A. The Baseline Model without Ambiguity

1. The Project

An entrepreneur has access to an innovation project that, once initiated, generates an earnings before interest and taxes (EBIT) flow $ X(t):= X\left(t,\omega \right) $ defined on a probability space ( $ \Omega, \mathcal{F},{Q}^0) $ endowed with a standard complete filtration $ \mathbf{F}=\{{\mathcal{F}}_t|t\ge 0\} $ satisfying the “usual conditions.” We assume that the EBIT process is exogenous and independent of the entrepreneur’s investment decision. We specify $ {Q}^0 $ to be the risk-neutral probability measure, in line with the traditional EBIT-based capital structure model of Goldstein et al. (Reference Goldstein, Ju and Leland2001).Footnote ⁴ Under $ {Q}^0 $ , $ X(t) $ follows a geometric Lévy process:

(1) $$ \frac{dX(t)}{X\left({t}^{-}\right)}=\mu dt+\sigma dW(t)+{\int}_{\mathrm{\mathbb{R}}}\iota \left(t,u\right)\tilde{N}\left( dt, du\right), $$

where $ W(t) $ is a standard Brownian motion, $ \tilde{N}\left( dt, du\right) $ is a compensated Poisson random measure given by

$$ \tilde{N}\left( dt, du\right)=N\left( dt, du\right)-\nu (du) dt, $$

with $ \nu (du):= \unicode{x1D53C}\left[N\left(1, du\right)\right] $ being the Lévy measure, and $ \iota \left(t,u\right) $ being a square-integrable predictable process with respect to $ \nu (du) $ . It is worth noting that the specification of a compensated Poisson random measure under the risk-neutral reference measure follows the same rational expectation equilibrium pricing rule as in Goldstein et al. (Reference Goldstein, Ju and Leland2001).Footnote ⁵ Additionally, we assume that $ X(0)=x>0 $ , $ \mu $ , $ \sigma $ , and $ r $ are constants with $ \mu <r $ and $ r $ being the risk-free rate.

We specify the jump component as a compound Poisson process with intensity $ \lambda <+\infty $ and assume a double exponential distribution for the jump size (i.e., $ X(t) $ follows a double exponential jump-diffusion process, first introduced by Kou (Reference Kou2002)). Specifically, we can write the jump component explicitly as

$$ \iota \left(t,u\right)={e}^u-1,\hskip1em \mathrm{and}\hskip1em \nu (du)=\lambda f(du), $$

where the jump size density $ f $ is given by

(2) $$ {f}_u=p{\eta}_1{e}^{-{\eta}_1u}{\mathbf{1}}_{u\ge 0}+q{\eta}_2{e}^{\eta_2u}{\mathbf{1}}_{u<0},\hskip1em {\eta}_1>1,{\eta}_2>0,p,q\ge 0,p+q=1 $$

in which $ \mathbf{1} $ is an indicator function. In the specification, $ p $ and $ q $ are the conditional probabilities of upward and downward jumps, respectively, and $ 1/{\eta}_1 $ and $ -1/{\eta}_2 $ are the conditional mean log jump sizes of positive and negative jumps, respectively. This model specification has two merits. First, as shown by Kou and Wang (Reference Kou and Wang2004), it allows for analytical solutions to valuation problems with American-style perpetual options.Footnote ⁶ This property is crucial to our analyses of joint investment and financing decisions in the real options model. Second, the specification has the appealing feature of modeling positive and negative jumps separately, which generates a rich set of priors, as will be shown later.

2. Financing

We assume that the project requires external funding, as available cash is insufficient. Taxation on project cash flows at rate $ \phi \in \left(0,1\right) $ motivates firms to issue debt for tax shielding. The entrepreneur chooses between all equity financing and optimal financing, aiming to determine the optimal equity-debt mix.Footnote ⁷ Drawing on the established literature (e.g., Leland (Reference Leland1998), Goldstein et al. (Reference Goldstein, Ju and Leland2001)), we analyze debt contracts with perpetual coupon payments $ C $ in a time-homogeneous context. Debt issuance reduces tax by $ \phi C $ but also exposes the firm to potential bankruptcy costs.

We begin with the all-equity financing case. The equity value at the $ {\mathrm{\mathcal{F}}}_0 $ -measurable random investment time $ {\tau}_I\ge 0 $ is

(3) $$ {V}_e\left({\tau}_I\right)={\unicode{x1D53C}}_{\tau_I}\left[{\int}_{\tau_I}^{\infty }{e}^{-r\left(t-{\tau}_I\right)}\left(1-\phi \right)X(t) dt-I\right], $$

where the constant $ I>0 $ denotes the investment cost. At time 0, the entrepreneur chooses the optimal stopping time $ {\tau}_I^e $ to initiate the project; hence, her value function at time 0 is

$$ {V}_e(0)=\underset{\tau_I}{\sup}\;{\unicode{x1D53C}}_0\left[{e}^{-r{\tau}_I}{V}_e\left({\tau}_I\right)\right]={\unicode{x1D53C}}_0\left[{e}^{-r{\tau}_I^e}{V}_e\left({\tau}_I^e\right)\right]. $$

In the case of optimal financing, the entrepreneur can issue debt at the time of investment, $ {\tau}_I $ , to take advantage of the tax benefits associated with debt. Default occurs later, at the stopping time $ {\tau}_D $ under the assumption of optimal default, a standard assumption in the literature (Sundaresan and Wang (Reference Sundaresan and Wang2007), Strebulaev and Whited (Reference Strebulaev and Whited2012), and Sundaresan et al. (Reference Sundaresan, Wang and Yang2015)). As per the traditional trade-off theory of capital structure (e.g., Leland (Reference Leland1994), Goldstein et al. (Reference Goldstein, Ju and Leland2001)), default triggers liquidation under the absolute priority rule, with a fraction $ \alpha \in \left(0,1\right) $ of the asset value lost in the liquidation process. This assumption is standard in models that balance the benefits of debt (via tax shields) with the costs of potential default and liquidation. Figure 1 illustrates the timelines for both financing choices.

FIGURE 1 Timeline

The timeline in Figure 1 illustrates the investment option and potential default under debt financing.

Given the above, the firm value at the investment time $ {\tau}_I $ is

$$ {V}_{\ast}\left({\tau}_I\right)=E\left({\tau}_I\right)+D\left({\tau}_I\right)-I, $$

where $ D\left({\tau}_I\right) $ is the entrepreneur’s valuation of debt, and $ E\left({\tau}_I\right) $ is the value of (leveraged) equity, given by

(4) $$ {\displaystyle \begin{array}{c}E\left({\tau}_I\right)=\underset{\tau_D\ge {\tau}_I}{\sup}\;{\unicode{x1D53C}}_{\tau_I}\left[{\int}_{\tau_I}^{\tau_D}{e}^{-r\left(t-{\tau}_I\right)}\left(1-\phi \right)\left(X(t)-C\right) dt\right]\\ {}={\unicode{x1D53C}}_{\tau_I}\left[{\int}_{\tau_I}^{\tau_D^{\ast }}{e}^{-r\left(t-{\tau}_I\right)}\left(1-\phi \right)\left(X(t)-C\right) dt\right].\end{array}} $$

The above indicates that the equity value is the discounted net profit received by equity holders until default. At default, equity holders recover nothing. As is standard in the trade-off theory of capital structure, we assume that the entrepreneur maximizes equity value by choosing the default policy $ {\tau}_D $ . However, when choosing the optimal debt policy $ {C}^{\ast } $ , she maximizes firm value $ {V}_{\ast}\left({\tau}_I\right) $ , incorporating her expectation of debt value $ D\left({\tau}_I\right) $ .Footnote ⁸ In a symmetric rational expectation equilibrium, the entrepreneur’s valuation of debt, $ D\left({\tau}_I\right) $ , equals debt investors’ valuation and is given by

$$ D\left({\tau}_I\right)={\unicode{x1D53C}}_{\tau_I}\left[{\int}_{\tau_I}^{\tau_D^{\ast }}{e}^{-r\left(t-{\tau}_I\right)} Cdt+\left(1-\alpha \right){\int}_{\tau_D^{\ast}}^{\infty }{e}^{-r\left(t-{\tau}_I\right)}\left(1-\phi \right)X(t) dt\right]. $$

The first term inside the conditional expectation operator is the present value of the coupon collected while the firm remains solvent. The second term is the present value of the asset value recovered upon liquidation in bankruptcy.

Since $ {\tau}_D^{\ast } $ and $ {C}^{\ast } $ depend on $ {\tau}_I^{\ast } $ , these three decision variables are jointly determined at time 0 by maximizing the discounted firm value at investment.

$$ {V}_{\ast }(0)=\underset{\tau_I,{\tau}_D>{\tau}_I,C}{\sup}\;{\unicode{x1D53C}}_0\left[{e}^{-r{\tau}_I}{V}_{\ast}\left({\tau}_I\right)\right]=\underset{\tau_I,{\tau}_D>{\tau}_I,C}{\sup}\;{\unicode{x1D53C}}_0\left[{e}^{-r{\tau}_I}{\unicode{x1D53C}}_{\tau_I}\left[{V}_{\ast}\left({\tau}_I\right)\right]\right], $$

where the second equality follows from the property of conditional expectations, and the optimization is subject to the default policy determined in (4).

B. The Set of Priors

Agents face ambiguity about the EBIT reference model in equation (1) and consider “close” alternative models. With ambiguity aversion, they opt for worst-case EBIT dynamics. To capture jump ambiguity, we extend the utility framework of Chen and Epstein (Reference Chen and Epstein2002) to allow for uncertainty in both jump intensity and size distribution. Our extension builds on the general results of Quenez and Sulem (Reference Quenez and Sulem2013), (Reference Quenez and Sulem2014), who provide the comparison theorem for BSDEs under Lévy processes and general results for related optimal stopping problems under ambiguity.

