case.

Despite accounting for the agents ambiguity aversion, these frameworks do not account for

the agents risk aversion.

In this work, we develop an approach that allows us to consider simultaneously bothrisk aversion and ambiguity aversion. Our approach is consistent with, but distinct from,

the robust portfolio optimization of Anderson, Hansen, and Sargent (1999), Uppal and Wang

(2003), and Maenhout (2004). Specifically we develop a robust indifference pricing framework,

as Jaimungal and Sigloch (2010) does in the context of a credit model. In this way, we account

for risk and ambiguity aversion in the agents behavior. In the next section, we provide more

details on the mathematical formulation of the problem.

2.2 Robust Investment Problem

To value the option to invest in the project, we invoke the concept of certainty equivalence (or

indifference pricing), which requires us to solve two optimal investment problems: those in

the absence or presence of the option to invest. However, to account for ambiguity aversion,

we allow the agent to consider candidate measuresQ in the set of candidate measures Q

(see Appendix A), the elements of which are equivalent to the reference measureP. Further-

more, the agent is assumed to have preferences invoked by the robust optimization problem

(Anderson, Hansen, and Sargent (1999), Uppal and Wang (2003), and Maenhout (2004))

U(x,P,S) = supA

infQ

EQ

x,P,S

u(XT) +

1

h(Q|P)

. (4)

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The function u(x) is concave and represents the agents utility; function h penalizes candi-

date measures Q, which are very far from the reference measure P; and > 0 acts as the

penalization strength.A popular choice for the penalty functionh (e.g., Anderson, Hansen, and Sargent (1999))

is the entropic penalty function h(Q|P) = E

dQdP

ln dQdP

. As 0, the candidate measure is

pinned to the reference measure, and the robust portfolio optimization problem is reduced to

the usual portfolio optimization problem. As +, all candidate measures are considered

equally viable, and the agent acts as if the worst-case scenario prevails. Consequently, acts

as a measure of the agents level of ambiguity aversion and interpolates between the classical

portfolio optimization problem and the worst-case scenario.The robust optimization problem with entropic penalty is not solvable in general. How-

ever, Maenhout (2004) suggests a modification of the related HJB equation, which leads to

tractable solutions for the complete market case and shows that ambiguity aversion can be

absorbed by modifying the agents risk aversion. In an incomplete market model for credit

risk, Jaimungal and Sigloch (2010) introduces a robust optimization problem, which amounts

to a modification of the HJB equation artificially imposed by Maenhout (2004). The au-

thors further demonstrate that ambiguity aversion and risk aversion are quite distinct: it isthe presence of a nontraded index, similar to the project value model considered here, that

induces the distinction.

In the current setting of irreversible investment, the agent will be investing during two

distinct periods: (i) the pre-exercise period, during which the agent is exposed to risk in the

project through the option to invest or delay; and (ii) the post-exercise period, during which

the agent is exposed only to risk in the traded asset. As such, we denote the agents robust

value function5 pre-exercisebyU(t,x,P,S) and the agents robust value function post-exercise

byV(t,x,P,S).

Motivated by the robust optimization problem introduced by Jaimungal and Sigloch

(2010), we define the robust optimal control/stopping problem as follows.

5Here and in the sequel, x tracks the agents discounted wealth.

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Definition 1 Robust Value Function. The agents value function is given by the robust

optimal control/stopping problem:

U(t,x,P,S) = supTt

supA

infQQ

EQ

t,x,P,S

V( , X + (P I)+, P, S)

t

U(u, Xu , Pu, Su)vu

1vu du

,

(5)

where,= T andV(t,x,P,S) = sup

AinfQQ

EQ

t,x,P,S

1

e x

Tt

V(u, Xu , Pu, Su)vu

1vu du

. (6)

The appropriate set of admissible strategiesA, stopping timesT, and measuresQ are defined

in Appendix A.

dQ

dP= exp

1

2

T0

vu1vu du+

T0

u dBu+

T0

u dWu

, (7)

so that the project value and the traded asset satisfy the SDEs

dPt= (+ t) Pt dt+ Pt dWQ

t , and (8)

dSt= ( + t) St dt+ St dBQt , (9)

where WQt and BQt areQ-Brownian motions with correlation . Furthermore, the agents

discounted wealthXt under an admissible strategyt satisfies the SDE

dXt = ( + t r)t dt+ t dWQ

t , (10)

implied by the self-financing condition.

Note that the value functions can be visualized as in Figure 1. Once the agent exercises

an option, the agents wealth increases, (s)he is no longer exposed to the options risk, and

the agents value function is reduced to V. The penalty terms6 given by the integrals in (5)

6The sign flip on the penalty term is due to our use of the exponential utility function: u(x) = 1

ex .

