Abstract
Ever since the first attempts to model capital investment decisions as options, financial economists have sought more accurate, more realistic real options models. Strategic interactions and market incompleteness are significant challenges that may render existing classical models inadequate to the task of managing the firm’s capital investments. The purpose of this paper is to address these challenges. The issue of incompleteness comes in for the valuation of payoffs due to absence of a unique martingale measure. One approach is to valuate assets by considering a rational utility-maximizing consumer/investor’s joint decisions with respect to portfolio investment strategy and consumption rule. In our situation, we add the stopping time as an additional decision. We employ variational inequalities (V.I.s) to solve the optimal stopping problems corresponding to times to invest. The regularity of the obstacle (payoffs received at the decision time) is a major element for defining the optimal strategy. Due to the lack of smoothness of the obstacle raised by the game problem, the optimal strategy is a two-interval solution, characterized by three thresholds.
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Notes
- 1.
M ε is defined as: \(\displaystyle{M_{\varepsilon}=\frac{\big (\frac{\delta_2}{\varepsilon}-r+\alpha-\lambda\varsigma\rho\big )^2}{2\varsigma^2r^2\gamma(1-\rho^2)}}\). If r+λςρ−α>0, we can take M ε =0, hence \(\varepsilon =\frac{\delta_{2}}{r-\alpha+\lambda\varsigma\rho}\). Note that ε can be arbitrarily small.
- 2.
- 3.
For any test function Ψ(x,s), we have the formula:
(32) - 4.
See footnote 1 for the definition of M ε .
- 5.
Here Y y (t) is the process defined in (2).
- 6.
References
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Acknowledgement
The first author acknowledges support of National Science Foundation DMS-1303775 and of the Research Grants Council of HKSAR (CityU 500113).
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Bensoussan, A., Hoe, S.(. (2014). Real Options with Competition and Incomplete Markets. In: Kabanov, Y., Rutkowski, M., Zariphopoulou, T. (eds) Inspired by Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-02069-3_2
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DOI: https://doi.org/10.1007/978-3-319-02069-3_2
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