Let $ \Theta $ denote the set of density generators. Each density generator $ \theta \in \Theta $ is binary (i.e., $ \theta =\left({\theta}_W,{\theta}_N\right) $ , where $ {\theta}_W $ is for the Brownian motion and $ {\theta}_N $ for the jump component). For each $ \theta \in \Theta $ , let $ {Z}^{\theta }(t) $ be the solution to the (forward) SDE:

$$ {dZ}^{\theta }(t)={Z}^{\theta}\left({t}^{-}\right)\left(-{\theta}_W(t) dW(t)-{\int}_{\mathrm{\mathbb{R}}}{\theta}_N\Big(t,u\Big)d\tilde{N}( dt, du\Big)\right), $$

for $ t\in \left[0,T\right] $ with $ {Z}^{\theta }(0)=1 $ . For $ {\theta}_W(t) $ , we adopt the $ \kappa $ -ignorance specification of Chen and Epstein (Reference Chen and Epstein2002) (i.e., $ {\theta}_W(t)\in \left[-\kappa, \kappa \right],0<\kappa <\infty $ ). Based on the technical requirements in Quenez and Sulem (Reference Quenez and Sulem2013), (Reference Quenez and Sulem2014), we specify $ {\theta}_N\left(t,u\right) $ as:

$$ {\displaystyle \begin{array}{ll}& {\theta}_N\left(t,u\right)=1-{e}^{\theta_{N,1}(t)u}{\mathbf{1}}_{u\ge 0}-{e}^{\theta_{N,2}(t)u}{\mathbf{1}}_{u<0},\\ {}& {\theta}_{N,1}(t)\in \left[-{M}_1,0\right],\hskip1em {\theta}_{N,2}(t)\in \left[0,{M}_2\right],\hskip1em \mathrm{and}\hskip1em {M}_1,{M}_2>0.\end{array}} $$

This specification for $ {\theta}_N\left(t,u\right) $ follows from the jump-size distribution in equation (2), and we discuss its implications after constructing the set of priors.

To construct the set of priors, we define a probability measure $ {Q}^{\theta } $ on $ {\mathrm{\mathcal{F}}}_T $ , equivalent to $ {Q}^0 $ for $ \theta \in \Theta $ as

$$ {\unicode{x1D53C}}^{\theta}\left[{\mathbf{1}}_A\right]=\unicode{x1D53C}\left[{\mathbf{1}}_A{Z}_T^{\theta}\right],\hskip1em A\in {\mathrm{\mathcal{F}}}_T. $$

Hence, under $ {Q}^{\theta } $ ,

$$ {dW}^{\theta }(t)= dW(t)+{\theta}_W(t) dt, $$

which represents the Brownian risk, and

(5) $$ {\tilde{N}}^{\theta}\left( dt, du\right)=\tilde{N}\left( dt, du\right)+{\theta}_N\left(t,u\right)\nu (du) dt=N\left( dt, du\right)-\left(1-{\theta}_N\left(t,u\right)\right)\nu (du) dt, $$

which is a compensated Poisson random measure by Girsanov’s theorem; see Øksendal and Sulem ((Reference Øksendal and Sulem2019), Chapter 1.4). Taken together, under $ {Q}^{\theta } $ , $ X(t) $ is given by

$$ \frac{dX(t)}{X\left({t}^{-}\right)}={\displaystyle \begin{array}{l}\left(\mu -{\theta}_W(t)\sigma -{\int}_{\mathrm{\mathbb{R}}}\left({e}^u-1\right){\theta}_N\left(t,u\right)\nu (du)\right) dt+\sigma {dW}^{\theta }(t)\\ {}+\hskip2px {\int}_{\mathrm{\mathbb{R}}}\left({e}^u-1\right){\tilde{N}}^{\theta}\left( dt, du\right).\end{array}} $$

Our method of distorting the reference measure can attenuate the impact of positive or negative jumps, depending on the monotonicity of the entrepreneur’s value function in the state variable. If the value function increases with the state variable, as in the baseline model without ambiguity, the worst-case measure leads to the smallest unconditional probability and mean size for positive jumps, and the largest values for negative jumps—all constrained within the set of priors. To see this, equation (5) shows that $ {N}^{\theta}\left( dt, du\right) $ has a Lévy measure $ {\nu}^{\theta }(du)=\left(1-{\theta}_N\left(t,u\right)\right)\nu (du) $ under $ {Q}^{\theta } $ . We can express $ {\nu}^{\theta }(du) $ as

$$ {\nu}^{\theta }(du)={\lambda}_t^{\theta }{f}_t^{\theta }(du), $$

where $ {\lambda}_t^{\theta } $ is the distorted jump intensity given by

$$ {\lambda}_t^{\theta }=\lambda {\int}_{\mathrm{\mathbb{R}}}\left(1-{\theta}_N\left(t,u\right)\right){f}_u du=\lambda \left(\frac{p{\eta}_1}{\eta_1-{\theta}_{N,1}(t)}+\frac{q{\eta}_2}{\eta_2+{\theta}_{N,2}(t)}\right), $$

and the distorted jump size density is given by

$$ {\displaystyle \begin{array}{c}{f}_{u,t}^{\theta }=\frac{\left(1-{\theta}_N\left(t,u\right)\right){f}_u}{\int_{\mathrm{\mathbb{R}}}\left(1-{\theta}_N\left(t,u\right)\right){f}_u du}\\ {}={p}_t^{\theta}\left({\eta}_1-{\theta}_{N,1}(t)\right){e}^{-\left({\eta}_1-{\theta}_{N,1}(t)\right)u}{\mathbf{1}}_{u\ge 0}+{q}_t^{\theta}\left({\eta}_2+{\theta}_{N,2}(t)\right){e}^{\left({\eta}_2+{\theta}_{N,2}(t)\right)u}{\mathbf{1}}_{u<0},\end{array}} $$

with $ {p}_t^{\theta } $ and $ {q}_t^{\theta } $ being the distorted probabilities of upward and downward jumps given by

$$ {\displaystyle \begin{array}{l}{p}_t^{\theta }=\frac{p{\eta}_1\left({\eta}_2+{\theta}_{N,2}(t)\right)}{p{\eta}_1\left({\eta}_2+{\theta}_{N,2}(t)\right)+q{\eta}_2\left({\eta}_1-{\theta}_{N,1}(t)\right)},\ \mathrm{and}\\ {}{q}_t^{\theta }=\frac{q{\eta}_2\left({\eta}_1-{\theta}_{N,1}(t)\right)}{p{\eta}_1\left({\eta}_2+{\theta}_{N,2}(t)\right)+q{\eta}_2\left({\eta}_1-{\theta}_{N,1}(t)\right)}.\end{array}} $$

Table 1 summarizes the Lévy measures for $ {N}^{\theta}\left( dt, du\right) $ under the set of priors specified $ {\theta}_{N,1}(t)\in \left[-{M}_1,0\right] $ and $ {\theta}_{N,2}(t)\in \left[0,{M}_2\right] $ . In the extreme case where only positive jumps occur ( $ p=1 $ ), the jump intensity ranges from $ \left[{\lambda \eta}_1/\left({\eta}_1+{M}_1\right),\lambda \right] $ , with the conditional probability of positive jumps fixed at 1. If the worst-case scenario corresponds to $ {\theta}_{N,1}(t)=-{M}_1 $ (or 0) for all $ t\ge 0 $ , the jump intensity is minimized (or maximized). A similar observation holds for the other extreme case ( $ q=1 $ ).

TABLE 1 Characteristics of the Lévy Measure Under Multiple Priors

C. Innovation with All-Equity Financing under Ambiguity

With the set of priors, the entrepreneur’s value function under all-equity financing at the investment time $ {\tau}_I $ and under a candidate measure $ {Q}^{\theta } $ is

(6) $$ {V}_e\left({\tau}_I;\theta \right)={\unicode{x1D53C}}_{\tau_I}^{\theta}\left[{\int}_{\tau_I}^{\infty }{e}^{-r\left(t-{\tau}_I\right)}\left(1-\phi \right)X(t) dt-I\right]. $$

At time 0, she chooses the optimal investment time $ {\tau}_I^e $ to start the project. The ambiguity-averse entrepreneur chooses the worst-case measure $ {Q}^{\theta^{\ast }} $ from the set of priors. As long as $ {Q}^{\theta^e}\left\{{\tau}_I^{\ast }<+\infty \right\}=1 $ , her value function at time 0 is

(7) $$ {V}_e\left(0;{\tau}_I^e,{\theta}^{\ast}\right)=\underset{\tau_I\ge 0}{\sup}\;\underset{Q^{\theta }}{\operatorname{inf}}\;{\unicode{x1D53C}}_0^{\theta}\left[{e}^{-r{\tau}_I}{V}_e\left({\tau}_I;\theta \right)\right]. $$

It is necessary to discuss a few technical details of the optimal stopping problem under ambiguity, given by equations (6) and (7). First, Quenez and Sulem (Reference Quenez and Sulem2014) prove that the nonlinear expectation in (7) admits the minmax relation. Thus, we can first solve the minimum expectation problem to find the worst-case measure $ {Q}^{\theta^{\ast }} $ and then solve the optimal stopping problem for $ {\tau}_I^{\ast } $ under $ {Q}^{\theta^{\ast }} $ . Because the minimum expectation problem involves the application of Girsanov’s theorem and the comparison theorem for BSDEs under Lévy processes and because $ {Q}^{\theta^{\ast }} $ exists for any horizon in our model, we can solve the optimal stopping problem under $ {Q}^{\theta^{\ast }} $ for the infinite horizon. Second, Quenez and Sulem (Reference Quenez and Sulem2013) show that the minimum expectation is dynamically consistent:

$$ \underset{Q^{\theta }}{\operatorname{inf}}\;{\unicode{x1D53C}}_0^{\theta}\left[{e}^{-r{\tau}_I}{V}_e\left({\tau}_I;\theta \right)\right]=\underset{Q^{\theta^{\prime }}}{\operatorname{inf}}\;{\unicode{x1D53C}}_0^{\theta^{\prime }}\left[\underset{Q^{\theta^{{\prime\prime} }}}{\operatorname{inf}}\;{\unicode{x1D53C}}_{\tau_I}^{\theta^{{\prime\prime} }}\left[{e}^{-r{\tau}_I}{V}_e\left({\tau}_I;\left\{{\theta}^{\prime },{\theta}^{{\prime\prime}}\right\}\right)\right]\right]. $$

where $ {\theta}^{\prime } $ and $ {\theta}^{{\prime\prime} } $ deliver density generators for the decision intervals $ \left[0,{\tau}_I\right] $ and $ \left[{\tau}_I,T\right] $ with $ T\le +\infty $ , respectively. Dynamic consistency provides analytical convenience in that the worst-case density generators for $ \left[0,{\tau}_I\right] $ and $ \left[{\tau}_I,T\right] $ coincide with that for $ \left[0,T\right] $ . Hence, it suffices to begin with the following to find the worst-case measure:

(8) $$ {V}_e\left(0;{\tau}_I,{\theta}^{\ast}\right)=\underset{Q^{\theta }}{\operatorname{inf}}\;{\unicode{x1D53C}}_0^{\theta}\left[{e}^{-r{\tau}_I}{V}_e\left({\tau}_I;\theta \right)\right],\hskip1em \mathrm{for}\hskip0.42em \mathrm{any}\hskip0.42em {\tau}_{\mathrm{I}}. $$

In the Supplementary Material, we provide the proof, which relies on dynamic consistency and the comparison theorem for BSDEs under the Lévy process.

