Comment on "Generating Scenario Trees for Multistage Decision Problems"

In models of decision making under uncertainty, one typically has to approximate the uncertainties by a limited number of discrete outcomes. Høyland and Wallace (2001) formulate a nonlinear programming problem to generate such a limited number of discrete outcomes while satisfying specified statistical properties. They have developed and employed this method for a stochastic multistage asset-allocation problem. When the method is applied to such financial optimization problems under uncertainty, we argue here that it does not suffice to match statistical properties. To obtain realistic outcomes, the (limited) description of the uncertainty in such models should also exclude arbitrage opportunities, and thereby be consistent with financial asset pricing theory. We illustrate that the method proposed by Høyland and Wallace can result in arbitrage opportunities in the scenario tree if only statistical properties are imposed. We show how one can check ex post for the presence of arbitrage opportunities in a scenario tree by checking for the existence of solutions to sets of linear equations. Arbitrage opportunities can also be precluded ex ante in the scenario tree by adding constraints to the nonlinear programming problem of Høyland and Wallace.