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Time

ProjectValue

PreExercise Value Function U(t, Xt, P

t)

Exercise Value FunctionV(t, X

+ (P

I)

+)

MaturityExerciseTime ( )

ExerciseBoundary

Figure 1: Value functions in (5) and (6) are defined over the pre- and post-exercise regions.

Upon exercise, wealth increases by the intrinsic value of the option, and the preexercise value

function is reduced to the postexercise value function at exercise.

and (6) represent scaled versions of the entropic penalty7, as in Maenhout (2004). However,

here we introduce it directly in the definition of the value, rather than making an ad hoc

modification of the HJB equation. Moreover, the value function is defined recursively8, and

the entropic penalty is rescaled by the agents current utility, so that entropy is converted

to units of utility. The matrix is an ambiguity matrix, as introduced by Uppal and Wang

(2003), to account for varied levels of ambiguity aversion across subclusters of assets. In our

case with a traded and nontraded asset, the most general form is

= 1

SP1 +

1

S

12 00 0

+ 1P

0 00 1

2

. (11)In this manner, SP represents ambiguity on the joint distribution ofSandP,P represents

ambiguity on the marginal distribution ofS, and Srepresents ambiguity on the marginal

distribution ofP.

7It is not difficult to verify that, if under the measure Q, the processes (St, Pt) have drifts (+t, +t),

then the relative entropy EdQdP

ln dQdP

= 1

2EQ

T0

vs1vsds

. This relationship justifies our calling the

penalty in (5) and (6) the scaled relative entropy.8The value function can be viewed as the continuous time limit of a discrete, intertemporally additive,

robust expected utility, stylistically of the form Vt = infQ EQ

Vt+1+ 1

Vt+1EQ[ln dQ

dP]

. Alternatively, one can

view it as a stochastic differential utility, as in Duffie and Epstein (1992).

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Given the value function, we are in a position to define the robust indifference price of the

option to invest.

Definition 2 Robust Indifference Price. The robust indifference priceft= f(t, P)of the

option to invest is the solution to U(t, x ft, P , S ) =V(t,x,P,S).

As such, the robust indifference price ft can be interpreted as the amount of wealth that

the agent is willing to give up right now in exchange for receiving the value of the option,

without altering their robust utility. This value is the maximum price that the agent is willing

to pay for receiving the option, whereas the agent will purchase the option for any price

below this robust indifference price. In real-world scenarios, agents are typically comparingamong options to invest in projects. The robust utility indifference value provides a consistent

approach for incorporating the agents aversion to risk and ambiguity.

3 Optimal Behavior of the Ambiguity-averse Agent

Given the definitions provided in the previous section, we now proceed to finding the ambiguity-

averse agents optimal behavior. We find the solution to the value function in the postexercise

region and then derive an obstacle problem for the agents value function in the preexercise

(continuation) region. Finally, we characterize the agents indifference price and solve for it

analytically in the perpetual case.

Proposition 1 Postexercise Value Function. The postexercise value functionV(t,x,P,S)

is independent ofP andSand is explicitly given by

V(t,x,P,S) =1

e x+1

22(Tt) , (12)

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whereE =

1 0

and the ambiguity adjusted market price of risk9 = (r)2 + E1E .Furthermore, the optimal controls are

t

=1 , (13)

where=

2 EE

and = ( r) 0 0.Unlike the usual robust portfolio optimization problems addressed by Uppal and Wang

(2003) and Maenhout (2004), among others, our result contains a nontraded asset in the mix.

The existence of the nontraded asset results in some unique features. It is easiest to explore

the results under the limiting case, in which the ambiguity matrix = 1

1, so that the

agent expresses ambiguity equally on both the traded and nontraded assets. In this case, the

optimal controls reduce to

= 1

1 +

r

2

, =

1 + ( r) , and =

1 +

r

. (14)

There are a number of notable features of this result. First, as 0, the optimal investment

is the usual Merton investment, and drifts of the traded and nontraded assets are equal to

their real world drifts. Second, as the agent becomes extremely ambiguity-averse (i.e., as

+), (s)he no longer invests in the traded asset, and its drift is reduced to the risk-free

rate. Furthermore, the drift of the nontraded asset is reduced to r

, which equals

the drift under the MEMM, corresponding to the traded asset gaining drift corrections to

render it risk-neutral, while all orthogonal Brownian motions remain unchanged (see Rouge

and El Karoui (2000)). These optimal drifts result from the lack of confidence in the model on

part of the agent, who instead trades as if (s)he is risk-neutral. Interestingly, the nontraded

asset attains the MEMM drift for this extreme case of complete ambiguity. Finally, the

optimal investment and drifts decrease as ambiguity aversion increases.