An immediate implication is that under $ {Q}^{\theta^{\ast }} $ , $ X(t) $ follows the process

$$ \frac{dX(t)}{X\left({t}^{-}\right)}={\mu}^{\theta^{\ast }} dt+\sigma {dW}^{\theta^{\ast }}(t)+{\int}_{\mathrm{\mathbb{R}}}\left({e}^u-1\right){\tilde{N}}^{\theta^{\ast }}\left( dt, du\right) $$

where

$$ {\displaystyle \begin{array}{c}{\mu}^{\theta^{\ast }}=\mu -\kappa \sigma -{\int}_{\mathrm{\mathbb{R}}}\left({e}^u-1\right)\left(1-{e}^{-{M}_1u}{\mathbf{1}}_{u\ge 0}-{\mathbf{1}}_{u<0}\right)\nu (du)\\ {}=\mu +\underset{\mathrm{drift}\ \mathrm{ambiguity}\ \mathrm{discount}<0}{\underbrace{\left(-\kappa \sigma \right)}}+\underset{\mathrm{jump}\ \mathrm{ambiguity}\ \mathrm{discount}<0}{\underbrace{\left(-\frac{\lambda p}{\left({\eta}_1-1\right)}+\frac{\lambda p{\eta}_1}{\left({\eta}_1+{M}_1-1\right)\left({\eta}_1+{M}_1\right)}\right)}}.\end{array}} $$

The distorted process above indicates that drift ambiguity and jump ambiguity reduce the perceived mean return by $ \kappa \sigma $ and $ \lambda p/\left({\eta}_1-1\right)-\lambda p{\eta}_1/\left({\eta}_1+{M}_1-1\right)\left({\eta}_1+{M}_1\right) $ , respectively.

Before turning to the detailed discussion of $ X(t) $ under $ {Q}^{\theta^{\ast }} $ , it is important to note that jump ambiguity distortions remain independent of drift ambiguity. This property arises from the independence of the Brownian and jump terms in line with the Itô-Lévy Decomposition. In essence, the relative importance of the two drift distortions depends on the entrepreneur’s relative concern with diffusion ambiguity versus jump ambiguity.

Importantly, beyond reducing drift, jump ambiguity also distorts jump intensity and the size distribution. For instance, the conditional mean for positive jumps (in log units) reaches its minimum at $ 1/{\eta}_1^{\ast } $ , where $ {\eta}_1^{\ast }={\eta}_1+{M}_1 $ , while the conditional mean of negative jumps (in log units) attains its maximum at $ -1/{\eta}_2^{\ast } $ , with $ {\eta}_2^{\ast }={\eta}_2 $ . The conditional probability of positive jumps reaches its minimum at $ {p}^{\ast }=p{\eta}_1/\left(p{\eta}_1+q\left({\eta}_1+{M}_1\right)\right) $ , and that of negative jumps achieves its maximum at $ {q}^{\ast }=q\left({\eta}_1+{M}_1\right)/\left(p{\eta}_1+q\left({\eta}_1+{M}_1\right)\right) $ . The jump intensity in the worst-case scenario does not attain either its maximum or minimum value but instead equals $ {\lambda}^{\ast }=\lambda \left(p{\eta}_1/\left({\eta}_1+{M}_1\right)+q\right) $ .

These worst-case parameters are summarized in Table 1. The interpretation of these results follows from our earlier discussion of Table 1, where we considered the monotonicity of the value function with respect to the state variable. Specifically, the parameter $ {\lambda}^{\ast } $ takes an interior value, as the distortion minimizes the unconditional probability of positive jumps while maximizing that of negative jumps.

The expression for $ X(t) $ under $ {Q}^{\theta^{\ast }} $ reveals that jump ambiguity distorts the entire distribution of $ X(t) $ , whereas drift ambiguity distorts the mean only. This is evident by examining the moment generating function of $ Y(t)=\ln \left(X(t)/X(0)\right) $ as

$$ {\unicode{x1D53C}}^{\theta^{\ast }}\left[{e}^{vY(t)}\right]={e}^{t{\phi}^{\theta^{\ast }}(v)}, $$

where

$$ {\phi}^{\theta^{\ast }}(v)={\displaystyle \begin{array}{l}\frac{1}{2}{\sigma}^2{v}^2+\left({\mu}^{\theta^{\ast }}-\frac{1}{2}{\sigma}^2-{\lambda}^{\ast}\left(\frac{p{\eta}_1^{\ast }}{\eta_1^{\ast }-1}+\frac{q^{\ast }{\eta}_2^{\ast }}{\eta_2^{\ast }+1}-1\right)\right)v+\hskip2px {\lambda}^{\ast}\left(\frac{p^{\ast }{\eta}_1^{\ast }}{\eta_1^{\ast }-v}+\frac{q^{\ast }{\eta}_2^{\ast }}{\eta_2^{\ast }+v}-1\right).\end{array}} $$

It follows immediately that the variance of $ Y(t) $ is

$$ {\unicode{x1D53C}}^{\theta^{\ast }}\left[{\left(Y(t)-{\unicode{x1D53C}}^{\theta^{\ast }}\left[Y(t)\right]\right)}^2\right]={\sigma}^2t+2{\lambda}^{\ast}\left(\frac{p^{\ast }}{{\left({\eta}_1^{\ast}\right)}^2}+\frac{q^{\ast }}{{\left({\eta}_2^{\ast}\right)}^2}\right)t. $$

In contrast to drift ambiguity, jump ambiguity has a significant effect on the variance of $ Y(t) $ and higher moments, which we demonstrate numerically in Section III.B.

After finding the worst-case measure, we solve for the optimal investment policy and the value function in equation (7). Here, we can gain analytical tractability of the double exponential jump-diffusion process for optimal stopping problems. The following proposition characterizes the optimal investment policy and the associated value function under all-equity financing.

Proposition 2. Let $ {\eta}_1^{\ast }={\eta}_1+{M}_1 $ and $ {\eta}_2^{\ast }={\eta}_2 $ . For $ X(0)=x $ , the value function $ {V}_e\left(0;{\tau}_I^e,{\theta}^{\ast}\right) $ in (7) has the solution

$$ {V}_e\left(0;{\tau}_I^e,{\theta}^{\ast}\right)={\displaystyle \begin{array}{l}{A}_0{X}_I^e\left[{c}_{1,1}{\left(\frac{X_I^e}{x}\right)}^{-{\beta}_1}+{c}_{2,1}{\left(\frac{X_I^e}{x}\right)}^{-{\beta}_2}\right]-\hskip2px I\left[{c}_{1,0}{\left(\frac{X_I^e}{x}\right)}^{-{\beta}_1}+{c}_{2,0}{\left(\frac{X_I^e}{x}\right)}^{-{\beta}_2}\right],\end{array}} $$

where the optimal stopping time satisfies $ {\tau}_I^e={\operatorname{inf}}_t\;\left\{X(t)\ge {X}_I^e\right\} $ , and the optimal investment boundary $ {X}_I^e $ is given by

(9) $$ {A}_0{X}_I^e=\underset{option\ multiplier>1}{\underbrace{\frac{\beta_1{\beta}_2}{\left({\beta}_1-1\right)\left({\beta}_2-1\right)}\frac{\eta_1^{\ast }-1}{\eta_1^{\ast }}}}I,\hskip1em {A}_0=\underset{value\ multiplier}{\underbrace{\frac{1-\phi }{r-{\mu}^{\theta^{\ast }}}}}. $$

Here, $ {\beta}_1 $ , $ {\beta}_2 $ , $ {\beta}_3 $ , $ {\beta}_4 $ are constants satisfying $ -\infty <-{\beta}_4<-{\eta}_2^{\ast }<-{\beta}_3<0<{\beta}_1<{\eta}_1^{\ast }<{\beta}_2<\infty $ and are the four roots of the equation $ G\left(\beta \right)=r $ , with

(10) $$ G\left(\beta \right)=\frac{1}{2}{\sigma}^2{\beta}^2+\left[{\mu}^{\theta^{\ast }}-\frac{1}{2}{\sigma}^2-{\lambda}^{\ast}\left(\frac{p^{\ast }{\eta}_1^{\ast }}{\eta_1^{\ast }-1}+\frac{q^{\ast }{\eta}_2^{\ast }}{\eta_2^{\ast }+1}-1\right)\right]\beta +{\lambda}^{\ast}\left(\frac{p^{\ast }{\eta}_1^{\ast }}{\eta_1^{\ast }-\beta }+\frac{q^{\ast }{\eta}_2^{\ast }}{\eta_2^{\ast }+\beta }-1\right). $$

The constants $ {c}_{1,0},{c}_{2,0},{c}_{1,1} $ and $ {c}_{2,1} $ are given by

(11) $$ {c}_{1,0}=\frac{\eta_1^{\ast }-{\beta}_1}{\beta_2-{\beta}_1}\frac{\beta_2}{\eta_1^{\ast }},{c}_{2,0}=\frac{\beta_2-{\eta}_1^{\ast }}{\beta_2-{\beta}_1}\frac{\beta_1}{\eta_1^{\ast }},{c}_{1,1}=\frac{\eta_1^{\ast }-{\beta}_1}{\beta_2-{\beta}_1}\frac{\beta_2-1}{\eta_1^{\ast }-1},{c}_{2,1}=\frac{\beta_2-{\eta}_1^{\ast }}{\beta_2-{\beta}_1}\frac{\beta_1-1}{\eta_1^{\ast }-1}. $$

The investment payoff at $ {\tau}_I^e $ is $ {A}_0X\left({\tau}_I^e\right)-I $ . In equation (9), the term $ {A}_0{X}_I^e $ represents the equity value at $ {\tau}_I^e $ (in the absence of overshooting) and admits a Gordon growth-like interpretation.Footnote ⁹ The value multiplier (VM) reflects the project’s value when $ X(t)=1 $ , and it decreases as ambiguity increases. On the right-hand side, the equity value is expressed in terms of the cost $ I $ and the option multiplier (OM), which accounts for the effect of uncertainty on the investment decision. Here, uncertainty includes both diffusion and jump risks, characterized by the primitive parameters of the double-exponential jump-diffusion process, as well as drift and jump ambiguity.

