Terminal Value Modeling in Finance: Laplace and Z-Transforms with Real Options Based on Stochastic Cash Flow Parameters

The article extends a transform-based framework for terminal value valuation and develops a methodology for incorporating real options whose underlying assets are economic parameters determining cash flows. The proposed approach enables terminal value modeling under both continuous and discrete discount rates using a variety of cash flow functions, while providing a consistent framework for the valuation of abandonment and expansion decision options. The main contribution of the study is the introduction of a framework in which uncertainty is modeled directly at the level of economic parameters determining cash flows, rather than at the level of aggregate cash flows themselves. In this approach, the underlying asset of a real option is a parameter of the terminal value function on which the value of the remaining cash flows depends linearly. This enables direct modeling of uncertainty associated with specific economic variables, such as revenues, costs, commodity prices, or sales volumes. Terminal value is modeled using the Laplace transform in the case of continuous discounting and the Z-transform in the case of discrete discounting. The valuation of real options is separated from the terminal value model, making option pricing independent of the discounting method applied. The proposed methodology allows different stochastic processes to be used for modeling parameter uncertainty. In the illustrative examples, a driftless Geometric Brownian Motion process is applied. Any deterministic trend should be incorporated directly into the function describing terminal value. European call and put options are employed to represent project expansion and abandonment flexibility. A general procedure is proposed for linking the value of a real option defined on a model parameter with the value of the corresponding decision option embedded in the project. Illustrative examples demonstrate the applicability of the proposed framework and show how abandonment and expansion options can be consistently incorporated into terminal value valuation models under both continuous and discrete discount rates.