Better Than Pre-Commitment Mean-Variance Portfolio Allocation Strategies: Hamilton-Jacobi-Bellman Equations, Viscosity Solutions, and Semi-Self-Financing Strategies

Pre-commitment mean-variance (MV) criteria are very popular for portfolio optimization problems, due to their intuitive nature. It is standard that MV strategies for these problems are self-financing, i.e. no exogenous infusion or withdrawal of cash are allowed under any circumstances. Due to a well-known equivalence to the expected quadratic utility function approach, a known issue with pre-commitment MV criteria is that, during the investment horizon, if the portfolio wealth exceeds a certain threshold, the self-financing optimal control may suggest irrational behavior, namely to reduce the expected portfolio wealth. This irrationality contrasts to the natural desire of an investor, which is to achieve higher gains, i.e. the more the better. In such a case, the equivalent quadratic utility function of the portfolio wealth is considered poorly-behaved.

In this talk, I will explain a novel approach to address the aforementioned issue for the pre-commitment MV portfolio optimization problem under a very general setting, namely continuous or discrete re-balancing, jump-diffusions (with finite activity), and realistic portfolio constraints. The approach is built upon a Hamilton-Jacobi-Bellman (HJB) equation numerical framework for the solution of the portfolio allocation problem, combined with a semi-self-financing strategy which allows only cash withdrawals when the portfolio wealth exceeds a certain threshold. An important aspect of our approach is guaranteed convergence of the numerical results to the viscosity solutions of the HJB equations. Under our strategies, the equivalent quadratic utility function can be shown to be well-behaved. Numerical results confirming the superiority of the efficient frontiers produced by our strategies are presented.