4.2 Proposed Project Valuation Schedule
4.2.1 Ambiguity
First, let’s examine how the proposed valuation schedule behaves under ambiguity. Specifically, take the derivative of Ψ w.r.t. ambiguity,δ,
∂Ψ
∂δ = (ρ−µ)
(ρ−µ)−2(α−12)δσ2 ·2
α− 1 2
σ (4.20)
which, seeing that the fraction is always positive, the sign of the derivative is uniquely determined byα:
sgn ∂Ψ
∂δ
= sgn
α−1 2
(4.21) This result establishesattitude monotonicity, that is the decision maker follows a valuation schedule that is monotonically increasing in ambiguity when ambiguity loving, i.e. α > 12, monotonically decreasing in ambiguity when ambiguity averse, i.e. α < 12, and indifferent to any level of ambiguity in her environment when ambiguity neutral, i.e. α= 12. Thus, the proposed
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00 0.05 0.10 0.15 0.20 0.25
δ Ψ
α 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Figure 4.3: Knightian multiplier under Choquet-Brownian motion with neo-additive capacities.
specification also satisfiesinformed neutrality, and with that the desiderata triplet.
Consult Figure 4.3 to see how the proposed model mends the pecu-liarities of Schr¨oder (2011). In particular, ambiguity neutrality is defined absolutely and not comparatively as has so far been the methodology in the literature. As a result, informed neutrality and attitude monotonicity yield an intuitive valuation schedule, though it appears laborious in construction.
Certainly, the Choquet-Brownian motion is less intuitive than multiple pri-ors frameworks, but its final result in the proposed neo-additive specification carries the same intuitive specification and parameter setup while satisfy-ing all desiderata, emphasizsatisfy-ing a clear distinction between ambiguity and ambiguity attitudes, to which we turn next.
4.2.2 Optimism and Pessimism
Next, we investigate the relationship between the valuation schedule and ambiguity attitudes by varying α. Take the derivative of the Knightian multiplier with respect to the index of optimismα:
∂Ψ
∂α = ρ−µ
(ρ−µ)−2(α−12)δσ2 ·2δσ≥0 (4.22)
60 Proposed Project Valuation Schedule
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00 0.25 0.50 0.75 1.00
α Ψ
δ 0.25 0.20 0.15 0.10 0.05 0.00
Figure 4.4: Ambiguity attitude sensitivity of the Knightian multiplier under Choquet-Brownian motion with neo-additive capacities.
to immediately see that an increase in optimism is unequivocally positive in the domainδ, σ >0. Since the valuation scheme is linear in Ψ, this result translates to the effect from increased optimism is expected.
Note how the curvature of the valuation schedule increases with the level of ambiguity,δ, which translates into an amplified effect of changes in ambiguity attitude, α, see Figure 4.4. In essence, this means that in envi-ronments of high uncertainty, ambiguity attitude becomes a potent factor in the evaluation of payment streams. This gives a theoretical foundation for understanding abrupt changes in stock market prices when no other cir-cumstances than market sentiment seem to have changed significantly.
4.2.3 Risk
Less expected is the result that even though the decision maker is assumed to be risk neutral, a decision maker following an ambiguous valuation scheme is still sensitive to the risk parameterσ. This outcome conflicts with the tra-ditional Dixit and Pindyck (1994) solution in the confident valuation scheme (4.5) in which risk plays no role in determining valuation. When considered in the Choquet-Brownian framework, valuation of a payoff stream depends not only on ambiguity per se, but also on the risk of the payoff stream, notwithstanding risk-neutrality. Specifically, risk, i.e. volatilityσ, scales the
effect of ambiguity on the valuation scheme and in effect causes an amplified distortion of the prior already being caused by volatility. To see this, recall from Proposition 1 (3.57) that the Choquet-Brownian motion deforms the geometric Brownian motion of the payoff stream, so that movement of the engine, i.e. the Choquet-Brownian, becomes scaled by the constant volatil-ity of the GBM, so that the decision maker evaluates higher deviations from the trend of the GBM when considering an ambiguous environment.
