APPENDIX I: Optimal Investment Rule Under Uncertainty
Following Merton (1973b), the value of the option, W(V), satisfies the following linear partial differential equation:
where r is the riskless interest rate, and WV, WVV denote the first and second partial derivatives with respect to V. In addition, W(V) must satisfy the following boundary conditions:
The solution is
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The author is grateful to Kenneth Froot, Jeffrey Sachs, Peter Wickham and especially, Robert Merton for helpful suggestions. The author also thanks, Maria Orihuela for assistance in text preparation. The views expressed in this paper are those of the author and do not necessarily reflect the views of the International Monetary Fund.
Brennan and Schwartz (1985), McDonald and Siegel (1986), and Pindyck (1988) applied the methodology of options pricing to project valuation and capital investment decisions. Their work demonstrated that the option of waiting (i.e., the option of delaying a project) is more valuable to the firm the greater the underlying uncertainty so that optimal investment rules differ from the “net present value (NPV)” rule usually applied in the simple certainty case.
The rate of return, μ, is determined by the capital market equilibrium conditions. For example, it can be given by the classical Sharp-Lintner CAPM model.
Even if the value of the investment project is known with certainty, for example, when it is fixed at a constant level, V0, but the fixed cost of investment, is stochastic, the whole analysis below will still go through. In this case, the project is formally analogous to a financial option with stochastic strike price.
It is better known as the “smooth-pasting” condition in international economics, as, for example, In the recent literature on exchange rate target zones.
We also used returns on the Standard and Poor’s Composite Stock Index to estimate market volatility and the resulting volatility estimates are closely comparable to those obtained from the returns on the value-weighted NYSE composite portfolio.