Multiple criteria in financial models and an examination of nondominated surfaces and related topics

Abstract

This dissertation is devoted to the computational performance testing of two classes of interactive multiple objective programming procedures and the multiple criteria modeling in financial applications. In addition, a new option pricing model with correlated credit risk under a jump-diffusion process is presented.

One class of interactive multiple objective programming procedures is represented by the multiple-probing Tchebycheff Method of Steuer and Choo (1983) and the other by the Aspiration Criterion Vector Method developed by Wierzbicki (1977, 1982, 1986).

In the multiple criteria modeling of financial applications, the dissertation attempts to open up a new paradigm in portfolio optimization where objectives in addition to risk and return, such as skewness, are acknowledged and analyzed. However, the difficulty encountered as we transit from two to more than two criteria is that the Markowitz mean-variance efficient frontier, which we will call nondominated set, is no longer a frontier but a surface.

While it is possible to parametrize a frontier, it is not possible to parametrize a surface. As soon as a third objective is admitted, the resulting optimization problem can not be addressed by generalizing the parametric solution techniques of Markowitz theory. This then leads to the methods and techniques of multiple objective programming that attempt to intelligently probe and sample efficient surfaces in a convergent fashion. This dissertation reviews several prominent interactive multiple objective programming procedures and introduces them as a tool to explore nondominated surfaces.

We also discuss various risk measures and utility functions in portfolio optimization, present a multi-dimensional portfolio optimization model as an extension of the models developed by Konno-Yamazaki (1991), and demonstrate the usefulness of multiple objective methods to search over a nondominated surface.

Lastly, a new option pricing model is presented with correlated credit risk under a jump-diffusion process. In most option pricing models, it is assumed that the returns have normal distributions. The new option pricing model considers default risk and allows the asset to follow a jump-diffusion process which gives rise to fatter tails than do normal or lognormal distributions.