4.3 Investment Timing
Now we turn from the all-or-nothing decision to a flexible investment timing setting where the firm can postpone the decision to invest. At any point in timeτ, the firm evaluates the valuation schedule Vτ in relation to some con-stant investment costI. Seeing that the firm now considers an opportunity to invest, the firm can take the decision to investnow or towait. Hence, the firm is faced by an optimal stopping problem considered to hold an infinite maturity real (American) option to invest and the problem thus becomes to solve for the optimal stopping timeτ at which exercising the investment opportunity reaping the value
Vτ−I (4.23)
would be optimal. In order to do this, the firm must have in place (i) a decision rule (strategy), and (ii) a valuation of its options. In the classical McDonald and Siegel (1986) and Dixit and Pindyck (1994) models, the firm follows a trigger strategy by picking the first date τ at which Vτ is greater than some trigger (or threshold) levelV∗, constant in the infinite maturity setting. Now, the firm faces the trade off of waiting for an even bigger V and pay the sunk cost I later, or get V and pay I now. In order to make the evaluation complete, the firm thus needs a model for the movements in the investment value in order to make an assessment of the value of waiting.
Therefore, the firm considers the following Hamilton-Jacobi-Bellman equation, which in this setting is the value of the option to invest
J(Vt) = maxn
Vt−It, J(Vt) +EP[dJt|Ft]−ρJ(Vt)dto
(4.24) The firm considers the value of investing now, simply Vt−I, that is the
62 Investment Timing
difference between the valuation schedule and investment cost as opposed to the value of waiting, the argument to the right. Whenever the value of waiting exceeds that of the value of investing, investment is postponed. This motivates the definition of the continuation region Vt−It < EP[dJt|Ft] + Jt(Vt)−ρJ(Vt)dt, in which case the equationJ(Vt) =EP[dJt|Ft] +J(Vt)− ρJ(Vt)dt must hold, so that
EP[dJt|Ft] =ρJ(Vt)dt (4.25) In order to establish an expression for dJt, note that it is a function of the valuation schedule, whose continuous-time process (Vt)t≥0 in a Choquet-Brownian motion setting given by (3.54) follows the law of motion
dVt= (µ+mσ)Vtdt+sσVtWt (4.26)
The behavior of the Bellman-equation over time can thus be derived by invoking Itˆo’s Lemma 3.1 in an expansion of dJt:
dJt= ∂J
∂Vt(µ+mσ)Vt+∂J
∂t +1 2
∂2J
∂Vt2(sσVt)2
dt+ ∂J
∂VtsσdWt
= ∂J
∂Vt
[(µ+mσ)Vtdt+sσVtdWt] +1
2(sσ)2Vt2∂2J
∂Vt2dt (4.27) In expectation conditional atFt, substitute in (4.25) to establish the second-order ordinary differential equation (ODE) in the continuation region
1
2(sσ)2Vt2∂2J
∂Vt2dt+ (µ+mσ)Vt
∂J
∂Vt−ρJ(Vt) = 0 (4.28) which yields the law of motion of option valueJ. Requiring (i) the absorb-ing barrier condition that if the project value has no value, the investment opportunity also has no valueJ(O) = 0, (ii) the value matching condition thatJ(V∗) =V∗−I and (iii) the smooth pasting conditionJ0(V∗) = 1 this ODE has a well-known solution18and we can thus finally write up the value
18For the general proof, see Dixit and Pindyck (1994).
of the option to invest
J(Vt) =
I
β−1
1−β
β−βVtβ in the continuation region Vt< V∗ Vt−I in the stopping region Vt≥V∗
(4.29) where we define B:=
I β−1
1−β
β−β, and let β be a constant defined
β:=
−
(µ+mσ)−12(sσ)2 +
q
(µ+mσ)−12(sσ)22
+ 2ρ(sσ)2
(sσ)2 (4.30)
for m = 2c−1 = 2[(1−δ)12 +δα]−1 = 2(α− 12)δ, and s2 = 4c(1− c). Finally, we need to determine the reservation value V∗ that completes characterization of the decision rule
V∗ = β
β−1
I (4.31)
4.3.1 Risk in Traditional Non-Ambiguous Environments In the traditional McDonald and Siegel (1986) and Dixit and Pindyck (1994) non-ambiguous environment, δ = 0, recall from the valuation schedule in Proposition 2 that once the option is exercised, the value of the project is indifferent to risk. However, preceding option exercise, an increase in risk increases the value of the option to invest in the continuation region as well as it increases the reservation value V∗.
