Incorporating Managerial Information into Real Option Valuation

Abstract

The adoption of real options analysis (ROA) by practitioners, despite being widely viewed as a superior method for valuing managerial flexibility, remains limited due to varied difficulties in its implementation. In this work, we propose an approach that utilizes cash-flow estimates from managers as key inputs and results in project value cash-flows that exactly match the arbitrarily distributed estimates. We achieve this through the introduction of an observable, but not tradable, market stochastic driver process which drives the project’s cash-flow, rather than modeling the project value directly. Our framework can be used to value managerial flexibilities and obtain hedges in an easy to implement manner for a variety of real options such as entry/exit, multistage, abandonment, etc. As well, our approach to ROA provides a co-dependence between cash-flows, is consistent with financial theory, requires minimal subjective input of model parameters, and bridges the gap between theoretical ROA frameworks and practice.

Appendix: Proof of Results

In this Appendix we provide concise proofs of the main results.

1.1 Proof of Proposition 1

We seek $\varphi (.)$ such that $\mathbb{P}(\varphi (S_{T}) \leq v\vert \mathcal{F}_{0}) = F^{{\ast}}(v)$. Since,

$$\displaystyle{S_{T}\vert _{\mathcal{F}_{0}}\stackrel{d}{=}S_{0}\exp \left \{(\nu -\tfrac{1} {2}\eta ^{2})T +\eta \sqrt{T}Z\right \}\quad \text{where}\quad Z\mathop{ \sim }\limits_{\mathbb{P}}\mathcal{N}(0,1),}$$

we have that

$$\displaystyle{ \mathbb{P}(\varphi (S_{T}) \leq v\vert \mathcal{F}_{0}) =\varPhi \left (\frac{\ln \frac{\varphi ^{-1}(v)} {S_{0}} - (\nu -\frac{1} {2}\eta ^{2})T} {\eta \sqrt{T}} \right ) \triangleq F^{{\ast}}(v)\,. }$$

(40)

Consequently, if F ^∗(. ) is invertible then

$$\displaystyle{ \varphi (S) = F^{{\ast}-1}\left (\varPhi \left (\frac{\ln \frac{S} {S_{0}} - (\nu -\frac{1} {2}\eta ^{2})T} {\eta \sqrt{T}} \right )\right ) }$$

(41)

and the proof is complete. □ 

1.2 Proof of Proposition 2

Here, we prove that the co-dependence structure of the cash-flow distribution is governed by a Gaussian copula. We require the following joint distribution function:

$$\displaystyle\begin{array}{rcl} & & \mathbb{P}(V _{1} < v_{1},\ldots,V _{n} < v_{n}) {}\\ & & = \mathbb{P}(\varphi _{1}(S_{T_{1}}) < v_{1},\ldots,\varphi _{n}(S_{T_{n}}) < v_{n}) {}\\ & & = \mathbb{P}(F_{1}^{{\ast}-1}\left (\varPhi (z(T_{ 1},S_{T_{1}}))) < v_{1},\ldots,F_{n}^{{\ast}-1}(\varPhi (z(T_{ n},S_{T_{n}}))) < v_{n}\right )\,, {}\\ \end{array}$$

where $z(T,S) = \frac{1} {\eta \sqrt{T}}\ln \frac{S} {S_{0}} -\frac{\nu -\frac{1} {2} \eta ^{2}} {\eta } \sqrt{T}$. Clearly,

$$\displaystyle{S_{T_{k}}\stackrel{d}{=}S_{0}\,\exp \left \{(\nu -\frac{1} {2}\eta ^{2})T_{ k} +\eta \sqrt{T_{k}}\,Z_{k}\right \}}$$

where $\{Z_{1},\ldots,Z_{k}\}$ are jointly normal with mean zero and covariance matrix $\varOmega _{\mathit{ij}} = \sqrt{T_{\min (i,j) } /T_{\max (i,j)}}$.

Since each distribution function F _k ^∗ is assumed invertible, we then have

$$\displaystyle\begin{array}{rcl} \mathbb{P}(V _{1} < v_{1},\ldots,V _{n} < v_{n})& =& \mathbb{P}\left (Z_{1} <\varPhi ^{-1}\left (F_{ 1}^{{\ast}}(v_{ 1})\right ),\ldots,Z_{n} <\varPhi ^{-1}\left (F_{ n}^{{\ast}}(v_{ n})\right )\right ) {}\\ & =& \varPhi _{\varOmega }\left (\varPhi ^{-1}\left (F_{ 1}^{{\ast}}(v_{ 1})\right ),\ldots,\varPhi ^{-1}\left (F_{ n}^{{\ast}}(v_{ n})\right )\right )\;. {}\\ \end{array}$$

