On the investment–uncertainty relationship in a real option model with stochastic volatility

We consider the classical investment timing problem in a framework where the instantaneous volatility of the project value is itself given by a stochastic process, hence lifting the old question about the investment–uncertainty relationship to a new level. Motivated by the classical cases of Geometric Brownian Motion (GBM) and Geometric Mean Reversion (GMR), we consider processes of similar functional form, but with Heston stochastic volatility replacing the constant volatility in the classical models. We refer to these processes as Heston-GBM and Heston-GMR. For these cases we derive asymptotic solutions for the investment timing problem using the methodology introduced by Fouque et al. (2000). In particular we show that compared to the classical cases with constant volatility, the question of whether additional stochastic volatility increases or decreases the investment threshold depends on the instantaneous correlation between the project value and the stochastic volatility. For the case of Heston-GBM we provide a closed form expression that measures this effect quantitatively; for the case of Heston-GMR we derive the sign of the effect analytically, using a type of maximum principle for ODEs. Various numerical examples are discussed and a comparative analysis is provided.