The stochastic-volatility, jump-diffusion optimal portfolio problem with jumps in returns and volatility

The risk-averse optimal portfolio problem is treated with consumption in continuous time for a stochastic-jump-volatility-jump-diffusion (SJVJD) model for both the risky asset and the volatility. The new developments are the SJVJD model with double-uniform jump-amplitude distributions and time-varying market parameters for the optimal portfolio problem. Although unlimited borrowing and short-selling are important in pure diffusion models, borrowing and short-selling are constrained for jump-diffusions. Finite-range jump-amplitude models allow very large constraints 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 are due to jumps in the wealth argument for stochastic dynamic programming jump integrals to remove a singularity in the stock fraction in the limit of vanishing volatility. The main modifications for constant relative risk aversion power utility models are for handling the partial integrodifferential equation resulting from the additional variance independent variable, instead of the ordinary integrodifferential equation found for the pure jump-diffusion model of the wealth process. Other constraints are considered for finite market conditions. Computational results are presented for optimal portfolio values, stock fraction and consumption policies. The main contribution of this paper is the extension of computational solutions for the optimal portfolio and consumption problem to include jumps in stochastic volatility.