Optimal Portfolio Selection with Consumption under Stochastic Volatility

Abstract

Optimal asset allocation policies and optimal consumption policies have been important issues in finance in the past decades. These topics are important not only for theoretical consideration, but also for applications in financial industry. Empirical studies have shown that volatilities of risky assets should be modeled as stochastic rather than deterministic. This adds further complication to the problems. We study these issues under constant relative risk aversion (CRRA) utility functions in a general and practical setting: stochastic volatility, incomplete markets and finite investment horizons. So far, there have been no numerical solution available in the literature, not to mention analytical solutions under this general setting.

In this thesis, we present closed-form approximate solutions for this dynamic optimization problem. We further develop an accurate and efficient numerical algorithm for solving this problem, which is governed by a nonlinear partial differential equation (the Hamilton-Jacobi-Bellman equation), and generates the first set of highly accurate numerical solutions for this problem.

We show that our theoretical predictions are in excellent agreement with the numerical solutions. The approximation errors in the analytical solutions are smaller than the model-parameter-estimation errors. This is true even for an extremely long investment horizon. Therefore, for practical purposes, one can treat our closed-form formulas as "exact" and use them to avoid performing cumbersome numerical computations. For a market with a big number of risky assets, such numerical computations are even impossible to conduct, while our approximate solutions are still applicable.

Date of Award	2 Aug 2017
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Qiang ZHANG (Supervisor)