Mathematically, see that
∂β
∂σ2 δ=0
= ∂
∂σ2
−[µ−12σ2] + q
[µ−12σ2]2+ 2ρσ2 σ2
<0 (4.32) and∵∂V∗/∂β|δ=0<0 we have∂V∗/∂σ2|δ=0>0. Then, since∂J/∂β|δ=0<
0 in the continuation regionVt< V∗ we must have that∂J/∂σ2|δ=0 >0. In the stopping region, the option value remains indifferent to changes in risk.
These are long-established results in real options theory, which we will see are nested as a special case of the proposed model.
Proposition 3(Real Option Risk Impact underδ= 0). An increase in risk
64 Investment Timing
in the δ = 0 non-ambiguous environment regardless of ambiguity attitude α ∈ [0,1] results in (i) an increase in the value of the investment option in the continuation regionVt< V∗, (ii) no change in value in the stopping regionVt≥V∗, and (iii) an increase in the reservation investment valueV∗. Altogether, an increase in risk leads to an increase in the value of waiting, and thus amplifies the tendency to delay investment.
4.3.2 Real Option Value in Ambiguous Environments
This subsection analyzes the firm’s optimal stopping problem when embed-ded in an ambiguous environment as modeled by the proposed specification under Choquet-Brownian motions with neo-addtive capacities.
When introducing ambiguityδ∈(0,1], the preliminary analysis of valu-ation schedules revealed that the impact on the project valuvalu-ation schedule of increased ambiguity is contingent on the attitude towards it. Therefore, while separability of the proposed model allows for a more refined analysis than that of Roubaud et al. (2010)—who define the level of ambiguity as c-ignorance deviations from the probabilistic case c = 12 and let the direc-tion denote taste—the introducdirec-tion of separability adds further complexity to the model by adding a new analytic dimension, and thus an analytic sen-sitivity approach becomes unfeasible. In consequence, the following analysis makes use of numerical simulations in developing its propositions. Again, we will use the valuesρ= 0.10, µ= 0.05, σ= 0.20 from Schr¨oder (2011), but we now changeπt to 1.5 instead of 1.0 as was used in the previous analysis, becauseπt= 1.5 will allow a clearer sensitivity analysis on the (δ, α)-space.
Characterization of the real option valuation commands an analysis of option value in the stopping region, the continuation region, and of the determination of which reservation value that should be set to trigger option exercise.
Stopping Region Vt≥V∗
Recall from (4.29) that in the stopping region the value of the real option is simply the valuation schedule minus the investment cost. As the investment cost is constant over time and in this setting unambiguous, we can draw full attention to the behavior of the valuation schedule. This analysis was already conducted in the previous section. Figure 4.5 provides the numerical
Figure 4.5: Option value in the stopping region.
simulation of the behavior of the valuation schedule (4.18) suspended in the (δ, α)-space.
These conclusions bring new insights to the real options literature by refining the findings of Roubaud et al. (2010). In ordinary McDonald and Siegel (1986) and Dixit and Pindyck (1994) inspired real option models, risk does not impact real option value in the stopping region (see Proposition 3).
The proposed model stipulates that what matters to firms in the stopping region is not risk, which can be diversified away, but ambiguity which can not. Furthermore, the direction of the impact of ambiguity is determined by the tastes of the decision maker and is amplified by higher degrees of ambiguity in the economy. Therefore, the model predicts that in highly ambiguous environments, large swings in project valuations can occur for small shifts in attitude directions, even when no other economic variables than market sentiment have changed.
66 Investment Timing
Figure 4.6: Option value in the continuation region.
Continuation Region Vt< V∗
We now turn option value in the continuation region. Again, the sensitivity analysis towards ambiguity becomes too elaborate to conduct analytically, so we turn to numerical simulation, with results illustrated in Figure 4.6.
We see that the ambiguity impact in the continuation region is similar in direction to that in the stopping region. For an ambiguity averse decision maker α < 12 higher ambiguity decreases continuation value, whereas the opposite holds for an ambiguity lover withα > 12. This result comes about because the Choquet-Brownian motion governing the payoff flow becomes less favorable under ambiguity aversion in an ambiguous environment, and more favorable under if the firm is ambiguity loving. Therefore, the direction of ambiguity attitude determines the scenarios that will be considered under option continuation.
This result is in sharp contrast with the positive effect on real option value in the continuation region for increasing risk. As was shown in Propo-sition 3, an increase in risk under no ambiguity unequivocally increases op-tion value in the continuaop-tion region, because if the firm waits it can choose
Figure 4.7: Reservation project value.
to invest contingent on more favorable market conditions. Under the ambi-guity specification proposed by this thesis, when adding a layer of ambiambi-guity to the environment, real option value skews the Choquet-Brownian motion that generates the values the payoff process is believed to take if the real option is continued without exercise.
