Warren J. Hahn

Using Binomial Decision Trees to Solve Real-Option Valuation Problems

Traditional decision analysis methods can provide an intuitive approach to valuing projects with managerial flexibility or real options. The discrete-time approach to real-option valuation has typically been implemented in the finance literature using a binomial lattice framework. Instead, we use a binomial decision tree with risk-neutral probabilities to approximate the uncertainty associated with the changes in the value of a project over time. Both methods are based on the same principles, but we use dynamic programming to solve the binomial decision tree, thereby providing a computationally intensive but simpler and more intuitive solution. This approach also provides greater flexibility in the modeling of problems, including the ability to include multiple underlying uncertainties and concurrent options with complex payoff characteristics.

Discrete time modeling of mean-reverting stochastic processes for real option valuation

Abstract In this paper the recombining binomial lattice approach for modeling real options and valuing managerial flexibility is generalized to address a common issue in many practical applications, 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 paper 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 dual correlated one-factor mean-reverting processes. These models facilitate the evaluation of options with early-exercise characteristics, as well as multiple concurrent options. The models we develop in this paper are tested by implementing the lattice in binomial decision tree format and applying to 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. For cases where the number of discrete time periods becomes too large to be solved using common decision tree software, we describe how recursive dynamic programming algorithms can be developed to generate solutions.

Volatility estimation for stochastic project value models

One of the key parameters in modeling capital budgeting decisions for investments with embedded options is the project volatility. Most often, however, there is no market or historical data available to provide an accurate estimate for this parameter. A common approach to estimating the project volatility in such instances is to use a Monte Carlo simulation where one or more sources of uncertainty are consolidated into a single stochastic process for the project cash flows, from which the volatility parameter can be determined. Nonetheless, the simulation estimation method originally suggested for this purpose systematically overstates the project volatility, which can result in incorrect option values and non-optimal investment decisions. Examples that illustrate this issue numerically have appeared in several recent papers, along with revised estimation methods that address this problem. In this article, we extend that work by showing analytically the source of the overestimation bias and the adjustment necessary to remove it. We then generalize this development for the cases of levered cash flows and non-constant volatility. In each case, we use an example problem to show how a revised estimation methodology can be applied.

A Discrete Time Approach for Modeling Two-Factor Mean-Reverting Stochastic Processes

Two-factor stochastic processes have been developed to more accurately describe the intertemporal dynamics of variables such as commodity prices. In this paper we develop an approach for modeling these types of stochastic processes in discrete time as two-dimensional binomial sequences. This approach facilitates the numerical solution of dynamic optimization problems such as investment decision making under uncertainty and option valuation related to commodities. We implement this approach in a two-dimensional lattice format, apply it to two hypothetical valuation problems discussed by Schwartz and Smith, and compare the results to those from simulation-and dynamic-programming-based methods.

Sensitivity Analysis of Model Input Dependencies Using Copulas

Many important decision and risk analysis problems are complicated by dependencies between input variables. In such cases, standard one-variable-at-a-time sensitivity analysis methods are typically eschewed in favor of fully probabilistic, or n-way, analysis techniques which simultaneously model all n input variables and their interdependencies. Unfortunately, much of the intuition provided by one-way sensitivity analysis and its associated graphical output displays such as Tornado diagrams may not be available in fully probabilistic methods. It is also difficult or impossible to isolate the marginal effects of variables in an n-way analysis. In this paper, we present a dependence-adjusted approach for identifying and analyzing the impact of the input variables in a model through the use of probabilistic sensitivity analysis based on copulas. This approach provides insights about the influence of both the input variables and the dependence relationships between the input variables. A key contribution of this approach is that it facilitates assessment of the relative marginal influence of variables for the purpose of determining which variables should be modeled in applications where computational efficiency is a concern, such as in decision tree analysis of large scale problems. In addition, we also investigate the sensitivity to the magnitude of correlation in the inputs.