9We call this the ambiguity adjusted market price of risk because plays the role that the market priceof risk plays in the Merton solution to the optimal investment problem. Indeed in the limit of zero ambiguity

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Having fully characterized the postexercise value function, we proceed to the more inter-

esting case of the preexercise value function U(t,x,P,S). In this region, the agent is exposed

to the risk embedded in the project value. Therefore, the value function must inherit a de-pendence on P. However, as in the postexercise region, the value function is independent of

S. Due to the embedded optimal stopping problem, there is no closed-form solution for finite-

time horizons; however, it is possible to show that the value function satisfies an equation

similar to the American option pricing problem under an ambiguity-adjusted MEMM for a

modified payoff.

Theorem 1 Preexercise Value Function. The value function in the pre-exercise region

admits the ansatz

U(t,x,P,S) =V(t,x,P,S) H(t, P), where =

1

1

21

1, (15)

and =

0 1

. The optimal controls describing the optimal investment and optimal

measure are given byt

=1 ( + P Pln H) . (16)

Moreover,Hsatisfies the linear-complementarity problem

tH+LH 0, ln H

(P K)+,

tH+LH ln H+ (P K)+ = 0, ln H(T, P) =

(P K)+,

(17)

where

L is the infinitesimal generator of the discounted project valueertPt under the ambi-

guity adjusted MEMMQ induced by the Radon-Nikodym derivativedQdP

=e1

2(

)2T+

BT . (18)

Specifically,L= ( r) P P+ 122 P2 P P, and the ambiguity-adjusted MEMM drift is= 1. (19)

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There are two types of adjusted project drifts appearing in this result. The first is the

optimal drift that arises in the optimal controls =+ , given by (16); the second is the

ambiguity-adjusted MEMM drift, which appears in the linear complementarity problem forH. The optimal drift does not appear explicitly in the linear complementarity problem;

however, the optimal stopping and investment problem are computed under this drift, after

accounting for ambiguity. In principle, the optimal drift is time- and state-dependent, even

though the drift is constant under the reference measure P.

Numerical exercises show that the measure under which the optimization is occurring

induces mean-reversion in the project value, even though the original dynamics was that of

a GBM. Moreover, the value function surprisingly is reduced to the case of no ambiguity butwith a modified level of risk aversion and an ambiguity-adjusted MEMM. Indeed, Henderson

(2007) identifies the analog10 of the obstacle problem (17), with the usual MEMM drift= r

and= (1 2)1. In this reduced valuation, the project value therefore behaves

as a GBM, even though the optimal measure is quite distinct from that of a GBM.

Our results reduce to those of Henderson (2007) in the limit of no ambiguity aversion,

but differ when ambiguity is present. As a simple illustration of the difference, consider the

limiting case of=

1

1

, such that the agent has equal ambiguity on both the traded andnontraded assets. Then,= ((1 2) +(1 +2))

1, and= r

. Interestingly, the

adjusted drift is the MEMM drift, and there are no corrections due to ambiguity; however,

the power is affected by ambiguity, which leads to a perturbation of the obstacle in the

PDE. For more general ambiguity matrices, will also depend on the various levels of am-biguity aversion. Figure 2 provides an illustration of these effects. As expected, the power

is decreasing in both P and S; however, somewhat surprisingly, the drift

is decreas-

ing in P

but increasing in S

. Furthermore, in the limiting case of S

0, is reducedto the constant MEMM drift (which equals 1% for the parameters used in this example),

10Henderson, however, writes the equation for the analog of the perpetual value function U directly, and

uses the boundary conditions of value-matching and smooth-pasting, rather than writing it as a linear com-

plementarity problem.

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0

5

10

0

2

4

6

8

10

1

1.2

1.4

1.6

1.8

2

Project Ambiguity (P)Asset Ambiguity (S)

PowerTransform

()

0

5

10

0

2

4

6

8

10

1

1.5

2

2.5

3

3.5

Project Ambiguity (P)Asset Ambiguity (S)

AmbiguityMEMM

Drfit

()

Figure 2: Sensitivity of the power transform and ambiguity-adjusted MEMMon the agentsaversion to ambiguity in the project value (P) and the traded asset (S). The remaining modelparameters are = 10%, = 15%, = 8%, = 30%,= 0.7 and r = 5%.

independent of ambiguity in the project and joint ambiguity. In contrast, is reduced to

((1 2) (1 +P/ ((1 2) +P/SP)))

1, which depends on all of the remaining levels of

ambiguity.

Proposition 2 Ambiguity-adjusted Drift Behavior. The ambiguity-adjusted MEMM

driftis increasing inSand decreasing inP.In the infinite-horizon case, Equation (17) is reduced to an ODE and admits an explicit

solution, which we study in the next subsection. However, it also generally admits a rep-

resentation in terms of an optimal stopping problem through a Feynman-Kac argument.