An immediate observation from the inequality:

$$ \frac{\beta_2}{\beta_2-1}I<{A}_0{X}_I^e<\frac{\beta_1}{\beta_1-1}I, $$

is that as jump ambiguity $ {M}_1\to \infty $ , $ {\beta}_2 $ approaches infinity, causing the lower bound of $ {A}_0{X}_I^e $ to decrease monotonically toward $ I $ . Although the limit of $ {\beta}_1 $ is harder to deduce, it is evident that the upper bound also decreases monotonically with $ {M}_1 $ . We conjecture that both bounds eventually converge, reducing the OM to 1 and driving the equity value at $ {\tau}_I^e $ to the net present value (NPV) threshold, $ I $ .

We deduce that $ {X}_I^e $ increases with $ {M}_1 $ . While the effect of $ {M}_1 $ on $ {X}_I^e $ is not immediately clear due to the simultaneous decrease in both the OM and VM, we can infer the relationship by comparing their rates of change. Computing $ \partial {X}_I^e/\partial {M}_1 $ is technically challenging, but the decline in $ {\mu}^{\theta^{\ast }} $ with $ {M}_1 $ follows a rate bounded by $ \tilde{C}/{M}_1^2 $ , leading the VM to decrease at the same rate, where $ \tilde{C}>0 $ denotes an arbitrary constant. Moreover, the upper and lower bounds for $ {A}_0{X}_I^e $ decrease at a rate bounded by $ \tilde{C}/{M}_1 $ . Thus, we numerically verify that $ {X}_I^e $ increases with $ {M}_1 $ at a rate bounded by $ \tilde{C}/{M}_1 $ .

Lastly, we anticipate that the expected investment time, $ {\tau}_I^e $ , also increases with $ {M}_1 $ for two reasons. First, investment occurs when $ X(t) $ reaches $ {X}_I^e $ from below, and under $ {Q}^{\theta^{\ast }} $ , $ X(t) $ has the least upward potential, leading to a longer average time to reach any given level compared to $ {Q}^0 $ . Second, since $ {X}_I^e $ is higher under $ {Q}^{\theta^{\ast }} $ than under $ {Q}^0 $ , the expected hitting time is further extended, reinforcing the delay in investment.

A fixed cash flow threshold policy is to invest when $ X(t)\ge \overline{X} $ for the first time, where $ \overline{X} $ is a fixed level.Footnote ¹⁰ In this case, the optimality condition (9) implies that the firm can maximize its equity value by choosing $ I $ optimally, satisfying $ {I}_e^{\ast }=\overline{X}\times VM/ OM $ . As discussed above, jump ambiguity lowers the VM at a rate bounded by $ \tilde{C}/{M}_1^2 $ and the OM at a rate bounded by $ \tilde{C}/{M}_1 $ . Hence, $ {I}_e^{\ast } $ decreases with $ {M}_1 $ at a rate bounded by $ \tilde{C}/{M}_1 $ .

Remark 2 parallels the insight of Campello and Kankanhalli (Reference Campello, Kankanhalli and Denis2024), who show that greater uncertainty reduces investment size in a two-period framework with mean-preserving spread.Footnote ¹¹ Our approach differs from that of Campello and Kankanhalli (Reference Campello, Kankanhalli and Denis2024), as in that model the ambiguity-neutral decision maker evaluates uncertainty through second-order stochastic dominance, whereas our continuous-time framework explicitly incorporates both diffusion and jump risks together with ambiguity.

D. Innovation with Optimal Financing

1. Optimal Default under Ambiguity

Because of the property $ {Q}^{\theta^{\ast }}\{{\tau}_I^{\ast }<+\mathrm{\infty}\}=1 $ in the all-equity financing case and dynamic consistency, we can first solve for the entrepreneur’s optimal default decision and then the optimal coupon to determine the optimal capital structure. Under the worst-case measure $ {Q}^{\theta^{\ast }} $ (Proposition 3), $ {Q}^{\theta^{\ast }}\{{\tau}_D^{\ast }<+\mathrm{\infty}\}=1 $ holds for our parameter requirements, and thus the levered equity value at $ {\tau}_I $ follows:

(12) $$ E\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },C\right)=\underset{\tau_D\ge {\tau}_I}{\sup}\;\underset{Q^{\theta }}{\operatorname{inf}}\;{\unicode{x1D53C}}_{\tau_I}\left[{\int}_{\tau_I}^{\tau_D}{e}^{-r\left(t-{\tau}_I\right)}\left(1-\phi \right)\left(X(t)-C\right) dt\right]. $$

Without ambiguity, the problem reduces to the baseline model in equation (4).

Using the same approach as in Section II.C, we begin by finding the density generator that delivers the worst-case measure:

(13) $$ E\left({\tau}_I;{\tau}_D,{\theta}^{\ast },C\right)=\underset{Q^{\theta }}{\operatorname{inf}}\;{\unicode{x1D53C}}_{\tau_I}\left[{\int}_{\tau_I}^{\tau_D}{e}^{-r\left(t-{\tau}_I\right)}\left(1-\phi \right)\left(X(t)-C\right) dt\right]. $$

Proposition 3. The density generator that gives the minimum expectation in (13) is $ {\theta}^{\ast }=\left({\theta}_W^{\ast },{\theta}_N^{\ast}\right)=\left(\kappa, 1-{e}^{-{M}_1u}{\boldsymbol{1}}_{u\ge 0}-{\boldsymbol{1}}_{u<0}\right) $ for all $ t\in \left[{\tau}_I,T\right] $ .

Proposition 3 establishes that the worst-case measure is the same under both optimal financing and all-equity financing. This result follows because, in both cases, equity holders receive net profit, and their main concern under ambiguity is that the perceived EBIT profile is the least favorable.

Given that we have determined the worst-case prior for equity holders, we can then solve for the optimal default timing $ {\tau}_D^{\ast } $ and the value function. The results are summarized in the proposition below.

Proposition 4. Let $ {\beta}_3 $ , $ {\beta}_4 $ , $ {\mu}^{\theta^{\ast }} $ , $ {\eta}_1^{\ast } $ , $ {\eta}_2^{\ast } $ , and $ {A}_0 $ be the same as in Proposition 2. The levered equity value $ E\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },C\right) $ in (12) has the following expression.

$$ {\displaystyle \begin{array}{c}E\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },C\right)={A}_0X\left({\tau}_I\right)-\frac{\left(1-\phi \right)C}{r}\left[1-{d}_{1,0}{\left(\frac{X_D^{\ast }}{X\left({\tau}_I\right)}\right)}^{\beta_3}-{d}_{2,0}{\left(\frac{X_D^{\ast }}{X\left({\tau}_I\right)}\right)}^{\beta_4}\right]\\ {}\hskip1.05em -{A}_0{X}_D^{\ast}\left[{d}_{1,1}{\left(\frac{X_D^{\ast }}{X\left({\tau}_I\right)}\right)}^{\beta_3}+{d}_{2,1}{\left(\frac{X_D^{\ast }}{X\left({\tau}_I\right)}\right)}^{\beta_4}\right],\end{array}} $$

where

(14) $$ {d}_{1,0}=\frac{\eta_2^{\ast }-{\beta}_3}{\beta_4-{\beta}_3}\frac{\beta_4}{\eta_2^{\ast }},{d}_{2,0}=\frac{\beta_4-{\eta}_2^{\ast }}{\beta_4-{\beta}_3}\frac{\beta_3}{\eta_2^{\ast }},{d}_{1,1}=\frac{\eta_2^{\ast }-{\beta}_3}{\beta_4-{\beta}_3}\frac{\beta_4+1}{\eta_2^{\ast }+1},{d}_{2,1}=\frac{\beta_4-{\eta}_2^{\ast }}{\beta_4-{\beta}_3}\frac{\beta_3+1}{\eta_2^{\ast }+1}, $$

and the optimal default policy is $ {\tau}_D^{\ast }={\operatorname{inf}}_t\left\{X(t)\le {X}_D^{\ast}\right\} $ with the optimal default boundary $ {X}_D^{\ast } $ given by

(15) $$ {A}_0{X}_D^{\ast }=\underset{option\ multiplier<1}{\underbrace{\frac{\beta_3{\beta}_4\left({\eta}_2^{\ast }+1\right)}{\left({\beta}_3+1\right)\left({\beta}_4+1\right){\eta}_2^{\ast }}}}\frac{\left(1-\phi \right)C}{r}. $$

At default, under liquidation bankruptcy, equity holders lose the firm, valued at $ {A}_0{X}_D^{\ast } $ —the left-hand side of equation (15). At the same time, they cease paying the tax deductible perpetual coupons, valued at $ \left(1-\phi \right)C/r $ . Thus, the NPV rule dictates that default occurs when $ {A}_0X(t)=\left(1-\phi \right)C/r $ for the first time. However, defaulting at the NPV threshold is suboptimal because equity holders have the option to inject additional equity to delay default, anticipating a future rebound in $ X(t) $ . Therefore, the optimal default threshold is set below the NPV threshold, discounted by the OM, which reflects the factors discussed earlier in relation to $ {X}_I^{\ast } $ in Proposition 2.Footnote ¹²

Remark 3. Jump ambiguity has a secondary effect on optimal default boundary $ {X}_D^{\ast } $ .

For $ {X}_D^{\ast } $ , we can establish its lower and upper bounds as follows:

$$ \frac{\beta_3+1}{\beta_3}\frac{\left(1-\phi \right)C}{r}<{A}_0{X}_D^{\ast }<\frac{\beta_4+1}{\beta_4}\frac{\left(1-\phi \right)C}{r}. $$

Since $ {\theta}_{N,2}=0 $ under $ {Q}^{\theta^{\ast }} $ , jump ambiguity governed by $ {M}_1 $ does not significantly affect $ {\beta}_3 $ and $ {\beta}_4 $ , implying that the bounds for $ {A}_0{X}_D^{\ast } $ do not converge in the same way as the bounds for $ {A}_0{X}_I^{\ast } $ . As a result, the influence of jump ambiguity on $ {X}_D^{\ast } $ stems primarily from the interaction between the VM and $ {C}^{\ast } $ , the endogenously determined optimal coupon. This suggests that both the OM and the default boundary are less sensitive to jump ambiguity than their counterparts for investment.

