## 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.

### Keywords

- Real Option
- Optimal Investment
- Geometric Brownian Motion
- Idiosyncratic Risk
- Employee Stock Option

*These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.*

This is a preview of subscription content, access via your institution.

## Buying options

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions## Notes

- 1.
This can be easily generalized to a project which pays out a continuous dividend.

- 2.
In earlier versions of this work, we referred to this process as the market sector indicator.

- 3.
In principle, it is possible to consider the cash-flow as a continuous stream of cash-flows, in which case (17a) would be modified to \(\mathit{dX}_{s} = (\mu -r)\pi _{s}\,\mathit{ds} +\sigma \,\pi _{s}\,\mathit{dB}_{s} +\varphi _{s}(S_{s})\,\mathit{ds}\). However, Managers rarely specify a continuous stream of cash-flows, and although operations can be viewed as providing income on a continuous basis, we opt to leave the cash-flows discrete as this is how managers typically estimate cash-flow streams.

- 4.
Since we have diffusion processes driving the relevant dynamics.

## References

Ané, T., Kharoubi, C.: Dependence structure and risk measure. J. Bus.

**76**(3), 411–438 (2003)Berk, J., Green, R., Naik, V.: Valuation and return dynamics of new ventures. Rev. Financ. Stud.

**17**(1), 1–35 (2004)Borison, A.: Real options analysis: where are the emperor’s clothes? J. Appl. Corp. Finance

**17**(2), 17–31 (2005)Carlos, J., Nunes, J.: Pricing real options under the constant elasticity of variance diffusion. J. Futur. Mark.

**31**(3), 230–250 (2011)Carmona, R. (ed.): Indifference Pricing: Theory and Applications. Princeton University Press, Princeton (2008)

Copeland, T., Antikarov, V.: Real Options: A Practitioner’s Guide. W. W. Norton and Company, New York (2001)

Datar, V., Mathews, S.: European real options: an intuitive algorithm for the black- scholes formula. J. Appl. Finance

**14**(1), 45–51 (2004)Davis, M.: Option pricing in incomplete markets. In:Mathematics of Derivative Securities, pp. 216–226. The Newton Institute, Cambridge University Press, Cambridge (1998)

Dixit, A., Pindyck, R.: Investment Under Uncertainty. Princeton University Press, Princeton (1994)

Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Davis, M.H., Elliott, R.J. (eds.) Applied Stochastic Analysis: Stochastics Monographs, pp. 389–414. Gordon and Breach, London (1991)

Grasselli, M.: Getting real with real options: a utility-based approach for finite-time investment in incomplete markets. J. Bus. Finance Account.

**38**(5&6), 740–764 (2011)Henderson, V.: Valuing the option to invest in an incomplete market. Math. Finan. Econ.

**1**, 103–128 (2007)Hugonnier, J., Morellec, E.: Corporate control and real investment in incomplete markets. J. Econ. Dyn. Control.

**31**, 1781–1800 (2007)Leung, T., Sircar, R.: Accounting for risk aversion, vesting, job termination risk and multiple exercises in valuation of employee stock options. Math. Financ.

**19(1)**, 99–128 (2009)Metcalf, G., Hasset, K.: Investment under alternative return assumptions. comparing random walk and mean revertion. J. Econ. Dyn. Control.

**19**, 1471–1488 (1995)Miao, J., Wang, N.: Investment, consumption, and hedging under incomplete markets. J. Financ. Econ.

**86**(3), 608–42 (2007)Nelsen, R.B.: An Introduction to Copulas. Springer, New York (2006)

Oriani, R., Sobrero, M.: Uncertainty and the market valuation of R&D within a real options logic. Strateg. Manag. J.

**29**(4), 343–361 (2008)Pham, H.: Continuous-time Stochastic Control and Optimization with Financial Applications. Springer, Berlin (2009)

Sarkar, S.: The effect of mean reversion on investment under uncertainty. J. Econ. Dyn. Control.

**28**, 377–396 (2003)Trigeorgis, L.: Real Options: Managerial Flexibility and Strategy in Resource Allocation. MIT Press, Cambridge (1996)

Tsekrekos, A.E.: The effect of mean reversion on entry and exit decisions under uncertainty. J. Econ. Dyn. Control.

**34**(4), 725–742 (2010)

## Acknowledgements

SJ would like to thank NSERC for partially funding this research.

## Author information

### Authors and Affiliations

### Corresponding author

## Editor information

### Editors and Affiliations

## Appendix: Proof of Results

### 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,

we have that

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

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:

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,

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

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

The HJB equations then reduce to

subject to the appropriate terminal conditions. Writing

the above PDE reduces to

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)

The DPEs then reduce to

and

subject to the appropriate terminal conditions. Writing

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*. □

## Rights and permissions

## Copyright information

© 2015 Springer Science+Business Media New York

## About this chapter

### Cite this chapter

Jaimungal, S., Lawryshyn, Y. (2015). Incorporating Managerial Information into Real Option Valuation. In: Aïd, R., Ludkovski, M., Sircar, R. (eds) Commodities, Energy and Environmental Finance. Fields Institute Communications, vol 74. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2733-3_8

### Download citation

DOI: https://doi.org/10.1007/978-1-4939-2733-3_8

Publisher Name: Springer, New York, NY

Print ISBN: 978-1-4939-2732-6

Online ISBN: 978-1-4939-2733-3

eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)