The pricing and timing of the option to invest for cash flows with partial information

This paper extends real options theory to consider the situation where the mean appreciation rate of cash flows generated by an irreversible investment project is not observable and governed by an Ornstein-Uhlenbeck process. The main purpose of this study is to analyze the impact of the uncertainnty of the mean appreciation rate on the pricing and investment timing of the option to invest under incomplete markets with partial information. We assume that an investor aims to maximize expected discounted utility of lifetime consumption. Based on consumption utility indifference pricing method, stochastic control and filtering theory, under constant absolute risk aversion (CARA) utility, we derive the implied value of cash flows after investment, and then obtain the implied value and the optimal investment threshold of the option to invest, which are determined by a semi-closed-form solution of a free-boundary partial differential equations (PDE) problem. We show that the solutions are independent of the utility time-discount rate. We provide numerical results by finite difference methods and compare the results with those under a fully observable case. Numerical calculations show that partial information leads to a significant loss of the implied value of the option to invest. This loss increases with the uncertainty of the mean appreciation rate. In contrast to standard real options theory, a high volatility of cash flows decreases the implied value of the option to invest as well as the implied information value.