Valuing pilot projects in a learning by investing framework: An approximate dynamic programming approach

We introduce a general discrete time dynamic framework to value pilot project investments that reduce idiosyncratic uncertainty with respect to the final cost of a project. The model generalizes different settings introduced previously in the literature by incorporating both market and technical uncertainty and differentiating between the commercial phase and the pilot phase of a project. In our model, the pilot phase requires N$N$ stages of investment for completion. With this distinction we are able to frame the problem as a compound perpetual Bermudan option. We work in an incomplete markets setting where market uncertainty is spanned by tradable assets and technical uncertainty is idiosyncratic to the firm. The value of the option to invest as well as the optimal exercise policy are solved by an approximate dynamic programming algorithm that relies on the independence of the state variables increments. We prove the convergence of our algorithm and derive a theoretical bound on how the errors compound as the number of stages of the pilot phase is increased. We implement the algorithm for a simplified version of the model where revenues are fixed, providing an economic interpretation of the effects of the main parameters driving the model. In particular, we explore how the value of the investment opportunity and the optimal investment threshold are affected by changes in market volatility, technical volatility, the learning coefficient, the drift rate of costs and the time to completion of a pilot stage.