Specifically,

H(t, P) = supT

EQ

t,Pexp

(P K)+

. (20)

This expression is precisely that of an American option written on the project, with the

exercise value = exp{

(P K)+} and the pricing measure given by the ambiguity-

adjusted MEMM. This representation is useful in interpreting the robust indifference price of

the real option, which now follows as a simple consequence of Theorem 1.

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Corollary 2 Robust Indifference Price. The robust indifference pricef(t, P) of the real

option is given by

f(t, P) =

ln H(t, P) . (21)

Moreover,fsatisfies the semilinear complementarity problem

tf+Lf 122 (P Pf)2 0,f (P K)+,

tf+Lf 122 (P Pf)2 (f (P K)+) = 0,f(T, P) = (P K)+.

(22)

Proof. The robust indifference pricef satisfiesU(t, x f , P, S ) =V(t,x,P,S) and is clearly

given by (21). Substituting H=e

f into (17) leads directly to (22).

As 0, the nonlinear term disappears, and f is reduced to an American option price

under the ambiguity-adjusted MEMM. This limit is also known as the marginal price or Davis

(1997) price, although here we have an ambiguity-adjusted drift of, which depends explicitlyon the ambiguity-aversion matrix . Consequently, even risk-neutral agents are exposed to

the effect of ambiguity. As shown in Figure 2, the drift increases with ambiguity in thetraded asset, but decreases with ambiguity in the project-value. Later, we will show that the

boundary and prices react accordingly.

Before proceeding to examples, we rewrite the robust indifference price in terms of an

expectation-like result. For this purpose, we use the representation (20) forHand insert it

into (21), which yields

f(t, P) =

ln sup

T

EQ

t,P exp

(P K)+

. (23)

In the limit of zero risk aversion, as expected, this relationship is reduced to the usual optimal

stopping problem for an American option. However, the problem also is reduced to the usual

expectation result in the limit of 1

0. This limit is achieved when || 1: i.e., when the

nontraded project is exactly (positively or negatively) correlated with the traded asset. This

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point is noted by others, including Henderson (2007) and Grasselli (2011), without ambiguity

aversion. In this limit, the nontraded asset Pt effectively becomes traded, because its risk can

be perfectly hedged by trading in the risky asset St. Therefore, the price is reduced to a risk-neutral expectation, even for a risk-averse agent. The agent may still be ambiguity-averse in

this limit; however, as|| 1, the ambiguity-adjusted drift r

and is independent

ofSP,S, andP. It is appealing that the level of ambiguity aversion does not appear in the

drift in this limit, because even if the agent has ambiguity, (s)he can locally hedge away the

risk, regardless of whether (s)he has a correct estimate of the drift.

4 The Perpetual Case

In the case of the infinite-time horizon (i.e., the perpetual option to invest), the value function

Uand the robust indifference price can be solved exactly, much like in the usual case of the

perpetual option to invest. In this limit, by time homogeneity11, it is easy to see that the

functionHis independent of time, and the optimal stopping time collapses to a fixed threshold

in the project value, i.e.,= inf{t: Pt P} for someP K. Furthermore, (17) is reduced

to the ODE

LH= 0 , (24a)with boundary, value-matching, and smooth-pasting conditions

H(0) = 1, (24b)

H(P) = e(PK)+, (24c)

PH(P) =

e(PK)+ IP>K. (24d)

11There is a small issue here. The value function V should be scaled by e122T in order for it not to

vanish in the limit as T +. This is distinct from introducing horizon unbiased utilities as in Henderson

(2007) which is not necessary here since utility feeds back from the horizon Tback to the date of investment.

However, regardless of this scaling, the robust indifference value and the function H(t, P) remain finite.

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These equations admit a closed-form solution, which leads to a closed-form solution for the

robust indifference price. We record the results in the proposition below.

Proposition 3 Perpetual Value Function and Robust Indifference Value. Whenever

r

< 12

, (25)

the solution to the perpetual value functionHsatisfying (24) is given by

H(P) =

1

1 e

(PK)

P

P

, P [0, P],

e(PK), P > P ,

(26)

where= r 122 2 and the optimal thresholdP is the positive root of the equa-tion12

P K=

ln

1 +

P

. (27)

Moreover, the agents robust indifference value is

f(P) =

ln

1

1 e(PK)

PP

, P [0, P],

P K, P > P

.

(28)

Risk aversion and ambiguity aversion play distinct roles in this equation. Both aversions

affect the multiplicative coefficients, whereas ambiguity aversion also affects the power of

dependence on P (through ). Thus, ambiguity cannot simply be rolled into a modification

of the risk aversion coefficient, as it is in the case of portfolio optimization in complete market

settings (cf Maenhout (2004)).

Proposition 4 Risk Aversion Dependence. The optimal threshold P is a decreasing

function of.

Proof. Straightforward differentiation of (27) leads to the result.