This completes the proof. □ 

1.3 Proof of Theorem 1

Here we provide a sketch of the proof. The first order condition in the HJB equations (18) and (19) provide the optimal investment policy in feedback control form as

$$\displaystyle{ \pi ^{(a)} = -\frac{(\mu -r)\partial _{x}V ^{(a)} +\rho \eta \sigma S\partial _{\mathit{ xS}}V ^{(a)}} {\sigma ^{2}\partial _{\mathit{xx}}V ^{(a)}}. }$$

(42)

The HJB equations then reduce to

$$\displaystyle{ \left (\partial _{t} + \mathcal{L}_{S}\right )V ^{(a)} -\dfrac{1} {2} \frac{\left [(\mu -r)\partial _{x}V ^{(a)} +\rho \eta \sigma S\partial _{\mathit{xS}}V ^{(a)}\right ]^{2}} {\sigma ^{2}\partial _{\mathit{xx}}V ^{(a)}} = 0, }$$

(43)

subject to the appropriate terminal conditions. Writing

$$\displaystyle{V ^{(a)}(t,x,S,I) = V ^{(0)}(t,x,I)h^{(a)}(t,S),}$$

the above PDE reduces to

$$\displaystyle{ \left (\partial _{t} + \mathcal{L}_{S}\right )h^{(a)}(t,S) +\rho \eta \lambda \, S\partial _{ S}h^{(a)}(t,S) + (\rho \eta )^{2}\frac{(S\partial _{S}h^{(a)}(t,S))^{2}} {h(t,S)} = 0, }$$

(44)

Now setting $h^{(a)}(t,S) = \left (H^{(a)}(t,S)\right )^{\beta }$ after some tedious computations, the above non-linear PDEs for h ^(a) reduces into the linear PDEs (23) and (24) for H ^(a). Moreover, the boundary conditions for V ^(a) become the stated boundary conditions for H ^(a). Since classical solutions exist for the linear PDE system (23) and (24), and the resulting feedback controls are admissible, the usual arguments imply that the solution of DPE is the solution to the original optimal control problem. The uniqueness of S ^∗ follows from the fact the terminal conditions and the subsequent pasting conditions are decreasing functions of S. Hence, the H function inherits this property and, therefore, the solution to (25) is unique. □ 

1.4 Proof of Theorem 2

Here we provide a sketch of the proof. The first order condition in the HJB equations (46) and (35a) provide the optimal investment policy in feedback control form as (a = 3, 4)

$$\displaystyle{ \pi ^{(a)} = -\frac{(\mu -r)\partial _{x}V ^{(a)} +\rho \eta \sigma S\partial _{\mathit{ xS}}V ^{(a)}} {\sigma ^{2}\partial _{\mathit{xx}}V ^{(a)}}. }$$

(45)

The DPEs then reduce to

$$\displaystyle{ \begin{array}{ll} &\max \Bigg\{\left (\partial _{t} + \mathcal{L}_{S}\right )V ^{(4)} -\dfrac{1} {2} \frac{\left [(\mu -r)\partial _{x}V ^{(4)}+\rho \eta \sigma S\partial _{\mathit{ xS}}V ^{(4)}\right ]^{2}} {\sigma ^{2}\partial _{\mathit{xx}}V ^{(4)}} \,; \\ &\qquad \qquad \qquad \qquad \qquad \qquad V ^{(0)}(t,x - C,S,I) - V ^{(4)}(t,x,S,I)\Bigg\} = 0, \end{array} }$$

(46)

and

$$\displaystyle{ \left (\partial _{t} + \mathcal{L}_{S}\right )V ^{(3)} -\dfrac{1} {2} \frac{\left [(\mu -r)\partial _{x}V ^{(3)} +\rho \eta \sigma S\partial _{\mathit{xS}}V ^{(3)}\right ]^{2}} {\sigma ^{2}\partial _{\mathit{xx}}V ^{(3)}} = 0, }$$

(47)

subject to the appropriate terminal conditions. Writing

$$\displaystyle{V ^{(a)}(t,x,S,I) = V ^{(0)}(t,x,I)h^{(a)}(t,S),}$$

and setting $h^{(a)}(t,S) = \left (H^{(a)}(t,S)\right )^{\beta }$ after some tedious computations, the above non-linear DPEs for h ^(a) reduce into the linear DPEs (36) and (37) for H ^(a). Moreover, the boundary conditions for V ^(a) become the stated boundary conditions for H ^(a). Standard results imply that the viscosity solution of the linear DPE system (36) and (37) is the solution to the original optimal control problem. The exercise point S ^∗ is unique once again due to the boundary conditions and pasting conditions being decreasing in S. □