2. Optimal Financing Choice

To find the optimal financing choice, we seek the debt value or optimal coupon that maximizes firm value upon debt issuance. We consider a symmetric equilibrium in which debt investors share the same preferences and set of priors as equity investors. Furthermore, debt and equity holders are mutually aware of each other’s decision rules. Thus, for any investment policy $ {\tau}_I $ and coupon policy $ C $ chosen at the stopping time $ {\tau}_I $ , debt holders’ valuation is given by

(16) $$ D\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },C\right)=\underset{Q^{\theta }}{\operatorname{inf}}\;{\unicode{x1D53C}}_{\tau_I}^{\theta}\left[{\int}_{\tau_I}^{\tau_D^{\ast }}{e}^{-r\left(t-{\tau}_I\right)} Cdt+\left(1-\alpha \right){\int}_{\tau_D^{\ast}}^{\infty }{e}^{-r\left(t-{\tau}_I\right)}\left(1-\phi \right)X(t) dt\right]. $$

Proposition 5. The density generator that gives the minimum expectation in (16) is $ {\theta}^{\ast }=\left({\theta}_W^{\ast },{\theta}_N^{\ast}\right)=\left(\kappa, 1-{e}^{-{M}_1u}{\boldsymbol{1}}_{u\ge 0}-{\boldsymbol{1}}_{u<0}\right) $ for all $ t\ge {\tau}_I $ . Furthermore, the value of debt has the expression.

$$ D\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },C\right)={\displaystyle \frac{C}{r}\left[1-{d}_{1,0}{\left(\frac{X_D^{\ast }}{X\left({\tau}_I\right)}\right)}^{\beta_3}-{d}_{2,0}{\left(\frac{X_D^{\ast }}{X\left({\tau}_I\right)}\right)}^{\beta_4}\right]}\hskip0ex \hspace{5.5pc}+\hskip2px {A}_1{X}_D^{\ast}\left[{d}_{1,1}{\left(\frac{X_D^{\ast }}{X\left({\tau}_I\right)}\right)}^{\beta_3}+{d}_{2,1}{\left(\frac{X_D^{\ast }}{X\left({\tau}_I\right)}\right)}^{\beta_4}\right], $$

where $ {d}_{i,j} $ is given by (14), $ {X}_D^{\ast } $ is given by (15), and $ {A}_1=\left(1-\alpha \right){A}_0 $ .

The result above suggests that debt investors use the same pricing measure as the entrepreneur and external equity investors do. This alignment is intuitive: in liquidation, debt investors would effectively become equity holders of the unleveraged firm, capturing the stochastic cash flows produced by the project. In addition, the debt value prior to default, which is represented by the first term in the integrand in equation (16), increases with the state variable. Consequently, their worst-case scenario coincides with that of the equity holders.

Now we can find the optimal capital structure by choosing the optimal coupon $ {C}^{\ast } $ to maximize the firm value at the investment time $ {\tau}_I $ ,

(17) $$ {V}_{\ast}\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },{C}^{\ast}\right)=\underset{C}{\max}\;\left\{\left(E\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },C\right)+D\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },C\right)-I\right)\right\}. $$

Unlike in the pure diffusion case (e.g., Sundaresan and Wang (Reference Sundaresan and Wang2007), Sundaresan et al. (Reference Sundaresan, Wang and Yang2015)), a closed-form solution for $ {C}^{\ast } $ is not available for Problem (17). Nevertheless, the following proposition shows that $ {C}^{\ast } $ remains a linear function of $ X\left({\tau}_I\right) $ as in the pure diffusion case.

Proposition 6. The optimal coupon $ {C}^{\ast } $ that maximizes $ {V}_{\ast}\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },{C}^{\ast}\right) $ in (17) has the form $ {C}^{\ast }=\psi X\left({\tau}_I\right) $ , where $ \psi $ is the unique positive solution to the following equation:

$$ \frac{\phi }{r}={A}_2{\psi}^{\beta_3}+{A}_3{\psi}^{\beta_4}, $$

where

$$ {\displaystyle \begin{array}{l}{A}_2=\frac{\phi }{r}{d}_{1,0}\left(1+{\beta}_3\right){\gamma}^{\beta_3}+\frac{\alpha \left(1-\phi \right){d}_{1,1}\left(1+{\beta}_3\right){\gamma}^{1+{\beta}_3}}{r-{\mu}^{\theta^{\ast }}},\\ {}{A}_3=\frac{\phi }{r}{d}_{2,0}\left(1+{\beta}_4\right){\gamma}^{\beta_4}+\frac{\alpha \left(1-\phi \right){d}_{2,1}\left(1+{\beta}_4\right){\gamma}^{1+{\beta}_4}}{r-{\mu}^{\theta^{\ast }}},\end{array}} $$

$ {\beta}_i $ , and $ {d}_{i,j} $ are given by (14), and $ \gamma $ satisfies $ {X}_D^{\ast }=\gamma C $ with $ \gamma =\frac{\beta_3{\beta}_4\left({\eta}_2^{\ast }+1\right)}{\left({\beta}_3+1\right)\left({\beta}_4+1\right){\eta}_2^{\ast }}\frac{r-{\mu}^{\theta^{\ast }}}{r} $ as in (15).

3. The Value of the Growth Option

Substituting the solutions to $ E\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },{C}^{\ast}\right) $ and $ D\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },{C}^{\ast}\right) $ into $ {V}_{\ast}\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },{C}^{\ast}\right) $ , we obtain the firm value at $ {\tau}_I $ as

(18) $$ {V}_{\ast}\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },{C}^{\ast}\right)={A}_4X\left({\tau}_I\right)-I, $$

where

$$ {A}_4={A}_0+\frac{\phi }{r}\psi \left[1-{d}_{1,0}{\left(\gamma \psi \right)}^{\beta_3}-{d}_{2,0}{\left(\gamma \psi \right)}^{\beta_4}\right]-\alpha {A}_0\gamma \psi \left[{d}_{1,1}{\left(\gamma \psi \right)}^{\beta_3}+{d}_{2,1}{\left(\gamma \psi \right)}^{\beta_4}\right] $$

for $ {A}_0 $ given in Proposition 2. Thus, the value of the growth option under optimal financing at time 0 is

(19) $$ {V}_{\ast}\left(0;{\tau}_I^{\ast },{\tau}_D^{\ast },{\theta}^{\ast },{C}^{\ast}\right)=\underset{\tau_I}{\sup}\;\underset{Q^{\theta }}{\operatorname{inf}}\;{\unicode{x1D53C}}_0^{\theta}\left[{e}^{-r{\tau}_I}{V}_{\ast}\left({\tau}_I;{\tau}_D^{\ast },{\theta}^{\ast },{C}^{\ast}\right)\right]. $$

Proposition 7. The density generator that gives the minimum expectation in (19) is $ {\theta}^{\ast }=\left({\theta}_W^{\ast },{\theta}_N^{\ast}\right)=\left(\kappa, 1-{e}^{-{M}_1u}{\boldsymbol{1}}_{u\ge 0}-{\boldsymbol{1}}_{u<0}\right) $ for all $ t\in \left[0,{\tau}_I^{\ast}\right] $ . Furthermore, for $ X(0)=x $ , the value of the growth option has the expression

$$ {V}_{\ast}\left(0;{\tau}_I^{\ast },{\tau}_D^{\ast },{\theta}^{\ast },{C}^{\ast}\right)={\displaystyle {A}_4{X}_I^{\ast}\left[{c}_{1,1}{\left(\frac{X_I^{\ast }}{x}\right)}^{-{\beta}_1}+{c}_{2,1}{\left(\frac{X_I^{\ast }}{x}\right)}^{-{\beta}_2}\right]}\hskip0ex \hspace{7pc}-\hskip2px I\left[{c}_{1,0}{\left(\frac{X_I^{\ast }}{x}\right)}^{-{\beta}_1}+{c}_{2,0}{\left(\frac{X_I^{\ast }}{x}\right)}^{-{\beta}_2}\right], $$

where the optimal stopping time satisfies $ {\tau}_I^{\ast }={\operatorname{inf}}_t\left\{X(t)\ge {X}_I^{\ast}\right\} $ , and the investment boundary $ {X}_I^{\ast } $ satisfies

(20) $$ {A}_4{X}_I^{\ast }=\underset{option\ multiplier>1}{\underbrace{\frac{\beta_1{\beta}_2}{\left({\beta}_1-1\right)\left({\beta}_2-1\right)}\frac{\eta_1^{\ast }-1}{\eta_1^{\ast }}}}I. $$

In the formulation above, $ {c}_{i,j} $ , $ {\beta}_i $ , and $ {\eta}_i^{\ast } $ are the same as in Proposition 2, and $ {A}_4 $ is from (18).

In equation (20), $ {A}_4 $ represents the VM under optimal financing, which is higher than the VM under equity financing, $ {A}_0 $ . The difference between these two multipliers reflects the net tax benefit of debt. The tax benefit (TB) and bankruptcy cost (BC) at $ {\tau}_I^{\ast } $ can be expressed as:

$$ {\displaystyle \begin{array}{l} TB=\frac{\phi {C}^{\ast }}{r}\left(1-{\unicode{x1D53C}}_{\tau_I^{\ast}}^{\theta^{\ast }}\left[{e}^{-r\left({\tau}_D^{\ast }-{\tau}_I^{\ast}\right)}\right]\right)=\frac{\phi \psi X\left({\tau}_I^{\ast}\right)}{r}\left[1-{d}_{1,0}{\left(\gamma \psi \right)}^{\beta_3}-{d}_{2,0}{\left(\gamma \psi \right)}^{\beta_4}\right],\\ {} BC=\alpha {A}_0{\unicode{x1D53C}}_{\tau_I^{\ast}}^{\theta^{\ast }}\left[X\left({\tau}_D^{\ast}\right){e}^{-r\left({\tau}_D^{\ast }-{\tau}_I\right)}\right]=\alpha {A}_0\gamma \psi X\left({\tau}_I^{\ast}\right)\left[{d}_{1,1}{\left(\gamma \psi \right)}^{\beta_3}+{d}_{2,1}{\left(\gamma \psi \right)}^{\beta_4}\right].\end{array}} $$

E. The Investment Acceleration Benefit of Debt

We use the analytical solutions to discuss the general implications of (optimal) debt financing relative to all-equity financing. Throughout the rest of the paper, we use the Arrow-Debreu (AD) price of investment, $ {\unicode{x1D53C}}_0^{\theta^{\ast }}\left[{e}^{-r{\tau}_I^j}\right] $ for $ j\in \left\{e,\ast \right\} $ , to capture the expected time until investment occurs. The AD price represents the time-0 value of a unit payoff made at the stopping time $ {\tau}_I^j $ , with higher AD prices indicating shorter expected times to investment.