12There is a closed-form solution of the negative root in terms of the Lambert-W function; however, there

does not appear to be an analog for the positive root.

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Figure 3: Region in which it is optimal to exercise early. Model parameters are the same as

in Figure 2, except that = 15%.

In the limit of zero risk aversion, or perfect correlation, the robust indifference value

collapses to the familiar-looking, but not entirely standard, result

f(P) 0

or||1

(P K)

P

P

.

In complete market settings, the option value is known to admit a power-like form, as above;

however, here, the agent still feels the effect of his/her aversion to ambiguity through the power. This is an important result, and it implies that even risk-neutral agents in the perpetual

case are prone to modifying their behavior due to ambiguity in the model. Moreover, the

parameter region in which an agent may exercise the option early is also modified, due to the

existence of both risk aversion and ambiguity aversion. Expansion of the parameter set due to

risk aversion is also noted by Henderson (2007); however, in the present case, the parameter

region is expanded even in the limit of a risk-neutral agent.

To explore this last point a little further, we note that the condition (25) (which dictateswhether the agent exercises his/her option early) is equivalent to > 0. Because isincreasing inS, but decreasing inP (see Proposition 2), is decreasing inSand increasing

in P. Consider settingP = 0 and Sat a level at which the agent does not invest in the

project. AsP increases (withSheld fixed), there may exist a critical level above which the

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agent will invest in the project. This feature is depicted in Figure 3. The shaded region in this

figure shows the parameter regime in which the agent invests in the project (at the threshold

P

). If the agent is reasonably confident in the project dynamics, but not very confidentin the traded asset dynamics, then it is clear that (s)he does not invest in the project. If,

however, the confidence in the project decreases, then the agent becomes more pessimistic

and the option to wait degrades, inducing the agent to invest early.

Figure 3 also shows that there is a region (corresponding to confidence in the traded asset)

in which Sis small enough that the agent will always invest early in the project, regardless

of ambiguity in the project value. Because the traded asset is correlated to the project value,

having confidence in the traded asset implies having partial confidence in the project value.Furthermore, even if the agent is completely unconfident in the remaining uncorrelated piece,

it is not sufficient to preclude early investment. Finally, increasing the projects reference drift

induces a downward parallel shift in the surface, implying that there is a critical reference

drift above which the agent never invests early in the project, regardless of the agents level

of ambiguity towards the dynamics. Thus, the reference drift can be so large that the agents

lack of confidence in their model does not induce them to invest early, because the option

itself is perceived as too valuable. This scenario is the analog of the usual statement thatearly investment occurs when < r+ 1

22, or alternatively when dividends are high enough.

In Figure 4, we illustrate how the option value and optimal exercise threshold behave as

ambiguities in the traded asset and project value increase. Increasing ambiguity in the project

value decreases the option value, draws the optimal threshold downwards, and accelerates

investment, consistent with the results of Nishimura and Ozaki (2007), Trojanowska and

Kort (2010), Roubaud, Lapied, and Kast (2010), and Miao and Wang (2011) for the lump-

sum case. The intuition behind this result is simply that the agent prefers to lock in gains

now, in lieu of greater gains in the future, because the probability that those large gains

will be realized is ambiguous. In contrast, increasing ambiguity in the traded asset increases

the option value, pushes the optimal threshold upwards, and delays investment. This is a

new result, which other works do not observe, because they do not distinguish between the

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0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Project Value ( P )

RobustOptionValue

(f)

Increasing ( P)

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Project Value ( P )

RobustOptionValue

(f)

Increasing ( S)

Figure 4: Sensitivity of the robust option value on the agents ambiguity towards the project

value (left panel) and traded asset (right panel). Vertical lines indicate the optimal project

value threshold to which the agent exercises the option. Model parameters are the same as

in Figure 2 and = 1.

hedging asset and project value. As ambiguity in the hedging asset increases, the agent is

willing to consider scenarios that lead to both more- and less-positive outcomes. As we show

in Proposition 2, increasing ambiguity in the hedging asset increases the effective drift

of the

project value; therefore, the option values and thresholds are pushed upwards. The overalleffect is therefore to delay investment and option values increase.

Another interesting feature to explore is how the optimal drift picked out via the robust

optimization problem varies as a function of ambiguity and time. As shown in Figure 5,

although the agents reference measure Phas a constant drift, the optimal measure contains

both time and state dependence. In particular, the drift can be positive or negative and

induces mean-reversion in the project value dynamics.

