Utility based pricing and exercising of real options under geometric mean reversion and risk aversion toward idiosyncratic risk

We study the classical real option problem in which an agent faces the decision if and when to invest optimally into a project. The investment is assumed to be irreversible. This problem has been studied by Myers and Majd (Adv Futures Options Res 4:1–21, 1990) for the case of a complete market, in which the risk can be perfectly hedged with an appropriate spanning asset, by McDonald and Siegel (Q J Econ, 101:707–727, 1986), who include the incomplete case but assume that the agent is risk neutral toward idiosyncratic risk, and later by Henderson (Valuing the option to invest in an incomplete market, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=569865, 2006) who studies the incomplete case with risk aversion toward idiosyncratic risk under the assumption that the project value follows a geometric Brownian motion. We take up Henderson’s utility based approach but assume as suggested by Dixit and Pindyck (Investment under uncertainty, Princeton University Press, Princeton, 1994) as well as others, that the project value follows a geometric mean reverting process. The mean reverting structure of the project value process makes our model richer and economically more meaningful. By using techniques from optimal control theory we derive analytic expressions for the value and the optimal exercise time of the option to invest.