Analysis of optimal portfolio on finite and small time horizons for a stochastic volatility market model

TL;DR

This paper derives an approximate closed-form solution for the optimal portfolio in a stochastic volatility market with jumps over small time horizons, providing a practical method for near-optimal investment strategies.

Contribution

It introduces a novel approximation method for the optimal portfolio in incomplete markets driven by stochastic volatility and jumps, with proven accuracy.

Findings

Derived a closed-form approximation for the optimal portfolio in small time horizons.

Proved the accuracy of the approximation formulas.

Provided a procedure to generate near-optimal portfolios for finite horizons.

Abstract

In this paper, we consider the portfolio optimization problem in a financial market under a general utility function. Empirical results suggest that if a significant market fluctuation occurs, invested wealth tends to have a notable change from its current value. We consider an incomplete stochastic volatility market model, that is driven by both a Brownian motion and a jump process. At first, we obtain a closed-form formula for an approximation to the optimal portfolio in a small-time horizon. This is obtained by finding the associated Hamilton-Jacobi-Bellman integro-differential equation and then approximating the value function by constructing appropriate super-solution and sub-solution. It is shown that the true value function can be obtained by sandwiching the constructed super-solution and sub-solution. We also prove the accuracy of the approximation formulas. Finally, we provide a procedure for generating a close-to-optimal portfolio for a finite time horizon.