Reservation Value V∗
The directional impact of ambiguity and ambiguity attitudes follows that of option value in the stopping and continuation region, see Figure 4.7. With this in mind, we can now construct the following two propositions to sum up the findings of the sensitivity analysis on option value.
Proposition 4 (Real Option Sensitivity to Ambiguity). For an ambiguity averse decision maker increased ambiguity decreases option value in both the stopping region and the continuation region as well reservation value, and increases for an ambiguity loving decision maker. An ambiguity neutral decision maker is indifferent to any level of ambiguity in the economy.
Proposition 5 (Real Option Risk Impact under Ambiguity). In an
am-68 Investment Timing
biguous environment δ ∈ (0,1], risk scales the effect of ambiguity so that real option value in both the stopping and in the continuation region as well as the reservation value decrease for higherδ under ambiguity aversion and increase for higherδ under ambiguity love.
This section established that the directional impact from ambiguity is identical on option value in the stopping region and the continuation region as well as on the reservation value. Now, in order to understand how ambi-guity impacts investment strategies, we turn to a sensitivity analysis of the value of waiting.
4.3.3 The Value of Waiting
Perhaps most central to the real options theory is the value of waiting, which is the parameter the firm uses in evaluating whether (i) to exercise the option and invest now, or (ii) to continue the life of the option and postpone investment. Transform the ambiguous valuation schedule (4.18) at the reservation valueV∗ to define the reservation profit
π∗ :=
(ρ−µ)−2
α− 1 2
δσ
V∗ (4.33)
This definition yields a simple trigger strategy Strategy =
( Invest if πt≥π∗
Wait if πt< π∗ (4.34) First, let’s examine the sensitivity of the investment timing decision to risk, σ, in a non-ambiguous environment, δ = 0. It is a well known result in the real options literature, see e.g. McDonald and Siegel (1986), that an increase in risk increases project option value in the continuation region as well as the reservation value of the project, while project value once the option has been exercised (i.e. the valuation schedule) does not change. As a consequence, the firm delays investment when faced by higher risk, because it requires a higher threshold level of profits.
Now, let’s reintroduce ambiguity to the economy so that δ ∈ (0,1].
Again, we draw attention to a numerical analysis and observe the simu-lated reservation profit surface on the (δ, α)-space in Figure 4.8. First, we see that an ambiguity neutral decision maker α = 12 does not change her
Figure 4.8: Reservation profit.
reservation profit and thus her strategy for any level of ambiguity in the economy.
By introducing attitudes towards the level of ambiguity, the proposed model reveals that ambiguity attitudes play a defining role in deciding the optimal moment of exercise of a real option to invest. Specifically, an am-biguity averse decision maker requires a high profit threshold to invest, and therefore has a high value of waiting and a consequent tendency to delay investment in ambiguous environments. On the other hand, an ambiguity loving decision maker an increase in ambiguity motivates early exercise of the real option to invest. Interestingly, note how the trigger strategy in-creases in sensitivity to changes in attitudes for higherδ. Intuitively, when ambiguity is injected to the economy, optimistic firms believe that the value of the investment project will increase, which increases the cost of waiting.
Pessimistic ambiguity averse firms believe that the value of the investment project will decrease, and hence it becomes an advantage to wait investing as opposed to receive the payoff stream from present onward. This result provides a theoretical foundation for understanding sudden changes in in-vestment patterns as induced by even small reversals in optimism in highly uncertain environments (in its Knightian sense) even when no observable
70 Investment Timing
economic variables have changed in magnitude.
Proposition 6 (Investment Timing under Ambiguity). All things equal, when considering a real option to invest, in an ambiguous environment, δ∈(0,1], an ambiguity averse decision maker,α ∈[0,12), will tend to delay investment, and an ambiguity loving decision maker, α(12,1], will tend to pursue an earlier exercise than if she was neutral toward ambiguity α = 12, in which case the value of waiting for the decision maker is indifferent to changes in ambiguity. These tendencies are amplified in potency as ambi-guity,δ, increases.
