Portfolio Choice with General Pricing Kernel

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.