Continuous Time Mean-Variance Optimal Portfolio Allocation Under Jump Diffusion: An Numerical Impulse Control Approach

Abstract

We present efficient partial differential equation (PDE) methods for continuous time mean-variance portfolio allocation problems when the underlying risky asset follows a jump-diffusion. The standard formulation of mean-variance optimal portfolio allocation problems, where the total wealth is the underlying stochastic process, gives rise to a one-dimensional (1-D) non-linear Hamilton-Jacobi- Bellman (HJB) partial integro-differential equation (PIDE) with the control present in the integrand of the jump term, and thus is difficult to solve efficiently. In order to preserve the efficient handling of the jump term, we formulate the asset allocation problem as a 2-D impulse control problem, one dimension for each asset in the portfolio, namely the bond and the stock. We then develop a numerical scheme based on a semi-Lagrangian timestepping method, which we show to be monotone, consistent, and stable. Hence, assuming a strong comparison property holds, the numerical solution is guaranteed to converge to the unique viscosity solution of the corresponding HJB PIDE. The correctness of the proposed numerical framework is verified by numerical examples. We also discuss the effects on the efficient frontier of realistic financial modeling, such as different borrowing and lending interest rates, transaction costs and constraints on the portfolio, such as maximum limits on borrowing and solvency.