On Stochastic Dominance and Decreasing Absolute Risk Averse Option Pricing Bounds

Merton, Perrakis and Ryan, Levy, and Ritchken have established option pricing bounds under first and second stochastic dominance preferences. These bounds are particularly important for valuing contingent claims when continuous trading in the claim and/or underlying security does not exist. This article provides option bounds under higher orders of dominance. Specifically, option bounds are obtained by solving mathematical programs where preference structures on prices are represented by constraints. For first, second, third and higher orders of stochastic dominance preferences, the special linear structure of the mathematical programs allow analytical solutions to be obtained for the bounds. For DARA preferences, third order stochastic dominance, while being necessary, is not sufficient and additional constraints must be imposed. Unfortunately these additional constraints are nonlinear. While in this case closed form analytical solutions for the option bounds are not obtained, numerical examples are presented to illustrate their strength.