Terminal Value Modeling in Finance: Laplace and Z-Transforms, Real Options, Stochastic Volatility

The article presents an innovative approach to the valuation of the terminal value of investment projects and enterprises, incorporating the use of real options. The proposed framework allows for the application of various stochastic processes to model the volatility of parameters affecting the value of cash flows. Based on the selected volatility model, the value of the option is calculated and subsequently incorporated into the terminal value valuation model. The option valuation process is separated from the terminal value model, which renders the option pricing independent of the discounting method—whether continuous or discrete—and imposes no restrictions on the stochastic processes used to model related uncertainties. The computed option value is then consistently integrated into the corresponding terminal value model. When a continuous discount rate is assumed, the model employs the Laplace transform; for discrete discounting, the Z-transform is used. The application of these transforms enables the transition from the time domain to the capital cost domain, allowing the specification of a fixed cost of capital and the expression of terminal value as a function of a parameter subject to uncertainty. This approach permits the modeling of complex time-domain functions for terminal value, which become analytically tractable after transformation. The method extends traditional terminal value models by enabling analytical treatment of nonlinear cash-flow structures and flexible stochastic modeling of uncertainty. To illustrate the practical application of the proposed approach, examples demonstrating its implementation are provided. The illustrative model employs a function representing the asymptotic decline of cash flows over time. Volatility is modeled using the Black-Scholes model, excluding drift.