Analysis of optimal portfolio on finite and small-time horizons for a stochastic volatility model with multiple correlated assets

TL;DR

This paper develops an approximation method for optimal portfolio selection in a stochastic volatility model with multiple correlated assets, analyzing the value function and generating near-optimal portfolios over finite and small time horizons.

Contribution

It introduces a novel approximation approach for portfolio optimization in multi-asset stochastic volatility models, including correlation effects and error control.

Findings

Derived a Hamilton-Jacobi-Bellman equation with correlations.

Provided an approximation scheme for the value function over time.

Generated near-optimal portfolios with controlled error.

Abstract

In this paper, we consider the portfolio optimization problem in a financial market where the underlying stochastic volatility model is driven by n-dimensional Brownian motions. At first, we derive a Hamilton-Jacobi-Bellman equation including the correlations among the standard Brownian motions. We use an approximation method for the optimization of portfolios. With such approximation, the value function is analyzed using the first-order terms of expansion of the utility function in the powers of time to the horizon. The error of this approximation is controlled using the second-order terms of expansion of the utility function. It is also shown that the one-dimensional version of this analysis corresponds to a known result in the literature. We also generate a close-to-optimal portfolio near the time to horizon using the first-order approximation of the utility function. It is shown that the error is controlled by the square of the time to the horizon. Finally, we provide an approximation scheme to the value function for all times and generate a close-to-optimal portfolio.