5 Numerical Experiments for Finite Time

In this section, we investigate some of the features of the finite-time horizon optimal exercise

policies. For this purpose, we can either discretize the linear PDE (17) with a nonlinear

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 520

15

10

5

0

5

10

Project Value ( P )

OptimalDrift(

*)

increasing P

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 55.5

6

6.5

7

7.5

8

Project Value ( P )

OptimalDrift(

*)

increasing S

Figure 5: Sensitivity of the optimal project drift to the agents ambiguity towards the

project value (left panel) or traded asset (right panel). Circles indicate the optimal project

value threshold, above which the agent exercises the option. Model parameters are the sameas in Figure 2 and= 1.

obstacle to obtain the value function and apply (21) to obtain the robust indifference value,

or we can discretize the nonlinear equation (22) and obtain the robust indifference value

directly. We implement both approaches, and their results are in agreement up to numerical

errors. Alternatively, a binomial tree approach can be developed along the lines of Grasselli

(2011).

In Figure 6, we show the effect of the various ambiguity parameters on the optimal exercise

boundary for three levels of risk aversion. In all three cases, increasing the level of ambiguity

in the traded asset forces the optimal boundary to move upwards, and the agent delays

investment. On the other hand, increasing the level of ambiguity aversion in the project

value itself causes the optimal boundary to move downwards, inducing the agent to invest

earlier. These results are consistent with the infinite time horizon results reported in the

previous section. As in the infinite time horizon case, the robust indifference value increases

with increasing ambiguity in the traded asset, but decreases with increasing ambiguity in

the project value. Moreover, as risk aversion increases, the optimal exercise boundaries move

downward, the robust indifference values decrease, and the variation in both the boundaries

and the values decreases. Thus, as an agent becomes more and more risk-averse, the effect

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of ambiguity weakens in a relative sense, because the agents risk aversion is already pushing

the agent towards the worst-case scenario.

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0 1 2 3 4 5 6 7 8 9 101

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

time (years)

ProjectValu

e

increasing P

0 1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

7

8

9

time (years)

ProjectValu

e increasing S

(a) Risk-Neutral = 0

0 1 2 3 4 5 6 7 8 9 101

1.2

1.4

1.6

1.8

2

2.2

2.4

time (years)

ProjectValue

increasing P

0 1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

time (years)

ProjectValue

increasing S

(b) Moderately Risk-Averse = 0.5

0 1 2 3 4 5 6 7 8 9 101

1.1

1.2

1.3

1.4

1.5

1.6

1.7

time (years)

ProjectValue

increasing P

0 1 2 3 4 5 6 7 8 9 101

1.1

1.2

1.3

1.4

1.5

1.6

time (years)

ProjectValue

increasing S

(c) Strongly Risk-Averse = 2

Figure 6: Effect of ambiguity aversion on the optimal exercise curves for various risk-averse

agents. The ambiguity aversion parameter being changed is set to 0.001, 0.01, 0.1, 1, and

10, and remaining ambiguities are set constant at SP = 1, S = 0.1, and P = 0.1. The

remaining model parameters are the same as in Figure 2.23

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103

102

101

100

101

102

0.275

0.28

0.285

0.29

0.295

0.3

ambiguityaversion ( P)

RealOptionValu

e

103

102

101

100

101

102

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

0.37

ambiguityaversion ( S)

RealOptionValu

e

(a) Risk-Neutral = 0

103

102

101

100

101

102

0.22

0.225

0.23

0.235

0.24

0.245

0.25

0.255

ambiguityaversion ( P)

RealOptionValue

103

102

101

100

101

102

0.23

0.24

0.25

0.26

0.27

0.28

0.29

ambiguityaversion ( S)

RealOptionValue

(b) Moderately Risk Averse = 0.5

103 102 101 100 101 1020.15

0.16

0.17

0.18

0.19

0.2

0.21

ambiguityaversion ( P)

RealOptionValue

103 102 101 100 101 102

0.185

0.19

0.195

0.2

0.205

0.21

0.215

ambiguityaversion ( S)

RealOptionValue

(c) Strongly Risk Averse = 2

Figure 7: Effect of ambiguity aversion on prices for various risk-averse agents. Remaining

model parameters are as in Figure 2.

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6 Conclusions

Investment opportunities are, more often than not, nontradable and only partially spanned by

traded assets. Moreover, agents are never fully confident in their base model for the projects

underlying value. Here, we provide an approach for valuing the option to invest irreversibly

in a project, which incorporates the agents aversions to risk and to model ambiguity. Our

results confirm those of other studies showing that increasing the agents aversion to risk

reduces the option value and accelerates investment. More importantly, we demonstrate

that model uncertainty can translate into delaying or accelerating investment in the project.

Specifically, as the agents uncertainty in the traded asset increases, the agent tends to delay

investment; and as the agents uncertainty in the project value increases, the agent tends to

invest earlier.