Using the results from Propositions 2 and 7, we have

(21) $$ {\displaystyle \begin{array}{rcl}\frac{X_I^e}{X_I^{\ast }}& =& \frac{A_4}{A_0}\\ {}& =& 1+\frac{\phi \left(r-{\mu}^{\ast}\right)}{\left(1-\phi \right)r}\psi \left[1-{d}_{1,0}{\left(\gamma \psi \right)}^{\beta_3}-{d}_{2,0}{\left(\gamma \psi \right)}^{\beta_4}\right]-\hskip2px \alpha \gamma \psi \left[{d}_{1,1}{\left(\gamma \psi \right)}^{\beta_3}+{d}_{2,1}{\left(\gamma \psi \right)}^{\beta_4}\right]\\ {}& =& 1+\frac{\phi \left(r-{\mu}^{\ast}\right)}{\left(1-\phi \right)r}\psi \left[{d}_{1,0}{\beta}_3{\left(\gamma \psi \right)}^{\beta_3}+{d}_{2,0}{\beta}_4{\left(\gamma \psi \right)}^{\beta_4}\right]+\hskip2px \alpha \gamma \psi \left[{d}_{1,1}{\beta}_3{\left(\gamma \psi \right)}^{\beta_3}+{d}_{2,1}{\beta}_4{\left(\gamma \psi \right)}^{\beta_4}\right]>1,\end{array}} $$

(22) $$ \frac{AD_{\ast }}{AD_e}={\left(\frac{A_4}{A_0}\right)}^{\beta_1}\left(1+{A}_5\right),\hskip1em {A}_5=\frac{c_{2,0}\left[{\left(\frac{x}{X_I^{\ast }}\right)}^{\beta_2-{\beta}_1}-{\left(\frac{x}{X_I^e}\right)}^{\beta_2-{\beta}_1}\right]}{c_{1,0}+{c}_{2,0}{\left(\frac{x}{X_I^e}\right)}^{\beta_2-{\beta}_1}}>0, $$

and

(23) $$ \frac{V_{\ast }}{V_e}={\left(\frac{A_4}{A_0}\right)}^{\beta_1}\left(1+{A}_6\right),\hskip1em {A}_6=\frac{\frac{c_{2,0}}{\beta_2-1}\left[{\left(\frac{x}{X_I^{\ast }}\right)}^{\beta_2-{\beta}_1}-{\left(\frac{x}{X_I^e}\right)}^{\beta_2-{\beta}_1}\right]}{\frac{c_{1,0}}{\beta_1-1}+\frac{c_{2,0}}{\beta_2-1}{\left(\frac{x}{X_I^e}\right)}^{\beta_2-{\beta}_1}}>0. $$

The third equality in (21) follows from the first-order condition for $ \psi $ (see the proof of Proposition 6). The second part of the above remark follows from $ {\beta}_1>1 $ and $ {A}_4>{A}_0 $ , which imply that a 1% reduction in the investment threshold leads to more than a 1% increase in the AD price and project value under debt financing.Footnote ¹³ Hence, $ {\beta}_1 $ captures the expected first-passage time effect, determined by the sample-path properties of $ X(t) $ . The second (or third) line in (21) defines the zero-leverage equivalent cash flow level—the level at which an unlevered firm has the same value as its optimally levered counterpart when the latter’s cash flow level is 1. Moreover, $ {A}_5 $ and $ {A}_6 $ are close to zero because $ x<{X}_I^{\ast }<{X}_I^e $ , $ {\beta}_2-{\beta}_1 $ is large, and the coefficients involving $ {c}_{1,0} $ and $ {c}_{2,0} $ remain moderate under realistic parameter values. Collectively, these observations imply that the acceleration benefit of debt reflects an interaction between the base ( $ {A}_4/{A}_0 $ ) and the power ( $ {\beta}_1>1 $ ). The benefit is further an interaction between $ {\beta}_3 $ , the primary factor for the base, and $ {\beta}_1 $ , the passage-time factor.

As discussed earlier, although ambiguity weakens the upside potential of $ X(t) $ and reduces the base, its impact on $ \gamma $ is secondary (see Remark 3). Furthermore, the concavity of $ {A}_4 $ in $ \psi $ (Proposition 6) implies that higher ambiguity reduces $ {A}_4 $ at a diminishing rate. Likewise, $ {A}_0 $ decreases as ambiguity increases, which slows the reduction of the base. Meanwhile, ambiguity raises $ {\beta}_1 $ , which can amplify $ {\left({A}_4/{A}_0\right)}^{\beta_1} $ for two reasons: the slowing decline of $ {A}_4/{A}_0 $ and the dominance of the power effect when $ {A}_4/{A}_0>1 $ . Overall, ambiguity enhances the gains in the AD price and project value by amplifying the power effect.

This follows from equations (9) and (20):

$$ {A}_0\overline{X}= OM\times {I}_e^{\ast },\hskip1em {A}_4\overline{X}= OM\times {I}_{\ast}^{\ast },\hskip1em \frac{I_{\ast}^{\ast }}{I_e^{\ast }}=\frac{A_4}{A_0}>1, $$

with $ {A}_4/{A}_0 $ decreasing in ambiguity at a diminishing rate.

A. Parameter Choices

Table 2 presents our parameter choices. The parameters in the first row correspond to the deterministic part of the model. These parameter values are drawn from the traditional literature on irreversible investment and capital structure (Leland (Reference Leland1998), Goldstein et al. (Reference Goldstein, Ju and Leland2001), Sundaresan and Wang (Reference Sundaresan and Wang2007), Chen and Kou (Reference Chen and Kou2009), and Sundaresan et al. (Reference Sundaresan, Wang and Yang2015)). To make the quantitative analysis more realistic, we estimate the parameters related to the stochastic part of our model $ \left\{\sigma, \lambda, {\eta}_1,{\eta}_2,p\right\} $ using DJX options data on the Dow Jones Industrial Average (DJIA) index. Because these parameters are inferred from option prices, they are risk neutral by construction and therefore align with our specification of the reference measure as risk neutral. Under the assumptions of the Gordon growth model, the cash flow and equity value processes share the same stochastic component; see Goldstein et al. (Reference Goldstein, Ju and Leland2001). This feature is also consistent with the equilibrium setting of Kou (Reference Kou2002), where the equilibrium asset price is related to cash flows through the utility-gradient pricing kernel.

TABLE 2 Benchmark Parameter Values

The DJIA index broadly represents the U.S. economy and includes large, industry-leading firms. These firms are not only well-established but also play a significant role in innovation. Moreover, DJX options are European-style and can only be exercised at maturity. This feature simplifies the pricing and allows for more accurate estimation of the model parameters. By contrast, pricing American options requires more complex numerical algorithms and accounting for the early-exercise premium, which may introduce additional numerical or estimation errors.

We use the full historical series of standardized DJX option contracts from the Ivy DB OptionMetrics volatility-surface data set. Our sample spans from the inception of these options on Oct. 6, 1997, through Aug. 31, 2023, which is the latest available date in the database. In addition, for each option on each trading date, we collect the corresponding closing index level, continuous dividend yield, and the term structure of risk-free interest rates provided by Ivy DB.Footnote ¹⁴ We focus on options with more than 30 days to expiration and moneyness in the range $ S/K\in \left(\mathrm{0.95,1.05}\right) $ , as these contracts tend to be more liquid. Moreover, since our model is based on an infinite horizon, incorporating longer-dated options is more appropriate for estimation.

We estimate the risk-neutral parameters $ \left(\sigma, \lambda, p,{\eta}_1,{\eta}_2\right) $ using a standard approach from the empirical option pricing literature (e.g., Bakshi, Cao, and Chen (Reference Bakshi, Cao and Chen1997), Huang and Wu (Reference Huang and Wu2004)). Specifically, for each trading day, we obtain parameter estimates by minimizing the sum of squared errors between model-implied and market-observed put option prices.Footnote ¹⁵ The parameter values reported are time-series averages of these daily estimates. A two-sided t-test against the null hypothesis of zero mean rejects the null at the 1% significance level for all parameters.

Panel B of Table 2 reports the first four central moments of $ Y(t) $ for $ t=1 $ .Footnote ¹⁶ The variance is 0.069, corresponding to a standard deviation of 0.26. Most of this variation arises from the jump component, as indicated by the relatively low diffusion volatility ( $ \sigma =0.118 $ ). The skewness and kurtosis are −0.824 and 4.93, respectively—values that are broadly consistent with those reported by Conrad, Dittmar, and Ghysels (Reference Conrad, Dittmar and Ghysels2013), who analyze the full universe of OptionMetrics data. In their Table 1, the median skewness ranges from −1.34 to −0.30, and the median kurtosis lies between 3.68 and 7.70 over the 1996–2005 period.

B. Quantifying Ambiguity

We employ relative entropy and detection-error probabilities to gauge ambiguity by comparing the likelihoods implied by an alternative model, $ {Q}^{\theta } $ , with those of the reference model, $ {Q}^0 $ . When ambiguity is small, the alternative model is close to the reference model in terms of relative entropy growth, and thus it is statistically challenging to differentiate them based on a finite sample of trajectory observations. Conversely, when ambiguity is large, it is relatively easy to distinguish them statistically. Thus, a plausible scope of ambiguity requires considering a set of alternative models that are statistically close to the reference model.

Formally, the relative entropy measures the distance between a pair of probability measures. Given $ {Q}^{\theta } $ and $ {Q}^0 $ , we can write the growth in entropy of $ {Q}^{\theta } $ relative to $ {Q}^0 $ over the time interval $ \left[t,t+\Delta t\right] $ , $ \mathcal{R}\left({\theta}_t\right) $ , as

$$ G\left(t,t+\Delta t\right)={\unicode{x1D53C}}_t^{\theta}\left[\ln \left(\frac{Z^{\theta}\left(t+\Delta t\right)}{Z^{\theta }(t)}\right)\right],\hskip1em \mathcal{R}\left({\theta}_t\right)=\underset{\Delta t\to 0}{\lim}\frac{G\left(t,t+\Delta t\right)}{\Delta t}\hskip1em t\ge 0. $$

We denote by $ h $ the relative entropy growth under the worst-case measure (i.e., $ h=\mathcal{R}\left({\theta}^{\ast}\right) $ ). The independence of the diffusion and jump components (per Itô-Lévy Decomposition) implies that diffusion ambiguity and jump ambiguity contribute additively to relative entropy growth (i.e., $ \mathcal{R}\left({\theta}^{\ast}\right)=\mathcal{R}\left({\theta}_W^{\ast}\right)+\mathcal{R}\left({\theta}_N^{\ast}\right) $ and $ {h}_W+{h}_N=h $ ). We use $ {h}_W $ and $ {h}_N $ to regulate the respective contributions of diffusion ambiguity and jump ambiguity to total relative entropy.

