Computational Methods for Portfolio and Consumption Policy Optimization in Log-Normal Diffusion, Log-Uniform Jump Environments

Computational methods for a jump-diffusion portfolio optimization application using a loguniform jump distribution are considered. In contrast to the usual geometric Brownian motion problem based upon two parameters, mean appreciation and diffusive volatility, the jumpdiffusion model will have at least five, since jump process needs at least a rate, a mean and a variance, depending on the jump-amplitude distribution. As the number number of parameters increases, the computational complexity of the problem of determining the parameter set of the underlying model becomes greater. In a companion stochastic parameter estimation paper, real market data, here a decade of log-returns for Standard and Poor’s 500 index closings, is used to fit the jump-diffusion parameters, with constraints based on matching the data mean and variance to keep the unconstrained parameter space to 3 dimensions. A weighted least squares method has been used. The jump-diffusion theoretical distribution and weights has been derived. In this computational paper, the computational features of a new multidimensional, derivative-less global search method used in the companion paper are discussed. The main part of this paper is to discuss the computational solution of an optimal portfolio and consumption finance application with these more realistic parameter results. The constant relative risk aversion (CRRA) canonical model is used to reduce the high dimensionality of the PDE of stochastic dynamic programming problem to something more reasonable. Many computational issues arise due to the jump process part of the model, since several jump integrals arise which are not present in the pure diffusion with drift model. The log-uniformly distributed jumps allow a wider range of portfolio policies than does previous work with normally distributed jumps.