The current study investigates the role of a lump-sum cash-flow upon investment. In real-

world situations, investment in the project typically leads to a series of cash-flows. Studying

this effect, Miao and Wang (2011) find that an investment leading to cash-flow streams (as

opposed to lump-sum payment) may induce a delay in investment. Here, we find that the agent

may delay investment, even if the investment is a lump-sum, once one distinguishes between

ambiguity in the hedging asset and project value. It would be interesting to extend the current

work to include a continuous cash-flow. Jaimungal and Lawryshyn (2010) demonstrate how

managerial cash-flow estimates (provided in the form of a sequence of optimistic, typical and

pessimistic scenarios) can be incorporated into the standard real options approach. Including

risk and ambiguity aversion into that framework would provide a direct tie to real-world cash-

flows. Moreover, testing the role of ambiguity on other options, such as switching or exiting

options, may provide insights into how various types of ambiguity affects decisions.

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7 Acknowledgements

This work was supported, in part, by NSERC of Canada. The author would like to thank

participants at the 6th World Congress of the Bachelier Finance Society, June 2010, and

participants at the 15th Annual Real Options Conference, June 2011, for useful comments

and feedback on earlier stages of this work.

A Admissible Sets, Stopping Times, and Measures

In this section, we provide appropriate classes of admissible trading strategies and candi-

date measures for the robust optimization problem. Although it is possible to widen the

class, bounded Markovian strategies suffice for this work. Consequently, we use the following

definitions:

Definition 3 Trading Strategies, Candidate Measures, and Stopping Times.

(a) An admissible trading strategy is a stochastic process t R that is almost certainly

bounded and of the form(t, Xt, Pt, St).

(b) A set of stopping times is given byT =

T,JTT,J, whereTT,J={ :T HVJ} and

HVJ =inf{u: Bu J}, as in Henderson (2007).

(c) A candidate measure is a measureQ P provided by the Radon-Nikodym derivative (7)

and for whichvt is almost certainly bounded and of the formvt= vt(t, Pt, St).

The set of admissible trading strategies will be denoted A and the set of candidate measures

will be denoted Q. Note that the pre- and post-exercise optimal strategy and candidate

measures given in (13) and (16) are in the set A and Q.

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B Proofs

B.1 Proof of Proposition 1

Because V is clearly independent of S and P, the dynamic programming principle implies

that it satisfies the HJB equationVt+ sup

inf,

( + r) Vx+

122 2 Vxx v

v V

= 0 ,

V(T , x, P, S ) =1

e x ,

(29)

where v= (, ). Because the boundary condition is independent ofP andS, the value

function V inherits this independence. By using the ansatz V(t,x,P,S) = 1

e xh(t), for

h(t) 0 and h(T) = 1, the HJB equation is reduced to

ht+h sup

inf,

( + r)() + 12

2()2 vv

= 0 . (30)

The first-order conditions lead to the optimal controls in (13). By substituting the feedback

controls back into the HJB equation, we find,

ht h

2

( r) 0 02 EE 1( r)0

0

= 0 . (31)This expression is simplified by using the following matrix inverse formulaA B

B D

1 = (A BD1B)1 A1B(D1 BA1B)1

D1B(A1 BD1B)1 (D BA1B)1

(32)whereA is ann nmatrix,B is ann mmatrix, andD is anm mmatrix. Consequently,(31) is reduced to ht

122h= 0, so that h(t) = e 12 2(Tt), and (12) is the unique solution

to the HJB equation.

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B.2 Proof of Theorem 1

BecauseU is clearly independent ofSbut not P, the dynamic programming principle implies

that it satisfies the HJB

Ut+ sup

inf,

( + r) Ux+

122 2 Uxx

+(+ r) P UP+1

22P2UP P+ U xP

vv U] 0 ,

V(t, x + (P K)+, P , S ) U(t,x,P,S)

(33)

where v= (, )

, and one of the inequalities is strict. Substituting the ansatz U(t,x,P) =V(t, x) G(t, P) into the above PDE, and after some tedious calculations, we find

Gt+ 122G+ ( r)P GP+ 122P2GP P

+sup

inf,

(G+ P GP) +

1

2

0 ,

e(PK)+ G(t, P)

(34)

where =

, =

( r) 0 0

, and =

0 1

. The first-order

conditions lead to the optimal controls in (16). Upon substituting the optimal controls back

into the PDE, we find thatGt+ ( r

1)P GP+1

22P2GP P

1

21

(P GP)2

G 0

e(PK)+ G(t, P)

(35)

Finally, setting G(t, P) = H(t, P) with =

1 121

1, and after some further

tedious computations, the nonlinear term drops out and we find that Ht+LH0, ln H(t, P)

(P K)+.(36)

Here,L= P P+ 122 P2P P. It is easy to see that this is theQ-infinitesimal generator ofthe process Pt, and the proof is complete.

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A verification theorem can be proven for the class of problems in Definition 1, along the

lines of the proof in Jaimungal and Sigloch (2010). For completeness, we provide the Theorem

and prove the first part, because the second follows analogously.

