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

Paper: arXiv 2302.06778

Authors: Minglian Lin, Indranil SenGupta

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical analysis using Hamilton-Jacobi-Bellman (HJB) equations and asymptotic expansions for multi-asset stochastic volatility models. However, it is entirely theoretical with no empirical validation, empirical data testing, or backtesting provided.

Research Flowchart

  flowchart TD
    A[Goal: Portfolio Optimization in Multi-Asset Stochastic Volatility Model] --> B[Inputs: n-Dimensional Correlated Brownian Motions & Utility Function]
    B --> C[Derive Hamilton-Jacobi-Bellman HJB Equation]
    C --> D[Apply First-Order Time-Horizon Expansion to Utility Function]
    D --> E[Control Error via Second-Order Terms & Bound to Square of Time Horizon]
    E --> F[Outcomes: Close-to-Optimal Portfolio & Generalized Value Function Scheme]