Now, let’s examine how the decision maker behaves at varying levels of riskσ when evaluating the timing of an investment opportunity. As it was indicated by Figure 4.8 the introduction of ambiguity and attitudes adds a new dimension to the impact of risk on the value of waiting. Numerical sim-ulations reveal an interesting pattern. We partition decision makers on the (δ, α)-space by combinations of beliefs an tastes that yield stop and invest versus delay and wait strategies. Proposition 3 stated that in the tradi-tional McDonald and Siegel (1986) non-ambiguous environment, increasing risk leads to a higher value of waiting, and thus a tendency to delay invest-ment. This finding is supported by the numerical analysis of the impact of risk on the timing of investment under ambiguity, illustrated in Figure 4.9.
The proposed model extends the traditional timing analysis by suspending the investment strategy in (δ, α). Indeed, also in ambiguous environments increasing risk leads to a tendency to delay investment by increasing the value of waiting for information.
What is new is that for different levels of risk, firms will pursue alternate strategies if heterogeneous in tastes and beliefs though faced by identical investment opportunities. In low risk environments despite heterogeneous beliefs, most firms will tend to invest, and in high risk environments most firms will tend to delay investment. However, at moderate levels of risk even small reversals in ambiguity attitudes make the difference between investing and waiting. As is clear from Figure 4.8 and Proposition 6 for a fixed level of risk, σ, if the firm is ambiguity averse, increased ambiguity will raise the value of waiting for the firm, and the firm will hence be more likely to postpone investment. On the other hand, if ambiguity loving, increased ambiguity will decrease the value of waiting, and the firm will hence be
Strategy: Stop Delay
Figure 4.9: Impact of risk on investment timing.
72 Investment Timing
more likely to invest now. Now, for increasing risk, the value of waiting will increasefor all firms in the (δ, α)-space, as is illustrated by larger delay regions in Figure 4.9. The following proposition summarizes the results of the analysis of risk impact on investment timing under ambiguity.
Proposition 7 (Impact of Risk on Investment Timing under Ambiguity).
In ambiguous environments, increasing risk will cause a larger part of firms in the (δ, α)-space to delay investment. In general, the more pessimistic the firm the higher the tendency to delay real option exercise.
In conclusion, both risk and ambiguity impact the value of waiting and the investment decision of the firm, but do so in critically different man-ners. In non-ambiguous environments,risk increases the option value in the continuation region, but leaves exercised project value unchanged, which altogether makes waiting the better option. Now, for an ambiguity averse firm, increasedambiguity reduces both the option value as well as the exer-cised project value, but does so more for the latter than the former, which altogether in this case also increases the value of waiting. Instead, for an ambiguity loving firm, increased ambiguity increases both the option value as well as the exercised project value, but does so more for the latter than the former, which altogether in this case increases the cost of waiting. Intu-itively, because of ambiguity, the optimist believes the value of the project to get higher than previous levels and it thus becomes costly to wait, whereas the pessimist believes that the value of the project will get lower and thus it becomes more beneficial to wait investing.
Conclusion
”Uncertainty is a quality to be cherished, therefore—if not for it, who would dare to undertake anything?”
– Auguste de Villiers de L’Isle-Adam
This thesis brings together decision making under uncertainty and continuous-time stochastic processes and examines the valuation schedule of a payoff process under ambiguity then applied to the classical McDonald and Siegel (1986) optimal stopping problem. The proposed model reveals that the implications of ambiguity on valuation schedules and real option valuations are drastically different than those of risk.
We begin with a review of the most commonly applied decision-theoretic approaches to choice under uncertainty and thus visit the representations of Gilboa and Schmeidler (1989) and Ghirardato et al. (2004) multiple priors preferences, Klibanoff et al. (2005) smooth ambiguity, and Schmeidler (1989) Choquet Expected Utility. Then, we introduce Chateauneuf et al.’s (2007) neo-additive capacities to the Choquet integral in order to arrive at an inte-gral that separates tastes and beliefs. Next, we take the preference relations and apply them to continuous-time stochastic processes and show how am-biguity leads to a deformation of the expectation of the law of motion of a payoff flow. As is customary in the literature (Dixit and Pindyck, 1994) we let a payoff flow be described by a geometric Brownian motion. Under the multiple priors approach, we follow Chen and Epstein (2002) and change the drifts of the law of motion through Girsanov’s Theorem and ensure dynamic consistency by restricting the family of possible motions byκ-ignorance. We
73
74
show, however, thatα-MEU preferences are dynamically inconsistent in its non-extreme cases. Further, we briefly sketch the Gindrat and Lefoll’s (2009) proof that even though the Klibanoff et al. (2005, 2009) smooth ambiguity representation allows for an elegant separation of ambiguity and ambigu-ity attitudes Skiadas (2013) proves that the sensitivambigu-ity toward ambiguambigu-ity of such preferences vanishes in the continuous-time limit. Finally, we construct the Kast and Lapied (2010) Choquet-Brownian motion for which dynamic consistency requires a constant capacity on succeeding nodes in the binomial lattice. In the continuous-time limit the Choquet-Brownian motion deforms both the drift and the volatility of the stochastic process under Choquet expectation. Chateauneuf et al.’s (2007) neo-additive capacities are intro-duced to the Choquet integral through the lattice and we arrive at a law of motion similar to that of Kast and Lapied (2010) but in whichc-ignorance is specified by the level of perceived ambiguity, δ, and the attitude toward it, α.
