dc.contributor.advisor Yang, Jie en_US dc.contributor.author Zhang, Zhifan en_US dc.date.accessioned 2014-06-20T15:33:51Z dc.date.available 2014-06-20T15:33:51Z dc.date.created 2014-05 en_US dc.date.issued 2014-06-20 dc.date.submitted 2014-05 en_US dc.identifier.uri http://hdl.handle.net/10027/18788 dc.description.abstract The portfolio choice optimization problem we study in this thesis is to construct a continuous-time portfolio which maximizes the probability of outperformance. In the literature of mathematical finance, this type of problem is typically solved by the quantile approach, which requires a non-atom pricing kernel. In real financial practice, the pricing kernel can be atomic, i.e., the probability that the pricing kernel equals to a constant can be positive. For example, an extreme case is that the pricing kernel equals to a constant with probability 1. Another example is the scenario analysis in risk management. Risk analysis is done by setting the asset price to be certain extreme values. In this case, the pricing kernel is atomic at those extreme values. In this thesis, we consider two portfolio choice optimization models, goal reaching model and Yaari's dual model, with more general pricing kernels which may allow the existence of atoms. For goal reaching model, we discuss the properties of the solution, and derive a modified optimization problem, which has a similar mathematical format to the optimal hypothesis test problems. Therefore, a general solution scheme for both non-atomic and atomic pricing kernel is derived based on a generalized Neyman-Pearson Lemma, which is famous in classical statistical theory. We also provide an example with pricing kernel follows geometric Brownian motion, to show the explicit solution based on our results. Our numerical experiments validate the optimal solution as well. For Yaari's dual model, we discuss the properties of optimal solution that is an optimal terminal cash flow which is nonincreasing with respect to the pricing kernel. The pricing kernel here could contain atoms and thus is more general than non-atomic ones. Under the assumption that probability distortion/weighting is differentiable, we derive a modified optimization problem that contains left-continuous quantile function of the pricing kernel and terminal case flow. A sub-optimization problem with Lagrange multiplier is studied. We propose an algorithm, called "Search-and-Cut" Algorithm to find the optimal solution, which is good for cases where the weighting/pricing-kernel ratio consists of a finite number of monotone pieces. We prove the existence and uniqueness of the optimal solution as well. Finally, we derive an optimal solution of Yaari's dual model for more general pricing kernels and probability distortions. The approaches we propose in this thesis could be used for other portfolio choice models, as well as for problems solved by non-atomic quantile approaches. en_US dc.language.iso en en_US dc.rights en_US dc.rights Copyright 2014 Zhifan Zhang en_US dc.subject portfolio choice optimization en_US dc.subject pricing kernel en_US dc.subject goal reaching model en_US dc.subject Yaari’s dual model en_US dc.subject probability distortion en_US dc.subject quantile approach en_US dc.subject generalized Neyman-Pearson Lemma en_US dc.title Portfolio Choice with General Pricing Kernel en_US thesis.degree.department Mathematics, Statistics, and Computer Science en_US thesis.degree.discipline Mathematics en_US thesis.degree.grantor University of Illinois at Chicago en_US thesis.degree.level Doctoral en_US thesis.degree.name PhD, Doctor of Philosophy en_US dc.type.genre thesis en_US dc.contributor.committeeMember Wang, Jing en_US dc.contributor.committeeMember Martin, Ryan en_US dc.contributor.committeeMember Ouyang, Cheng en_US dc.contributor.committeeMember Wang, Fangfang en_US dc.type.material text en_US
﻿