Optimal Consumption and Robust Portfolio Choice for the 3/2 and 4/2 Stochastic Volatility Models

Abstract

This manuscript derives optimal consumption and investment strategies for risk-averse investors under the 4/2 stochastic volatility class of models. We work under an expected utility (EUT) framework and consider a Constant Relative Risk Aversion (CRRA) investor, who may also be ambiguity-averse. The corresponding Hamilton–Jacobi–Bellman (HJB) and HJB–Isaacs (HJBI) equations are solved in closed-form for a subset of the parametric space and under some restrictions on the portfolio setting, for complete markets. Conditions for proper changes of measure and well-defined solutions are provided. These are the first analytical solutions for the 4/2 stochastic volatility model and the embedded 3/2 model for the type of excess returns established in the literature. We numerically illustrate the differences between the 4/2 model and the embedded cases of the 1/2 model (Heston) as well as the 3/2 model under the same data, and for two main cases: risk-averse investor in a complete market with consumption, and ambiguity-averse investor in a complete market with no consumption. In general, the 4/2 and 1/2 models recommend similar levels of consumption and exposure, while the 3/2 leads to significantly different recommendations.

Keywords

• 4/2 stochastic volatility
• CRRA (power) utility
• optimal portfolio investment and consumption
• Heston’s model
• 3/2 model