In the remaining part of the thesis we address the research question by conducting a sensitivity analysis of the valuation schedules proposed by the current specifications in the literature, after which we apply the specification proposed by this thesis to the optimal stopping problem. First, we show an-alytically that Schr¨oder’s (2011) α-MEU based specification yields peculiar implications that are at odds with economic intuition, specifically that for any fixed index of optimism,α∈(0,12), the firm at first assigns a discount to the valuation of a payoff flow, but then at some level of high ambiguity, the firm starts assigning high premia. Therefore, while Schr¨oder (2011) achieves separability, the specification satisfies neither informed ambiguity neutrality nor attitude monotonicity and accordingly confounds ”pessimism” and ”op-timism”. We showed that Roubaud et al.’s (2010)c-ignorance fails to achieve a proper separation of tastes and beliefs, so that while attitude monotonicity is obtained, the specification fails to satisfy informed ambiguity neutrality.
Therefore, we introduce the specification proposed by this thesis and see that it satisfies all three desiderata.
In order to answer the research question, the analysis proceeds with a series of numerical simulations which reveal that ambiguity has a different impact than that of risk on the valuation of a payoff flow, on the valuation of an investment opportunity, and on investment timing. First, whereas risk traditionally has no impact on value in the stopping region the proposed
model shows that for an ambiguity averse (loving) firm, ambiguity increases (decreases) the valuation schedule. Further, an ambiguous environment risk scales the effect of ambiguity on the valuation schedule in a direction gov-erned by ambiguity attitude. Thus, even if the firm is risk-neutral it will react to changes in risk through ambiguity. Second, we show that for an op-timistic (pessimistic) firm ambiguity (increases) decreases real option value in the continuation region, again scaled by risk. Finally, because the change in stopping value is larger than the change in continuation value, perceived ambiguity impacts the timing of investment. For an ambiguity averse firm increased ambiguity delays investment, whereas investment tends to be ex-ercised earlier if the decision maker is ambiguity loving. The delay of invest-ment under ambiguity aversion is critically different in construction than the delay of investment from risk. While risk leaves project value in the stopping region unchanged and increases value in the continuation region, ambigu-ity reduces option value in both regions, but does so more in the stopping region which results in a tendency to delay investment. Hence, the model offers explanatory power of observed investment patterns in seemingly iden-tical environments when firms are heterogeneous in their tastes and beliefs towards ambiguity. We note that reservation profit is highly sensitive to slight changes in attitudes. Therefore, in highly ambiguous environments even small changes in sentiment can change the decision to invest.
Our findings are similar in its extreme cases to that of the existing literature pioneered by Nishimura and Ozaki (2007), Schr¨oder (2011) and Roubaud et al. (2010). As pointed out by Eichberger and Kelsey (2009), a separation of tastes and beliefs toward ambiguity is still a not completely resolved issue in the ambiguity literature, which this thesis finds is indeed also the case for real option valuation under ambiguity. Therefore, to the knowledge of the authors, this thesis is the first to offer a dynamically consis-tent specification that clearly separates ambiguity and ambiguity attitudes satisfying the three desiderata. It may be argued that the results of the model analysis are self-evident, but then, if the results are trivial then in fact such critique exactly points to the need to include ambiguity in modeling frameworks. Hence, the proposed model delivers an intuitive and powerful framework within which to understand the effects of ambiguity and opti-mism and pessiopti-mism on investment value and timing.
This work can be extended in several directions. Though this paper