Optimal Portfolio Problem for Stochastic-Volatility, Jump-Diffusion Models with Jump-Bankruptcy Condition: Practical Theory

Abstract

This paper treats the risk-averse optimal portfolio problem with consumption in continuous time with a stochastic-volatility, jump-diffusion (SVJD) model of the underlying risky asset and the volatility. The new developments are the use of the SVJD model with double-uniform jump-amplitude distributions and time-varying market parameters for the optimal portfolio problem. Although unlimited borrowing and short-selling play an important role in pure diffusion models, it is shown that borrowing and short selling are constrained for jump-diffusions. Finite range jump-amplitude models can allow constraints to be very large in contrast to infinite range models which severely restrict the optimal instantaneous stock-fraction to [0,1]. The reasonable constraints in the optimal stock-fraction due to jumps in the wealth argument for stochastic dynamic programming jump integrals remove a singularity in the stock-fraction due to vanishing volatility. Main modifications for the usual constant relative risk aversion (CRRA) power utility model are for handling the partial integro-differential equation (PIDE) resulting from the additional variance independent variable, instead of the ordinary integro-differential equation (OIDE) found for the pure jump-diffusion model of the wealth process. In addition to natural constraints due to jumps when enforcing the positivity of wealth condition, other constraints are considered for all practical purposes under finite market conditions. Also, a computationally practical solution of Heston's (1993) square-root-diffusion model for the underlying asset variance is derived. This shows that the non-negativity of the variance is preserved through the proper singular limit of a simple perfect-square form. An exact, non\-singular solution is found for a special combination of the Heston stochastic volatility parameters.

Keywords: optimal-portfolio problem, stochastic-volatility, jump-diffusion, finite markets, jump-bankruptcy condition, double-uniform jump-amplitudes, non-negative-variance verification

JEL Classification: C61, C73, C63, D81, D84