For the double exponential jump-diffusion process, we derive the relative entropy growth $ \mathcal{R}\left({\theta}^{\ast}\right) $ as (see the Supplementary Material for the proof)

$$ \mathcal{R}\left({\theta}^{\ast}\right)=\frac{1}{2}{\kappa}^2+\left(\frac{1}{\eta_1}-\frac{2}{\eta_1^{\ast }}+\frac{1}{\eta_1^{\ast }+{M}_1}\right)\lambda p{\eta}_1-\left(\frac{M_1}{{\left({\eta}_1^{\ast}\right)}^2}-\frac{1}{\eta_1^{\ast }}+\frac{1}{\eta_1^{\ast }+{M}_1}\right){\lambda}^{\ast }{p}^{\ast }{\eta}_1^{\ast }, $$

where $ {\eta}_1^{\ast } $ , $ {p}^{\ast } $ , and $ {\lambda}^{\ast } $ are the same as in Proposition 2. Moreover, given that $ {h}_W+{h}_N=h $ , we have

$$ {h}_W=\frac{1}{2}{\kappa}^2,\hskip0.2em \mathrm{and}\hskip0.4em {h}_N=\left(\frac{1}{\eta_1}-\frac{2}{\eta_1^{\ast }}+\frac{1}{\eta_1^{\ast }+{M}_1}\right)\lambda p{\eta}_1-\left(\frac{M_1}{{\left({\eta}_1^{\ast}\right)}^2}-\frac{1}{\eta_1^{\ast }}+\frac{1}{\eta_1^{\ast }+{M}_1}\right){\lambda}^{\ast }{p}^{\ast }{\eta}_1^{\ast }, $$

by which we can recover values of $ \kappa $ and $ {M}_1 $ based on $ h $ and $ {h}_N $ (or $ {h}_W $ ).

The detection-error probability quantifies relative entropy growth ( $ h $ ) that seems plausible to the decision maker. Let $ {\zeta}^{\theta }(t)=\ln \left({Z}^{\theta }(t)\right) $ denote the log of the Radon-Nikodym derivative process. The detection-error probability is defined as (assuming an equal prior on $ {Q}^{\theta } $ and $ {Q}^0 $ )

(24) $$ \pi \left(t,T;h\right)=\frac{1}{2}\left[{Q}^0\left\{{\zeta}^{\theta }(T)>0|{\mathrm{\mathcal{F}}}_t\right\}+{Q}^{\theta}\left\{{\zeta}^{\theta }(T)<0|{\mathrm{\mathcal{F}}}_t\right\}\right]. $$

The first term within the right-hand side bracket represents the probability of the agent mistakenly choosing model $ {Q}^{\theta } $ over the reference model $ {Q}^0 $ , given a history of length $ T-t $ generated from the state process under $ {Q}^0 $ , while the second term is the probability of the agent erroneously favoring model $ {Q}^0 $ over model $ {Q}^{\theta } $ , given a history of length $ T-t $ produced from the state process under $ {Q}^{\theta } $ . In our analysis, we seek to identify the maximum level of ambiguity, $ {h}^{\ast } $ , such that $ \pi \left(t,T;{h}^{\ast}\right)=5\% $ . This corresponds to the decision maker using a 5% confidence level when rejecting implausible models. Anderson et al. (Reference Anderson, Hansen and Sargent2003) adopt a detection-error probability of 10%, while Bidder and Smith (Reference Bidder and Smith2012) and Croce, Nguyen, and Schmid (Reference Croce, Nguyen and Schmid2012) use lower values at 2.5% and 1.15%, respectively.

Aït-Sahalia and Matthys (Reference Aït-Sahalia and Matthys2019) provide a method for calculating the detection-error probability in the presence of jump ambiguity, based on the conditional Fourier transform of $ {\zeta}^{\theta^{\ast }}(t) $ . Given a sample of length $ n $ , the detection-error probability is

(25) $$ {\displaystyle \begin{array}{c}\pi \left(t,n;h\right)=\frac{1}{2}\left[{Q}^0\left\{\left.{\zeta}^{\theta^{\ast }}(n)>0\right|{\mathrm{\mathcal{F}}}_t\right\}+{Q}^{\theta^{\ast }}\left\{\left.{\zeta}^{\theta^{\ast }}(n)<0\right|{\mathrm{\mathcal{F}}}_t\right\}\right],\hskip1em t\ge 0,n= mT\\ {}=\frac{1}{2}-\frac{1}{2\pi }{\int}_{{\mathrm{\mathbb{R}}}^{+}}\left(\Re \left[\frac{{\hat{\zeta}}_{\theta^{\ast}}^{\theta^{\ast }}\left(u,t,n\right)}{iu}\right]-\Re \left[\frac{{\hat{\zeta}}_0^{\theta^{\ast }}\left(u,t,n\right)}{iu}\right]\right) du,\end{array}} $$

where $ T $ is the number of years, $ m $ is the sampling frequency (given that we use annualized parameter values), $ i\equiv \sqrt{-1} $ , $ \Re \left(\cdot \right) $ denotes the real part of a complex number, and the two conditional Fourier transforms are

$$ {\hat{\zeta}}_0^{\theta^{\ast }}\left(u,t,n\right)={\unicode{x1D53C}}_t\left[{e}^{iu{\zeta}^{\theta^{\ast }}(n)}\right],\hskip1em \mathrm{and}\hskip1em {\hat{\zeta}}_{\theta^{\ast}}^{\theta^{\ast }}\left(u,t,n\right)={\unicode{x1D53C}}_t^{\theta^{\ast }}\left[{e}^{iu{\zeta}^{\theta^{\ast }}(n)}\right]. $$

We explicitly calculate the two conditional Fourier transforms, $ {\hat{\zeta}}_0^{\theta^{\ast }}\left(u,t,n\right) $ and $ {\hat{\zeta}}_{\theta^{\ast}}^{\theta^{\ast }}\left(u,t,n\right) $ , in the Supplementary Material.

To implement the calibration, we need to specify the ratio $ {h}_W/h $ and set $ n= mT $ . In our baseline analysis, we use $ {h}_W/h=0.5 $ , implying that the decision maker places equal weight on jump ambiguity and drift ambiguity.Footnote ¹⁷ For the sample length used to calculate detection error probabilities, we assume access to 30 years of data ( $ T=30 $ ) to gauge the extent of sample length in determining the ambiguity level allowed for the EBIT dynamics. We choose this horizon because our focus is on growth options rather than assets in place, and extending the sample length beyond 30 years would draw on more distant periods that are less informative for our analysis. Additionally, we select a quarterly frequency ( $ m=4 $ ), which is the highest frequency available from Compustat.

Figure 2 presents the calibration results for detection-error probabilities. Graph A shows that as ambiguity ( $ h $ ) increases, $ \pi \left(n;h\right) $ decreases, indicating that it becomes easier to distinguish the worst-case model from the reference model. For our benchmark parameters, with $ {h}_W/h=0.5 $ , the critical value $ {h}^{\ast } $ satisfying the lower bound $ \pi \left(n;{h}^{\ast}\right)=0.05 $ is 0.049. Graphs B and C illustrate how the ambiguity parameters $ \kappa $ and $ {M}_1 $ vary with relative entropy growth. With $ {h}_W/h=0.5 $ , the maximum drift ambiguity ( $ {\kappa}^{\ast } $ ) is about 0.221, implying a drift reduction of $ {\kappa}^{\ast}\sigma =0.221\times 0.118=0.026 $ from 0.02 under the reference measure. On the other hand, the maximum jump ambiguity ( $ {M}_1^{\ast } $ ) is significant, around 4.142, lowering the conditional mean log positive jump size from 1/8.343 to about 1/12.485 (Graph D). Additionally, Graph E shows that the jump intensity ( $ {\lambda}^{\ast } $ ) decreases from approximately 1.986 to 1.912. This result is consistent with our discussion at the end of Section II.B and after Proposition 2 in Section II.C, where we show that $ {\lambda}^{\ast } $ attains an interior value because the worst-case distortion minimizes the unconditional probability of positive jumps while maximizing that of negative jumps. Lastly, Graph F shows that the conditional probability of positive jumps ( $ {p}^{\ast } $ ) drops to less than 0.078 from 0.112. Collectively, jump ambiguity minimizes the influence of positive jumps while amplifying that of negative jumps.

FIGURE 2 Calibration of Ambiguity

Figure 2 plots the detection-error probability ( $ \pi \left(n;h\right) $ ), the parameters governing ambiguity ( $ {\kappa}^{\ast } $ and $ {M}_1^{\ast } $ ), and the parameters of the double exponential jump-diffusion process under both the worst-case measure ( $ {Q}^{\theta^{\ast }} $ ) and the reference measure ( $ {Q}^0 $ ) across different levels of ambiguity. $ h $ denotes the total relative entropy growth bound. The maximum $ {h}^{\ast } $ is set such that $ \pi \left(n;{h}^{\ast}\right)=0.05 $ . The sample length for calculating detection-error probabilities is $ n=120 $ (quarters), and we set $ {h}_W/h=0.5 $ .

Figure 3 presents the first four central moments of $ Y(t) $ as a function of relative entropy growth ( $ h $ ), demonstrating how jump ambiguity affects the distribution of $ Y(t) $ in a way that is distinct from drift ambiguity. As relative entropy growth rises, the variance of $ Y(t) $ decreases and skewness becomes more negative. Jump ambiguity thins the right tail of the distribution while thickening the left tail, and as a result, higher levels of ambiguity increase kurtosis.

FIGURE 3 Central Moments Under the Worst-Case Measure

Figure 3 plots the first four central moments of $ Y(t) $ based on the calibrated parameters plotted in Figure 2. $ h $ denotes the total relative entropy growth bound. The maximum $ {h}^{\ast } $ is set such that $ \pi \left(n;{h}^{\ast}\right)=0.05 $ . The sample length for calibrating ambiguity is $ n=120 $ (quarters), and we set $ {h}_W/h=0.5 $ .

Our model features a path-dependent option whose value hinges on the sample-path characteristics of the underlying process. In particular, the first-passage time distribution—which captures the probability that the process reaches a given threshold for the first time—is the key determinant. In our model, ambiguity about jump risk shifts the entire distribution leftward and amplifies its asymmetry. To understand its implications for path-dependent decisions, we analyze how jump ambiguity affects the first-passage time probabilities of reaching levels above and below the current state, as shown in Figure 4.

FIGURE 4 First-Passage Time Under the Worst-Case Measure

Figure 4 plots the AD price of reaching a fixed upper level from below $ {AD}_u:= \unicode{x1D53C}\left[{e}^{-r{\tau}_u}\right] $ for $ {\tau}_u:= \operatorname{inf}\left\{t\ge 0;X(t)\ge u\right\} $ and the AD price of reaching a fixed lower level from above $ {AD}_d:= \unicode{x1D53C}\left[{e}^{-r{\tau}_d}\right] $ for $ {\tau}_d:= \operatorname{inf}\left\{t\ge 0;X(t)\le d\right\} $ across different ambiguity levels. We set $ u=1.2x $ and $ d=0.8x $ . $ h $ denotes the total relative entropy growth bound. The maximum $ {h}^{\ast } $ is set such that $ \pi \left(n;{h}^{\ast}\right)=0.05 $ . The sample length for calibrating ambiguity is $ n=120 $ (quarters), and we set $ {h}_W/h=0.5 $ .

