Numerical solution of the Hamilton–Jacobi–Bellman formulation for continuous-time mean–variance asset allocation under stochastic volatility

Abstract

We present efficient partial differential equation (PDE) methods for continuous-time mean-variance portfolio allocation problems when the underlying risky asset follows a stochastic volatility process. The standard formulation for mean-variance optimal portfolio allocation problems gives rise to a two-dimensional nonlinear Hamilton-Jacobi-Bellman (HJB) PDE. We use a wide stencil method based on a local coordinate rotation to construct a monotone scheme. Further, by using a semi-Lagrangian times stepping method to discretize the drift term, along with an improved linear interpolation method, accurate efficient frontiers are constructed. This scheme can be shown to be convergent to the viscosity solution of the HJB equation, and the correctness of the proposed numerical framework is verified by numerical examples. We also discuss the effects on the efficient frontier of the stochastic volatility model parameters.