Terminal Value Modeling in Finance: Laplace and Z-Transforms with Seasonality, and European Real Options

The article presents an analytical framework for modeling the terminal value of investment projects under conditions of seasonal or cyclical cash flows during the residual period. The proposed methodology employs the Laplace transform for continuous discount rates and the Z-transform for discrete discount rates, providing a unified approach to terminal value valuation under different discounting conventions. Seasonality is represented using periodic cash flow functions expanded in a Fourier series, allowing terminal value to be expressed analytically as a function of the amplitudes, frequencies, and phases of the underlying cyclical components. Consequently, the proposed framework enables the valuation of projects whose residual cash flows exhibit seasonal patterns without requiring the assumption of constant or perpetually growing cash flows. A key contribution of the study is the integration of transform-based terminal value modeling with real option valuation. The terminal value obtained from a broad class of periodic cash flow functions can be used directly within a real-option framework, enabling analytical valuation of decision options associated with project expansion, or abandonment. The results demonstrate that the phase of the seasonal cycle may significantly affect both terminal value and the value of decision options. The proposed methodology provides analytical solutions without the need for numerical simulations and offers a practical valuation framework for projects characterized by seasonal or cyclical cash flow structures, such as those encountered in the energy, agricultural, tourism, and infrastructure sectors.