Figure 4 illustrates how ambiguity in the return distribution affects first-passage dynamics. Specifically, ambiguity reduces the expected first-passage time to an upper level when approaching from below (Graph A) and increases it when approaching a lower level from above (Graph B). The results clearly indicate that as ambiguity $ h $ increases, $ {AD}_u $ drops sharply while $ {AD}_d $ rises substantially. These patterns form the basis for interpreting the subsequent results.

C. Investment and Financing Decisions

We next examine the impact of ambiguity on investment and financing decisions. Consistent with Remark 1, which shows that ambiguity reduces the investment payoff at $ {\tau}_I^e $ and prolongs the expected time to investment, Figure 5 demonstrates that ambiguity lowers both the project value and the AD price of investment while raising the investment threshold. The resulting decline in project value suggests that investors demand substantial compensation for bearing ambiguity. Notably, the effects of ambiguity stand in stark contrast to those of volatility, which typically boost project value. The difference arises because the ambiguity-averse entrepreneur bases her decision on the least favorable EBIT profile.

FIGURE 5 The Value of Investment

Graphs A–C of Figure 5 plot the the optimal investment boundary $ {X}_I^j $ , the AD price of investment $ {AD}_I^j $ , and project value $ {V}_j(0) $ against relative entropy growth, where $ j $ denotes equity financing ( $ e $ ) or optimal financing ( $ \ast $ ). $ h $ denotes the total relative entropy growth bound. The maximum $ {h}^{\ast } $ is set such that $ \pi \left(n;{h}^{\ast}\right)=0.05 $ . The sample length for calibrating ambiguity is $ n=120 $ (quarters). We set $ {h}_W/h=0.5 $ .

The results in Figure 5 also indicate the benefits of debt in terms of the investment boundary and project value, in line with the theoretical analysis in Section II.E. Figure 6 further illustrates the mechanism underlying the benefits of debt. In the absence of ambiguity, we find that $ {X}_I^e/{X}_I^{\ast }=1.063 $ , $ {AD}_{\ast }/{AD}_e=1.094 $ , and $ {V}_{\ast }/{V}_e=1.094 $ , which correspond to $ {\beta}_1=1.468 $ . Additionally, the values of $ {A}_6 $ on the order of $ {10}^{-8} $ , shown in Graph E, are consistent with our earlier discussion following Remark 4. Without ambiguity, the amplification effect through the power coefficient $ {\beta}_1 $ is small, and thus the benefit of debt financing in terms of enhancing the project value is very limited.

FIGURE 6 The Value-Enhancing Effect of Debt

Graphs A–C of Figure 6 plot $ {X}_I^e/{X}_I^{\ast } $ , $ {AD}_{\ast }/{AD}_e $ , and $ {V}_{\ast }/{V}_e $ against relative entropy growth $ h $ . The expressions of $ {X}_I^e/{X}_I^{\ast } $ , $ {AD}_{\ast }/{AD}_e $ , and $ {V}_{\ast }/{V}_e $ are given in equations (21), (22), and (23). Graph D plots $ {\beta}_1 $ , the smaller positive root of $ G\left(\beta \right)=r $ , against $ h $ . Graph E plots the coefficient $ {A}_6 $ in (23). Graph F plots $ {\beta}_3 $ , the larger negative root of $ G\left(\beta \right)=r $ , against $ h $ . The maximum $ {h}^{\ast } $ is set such that $ \pi \left(n;{h}^{\ast}\right)=0.05 $ . The sample length used for calibrating ambiguity is $ n=120 $ (quarters) and we set $ {h}_W/h=0.5 $ .

Graph A shows that the ratio $ {X}_I^e/{X}_I^{\ast } $ decreases with ambiguity at a diminishing rate, consistent with the concavity of the base (equation (21)) in ambiguity. This concavity can be further explained by the concave relationship between $ {\beta}_3 $ and ambiguity. More importantly, Graphs B and C highlight the dominance of the power effect. Although the base value in Graph A declines modestly from 1.063 to 1.048 as ambiguity increases, the power coefficient $ {\beta}_1 $ rises substantially from 1.468 to 2.687, as shown in Graph D. The substantial effect of ambiguity on $ {\beta}_1 $ drives the overall increase in both $ {AD}_{\ast }/{AD}_e $ and $ {V}_{\ast }/{V}_e $ . Notably, the change in $ {V}_{\ast }/{V}_e $ by varying the ambiguity level is quantitatively meaningful: without ambiguity, $ {V}_{\ast }/{V}_e-1=9.42\% $ , while at the highest attainable ambiguity level $ {h}^{\ast } $ , $ {V}_{\ast }/{V}_e-1=13.48\% $ —representing a 43% increase. This result suggests that, in contrast to the scenario without ambiguity, debt financing greatly enhances the project value under ambiguity.

Figure 7 shows the optimal capital structure. Because the default boundary ( $ {X}_D^{\ast } $ ), coupon ( $ {C}^{\ast } $ ), levered equity, debt, and net tax benefit depend on the realized cash flow at investment ( $ X({\tau}_I^{\ast }) $ ), we scale them by $ X({\tau}_I^{\ast }) $ to avoid the “overshooting issue” ( $ X({\tau}_I^{\ast })\ge {X}_I^{\ast } $ ). The AD price of default, leverage, and cost of debt ( $ {C}^{\ast }/D({\tau}_I^{\ast }) $ ) are already ratios and need no adjustment. These normalized quantities resemble those in the standard assets-in-place financing problem, where the cash flow at financing is exogenously fixed (e.g., Goldstein et al. (Reference Goldstein, Ju and Leland2001)). In our growth-option setting, by contrast, the cash flow at investment ( $ X({\tau}_I^{\ast }) $ ) is endogenously determined and varies with model parameters.

FIGURE 7 Optimal Capital Structure

Figure 7 plots the the scaled optimal default boundary $ {X}_D^{\ast } $ , the AD price of default, the scaled optimal coupon $ {C}^{\ast } $ , the scaled levered equity and debt, optimal leverage $ D/\left(D+E\right) $ , the scaled net tax benefit, and the cost of debt ( $ {C}^{\ast }/D({\tau}_I^{\ast }) $ ) against ambiguity. The scaled quantities are divided by $ X({\tau}_I^{\ast }) $ . The maximum $ {h}^{\ast } $ is set such that $ \pi \left(n;{h}^{\ast}\right)=0.05 $ . The sample length used for calibrating ambiguity is $ n=120 $ (quarters), and we set $ {h}_W/h=0.5 $ .

The results in Figure 7 are consistent with our earlier discussion on the effects of ambiguity and the behavior of the cash flow process under the worst-case scenario, which are elaborated in the remarks in Section II.D. As shown previously in Figure 4, ambiguity reduces the expected first-passage time to a fixed level from above ( $ {\tau}_d $ ). This helps explain the decline in the default threshold $ {X}_D^{\ast } $ and the increase in the AD price of default observed in Figure 7. Specifically, the optimal default boundary drops moderately from 0.275 to 0.209 as ambiguity increases from zero to its maximum, in line with Remark 3.

Moreover, as the optimal coupon $ {C}^{\ast } $ declines with ambiguity, the values of levered equity, debt, leverage, and the net tax benefit (NTB) of debt all decrease. This result contrasts sharply with the effect of volatility, which tends to increase the levered equity value while reducing the debt value. The cost of debt rises only slightly—from 6.25% to 6.97%—since both $ {C}^{\ast } $ and the debt value decline, with the latter falling at a marginally faster rate. These numerical results on the cost of debt are reasonable for an investment-grade firm, given that our model is calibrated using DJX option data.

An additional insight comes from the NTB. Without ambiguity $ h=0 $ , the NTB equals 1.687, and the total firm value is $ {A}_4={A}_0+ NTB=\left(1-\phi \right)/\left(r-\mu \right)+ NTB=28.354 $ , implying that the NTB accounts for about 6% of total firm value. In the presence of ambiguity, $ h={h}^{\ast } $ , this proportion drops to 4.6% moderately. The range of the NTB as a fraction of firm value implied by our model is broadly consistent with the empirical estimates reported by Korteweg (Reference Korteweg2010).

D. Optimal Project Size

To examine how ambiguity affects the optimal project size under the two financing scenarios, we assume that the firm follows a fixed cash flow threshold policy, as discussed in Remarks 2 and 6. Specifically, we set the cash flow threshold at $ \overline{X}=2X(0)=2x $ , so that investment occurs when the cash flow level doubles its initial value. Given this threshold, the firm chooses the optimal project size $ {I}_i^{\ast } $ (for $ i\in \left\{e,\ast \right\} $ ) to maximize firm value, where $ {I}_i^{\ast }=I\overline{X}/{X}_I^i $ . Figure 8 plots the optimal size as a function of relative entropy growth $ h\in \left[0,{h}^{\ast}\right] $ , in our baseline setting where ambiguity is equally shared across the two types.

FIGURE 8 The Optimal Project Size Choices under Ambiguity

Figure 8 plots the optimal project size $ {I}_i^{\ast } $ for $ i\in \left\{e,\ast \right\} $ under a fixed cash flow threshold policy. The fixed cash flow trigger $ \overline{X}=2X(0)=2x $ . The upper bound of $ h $ is set such that the detection-error probability is 5% with $ n=120 $ (quarters) and $ {h}_W/h=0.5 $ .

Overall, the results are consistent with Remarks 2 and 6. Graph A shows that in the absence of ambiguity, the optimal project size under equity financing is nearly halved relative to the baseline level ( $ I=100 $ ), whereas optimal financing supports a larger project, with an optimal size close to 56. Since the value function is linear in project size under both financing scenarios, this increase in size under optimal financing yields a proportional gain in project value. The net tax benefit of debt further amplifies this gain. Thus, the larger size under optimal financing results in a magnified improvement in firm value. As ambiguity rises, the optimal size declines under both financing modes. Graph B plots the ratio $ {I}_{\ast}^{\ast }/{I}_e^{\ast } $ , which mirrors the pattern of the ratio $ {X}_I^e/{X}_I^{\ast }={A}_4/{A}_0 $ in Figure 6, in line with Remark 6. These findings suggest that firms respond to ambiguity by downsizing their projects, while optimal financing mitigates the adverse effect.