Theorem 3 (a) Suppose that there exists a function V = V(t, x) that is a solution of

(29). Furthermore, suppose that for each (t, x) [0, T] R, there exist (t , t ,

t ) =

((t, x), (t, x), (t, x)) R3, which leads to

supR

inf(,)R2

( + r) Vx+

122 2 Vxx v

v V

. (37)

Assume thatt is admissible and there exists a measureQ Q, under which drifts ofSt

andPt are + t and+

t, respectively. ThenV(t, x) is a solution of equation (6).

(b) Suppose that there exists a functionU=U(t,x,P) that is a solution of (33). Further-

more, suppose that for each (t, x) [0, T] R, there exist (t , t ,

t ) = (

(t,x,P),

(t,x,P), (t,x,P)) R3, leading to

supR

inf(,)R2

( + r) Ux+

122 2 Uxx

+ (+ r) P UP+1

22

P2

UP P+ U xP v

v U.(38)

Assume that the trading strategyt defined by

t=

t , t < ,t , t is admissible and that there exists a measure Q Q , under which St and Pt have drifts

( + t, + t), where

t=

t , t < ,

t , t ,and t=

t , t < ,

t , t .

ThenU is a solution of (5).

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Proof.

We start with part (a).

Consider the measure Q

, and let Abe any admissible strategy. Itos lemma impliesthat

V(T, XT) =U(t, x) +

Tt

(tV + L,v

0 V) ds+

Tt

s wV dBs ,

where B is a Q standard Brownian motion. Because V is a solution of (29) and A

is an arbitrary strategy, tV + L,vV 12V (v

)v. Taking expectations and using

V(T, XT) =u(XT), we get

V(t, x) EQ

t u(XT) +12 T

t

V(s, Xs) (vs )

vs ds . (39)Because this inequality holds for all admissible trading strategies, it follows that

V(t, x) supA

EQ

t

u(XT) +

1

2

Tt

V(s, Xs) (vs )

vs ds

sup

Ainf

QQE

Qt

u(XT) +

1

2

Tt

V(s, Xs) (vs)vs ds

. (40)

Next, we fix the strategy t and let Q Q be any candidate measure under which Pt and

St have drifts + t and+ t, respectively . Because we always have tV + L,vV

1

2V (vs)

vs, we can argue as above:

V(t, x) EQt

u(X

T ) +1

2

Tt

V(s, X

s ) (vs)vs ds

.

Because this relation holds for any Q Q, we get

V(t, x) infQQ

EQt

u(X

T ) +1

2

Tt

V(s, X

s ) (vs)vs ds

sup

Ainf

QQE

Qt

u(XT) +

1

2

Tt

V(s, Xs) (vs)vs ds

. (41)

Part (a) of the theorem follows from the inequalities (40) and (41). For the strategy and

the measure Q, we have equality everywhere, i.e.,

V(t, x) =EQ

t

u(X

T ) +1

2

Tt

V(s, X

s ) (vs )

vs ds

.

Part (b) follows in a similar manner.

From arguments analogous to those above and in part (a), it follows that

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B.3 Proof of Proposition 2

Explicit but tedious computations show that

= (SP, S, P) r

(42)

where,

(SP, S, P) = (SP+P+SPP) S+ (1

2)SP2 +SPP

(SP+P+SPP+ (1 2)2SP) S+ (1 2)SP2 +SPP

. (43)

First, consider as a function ofS. In this case, the above expression can be written

A S+B

(A +C) S+B =

A

A +C+

CB

(A +C) ((A +C) S+B) ,

which is clearly decreasing in S, implying that is increasing in S. By straightforwardmanipulation, (43) can be rewritten in the form

(SP, S, P) = (SP+S+SPS) P+ (1

2)SP2 +SPS

(SP+S+SPS) P+ (1 2)SP2 +SPP+ (1 2)S2SP. (44)

Considering as a function ofP, the above expression can be written in the form

A P

+B

A P+B+C = 1 C

A P+B+C,

which is clearly increasing inP, implying thatis decreasing inP. Results 1, 2 and 3 followby direct computations.

B.4 Proof of Proposition 3

The characteristic equation corresponding to (24a) is

( r)z+ 12

2z(z 1) = 0. (45)

with roots z= 0 and z=( r 12

2)/2 =, so that H(P) =A+B andP is the most

general solution. If < 0, then the value function is infinite and the agent never exercises

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the option early. If 0, then the boundary condition H(0) = 1 implies A= 1, while the

value-matching and smooth-pasting conditions (24c-24d) imply

1 +B (P) =e(PK) (46a)

B (P)1 =

e

(PK) (46b)

Multiplying the second equation by P, substituting into the first, and rearranging leads to

the optimal threshold equation (27) for P. Moreover,B =(1 e(PK))(P), which

leads to (26). Finally, applying (21) leads directly to (28).

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