A discrete-time approach for valuing real options with underlying mean-reverting stochastic processes

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Hahn, Warren Joseph

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In this research the recombining binomial lattice approach for valuing real options is generalized to address a common issue in many real valuation problems, underlying stochastic processes that are mean-reverting. Binomial lattices were first introduced to approximate stochastic processes for valuation of financial options, and they provide a convenient framework for numerical analysis. Unfortunately, the standard approach to constructing binomial lattices can result in invalid probabilities of up and down moves in the lattice when a mean-reverting stochastic process is to be approximated. There have been several alternative methods introduced for modeling mean-reverting processes, including simulation-based approaches and trinomial trees, however they unfortunately complicate the numerical analysis of valuation problems. The approach developed in this research utilizes a more general binomial approximation methodology from the existing literature to model simple homoskedastic mean-reverting stochastic processes as recombining lattices. This approach is then extended to model a two-factor mean-reverting process that allows for uncertainty in the long-term mean, and to model two correlated one-factor mean-reverting processes. These models facilitate the evaluation of real options with early-exercise characteristics, as well as multiple concurrent options. The models developed in this research are tested by implementing the lattice in binomial decision tree format and applying to hypothetical real option examples with underlying mean-reverting commodity price. To specify the stochastic process for commodity price, different data analysis techniques such as Kalman filtering and seemingly unrelated regression are used. These different techniques are empirically tested to evaluate differences in the estimates and assess the tradeoffs in computational requirements. To validate the binomial model, results are compared to those from simulation-based methods for simple options. The convergence properties of the model and the relationship between length of time increment and accuracy of solutions obtained are also investigated. For cases where the number of discrete time periods becomes too large to be solved using common decision tree software, recursive dynamic programming algorithms are developed to generate solutions. Finally, we illustrate a real application by solving for the value of an oil and gas switching option which requires a binomial model of two correlated one-factor commodity price models.