A monotone numerical integration method for mean-variance portfolio optimization under jump-diffusion models

ArXiv ID: 2309.05977 “View on arXiv”

Authors: Unknown

Abstract

We develop a efficient, easy-to-implement, and strictly monotone numerical integration method for Mean-Variance (MV) portfolio optimization in realistic contexts, which involve jump-diffusion dynamics of the underlying controlled processes, discrete rebalancing, and the application of investment constraints, namely no-bankruptcy and leverage. A crucial element of the MV portfolio optimization formulation over each rebalancing interval is a convolution integral, which involves a conditional density of the logarithm of the amount invested in the risky asset. Using a known closed-form expression for the Fourier transform of this density, we derive an infinite series representation for the conditional density where each term is strictly positive and explicitly computable. As a result, the convolution integral can be readily approximated through a monotone integration scheme, such as a composite quadrature rule typically available in most programming languages. The computational complexity of our method is an order of magnitude lower than that of existing monotone finite difference methods. To further enhance efficiency, we propose an implementation of the scheme via Fast Fourier Transforms, exploiting the Toeplitz matrix structure. The proposed monotone scheme is proven to be both $\ell_{"\infty"}$-stable and pointwise consistent, and we rigorously establish its pointwise convergence to the unique solution of the MV portfolio optimization problem. We also intuitively demonstrate that, as the rebalancing time interval approaches zero, the proposed scheme converges to a continuously observed impulse control formulation for MV optimization expressed as a Hamilton-Jacobi-Bellman Quasi-Variational Inequality. Numerical results show remarkable agreement with benchmark solutions obtained through finite differences and Monte Carlo simulation, underscoring the effectiveness of our approach.

Keywords: Mean-Variance Optimization, Jump-Diffusion Dynamics, Fast Fourier Transforms (FFT), Monotone Numerical Integration, Portfolio Optimization, Portfolio Management

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper relies heavily on advanced mathematical concepts including jump-diffusion models, Hamilton-Jacobi-Bellman Quasi-Variational Inequalities, infinite series representations, and Fourier transforms, resulting in a high math complexity score. While it mentions numerical results and comparison with benchmarks, the abstract and excerpt focus on theoretical derivations and methodological innovations without presenting empirical backtests, statistical performance metrics, or implementation details like code or datasets, leading to a low empirical rigor score.

  flowchart TD
    A["Research Goal:<br>Develop efficient, monotone method for<br>Mean-Variance portfolio optimization<br>under jump-diffusion models"] --> B["Key Methodology:<br>Derive infinite series for<br>conditional density via<br>Fourier transform"]
    B --> C["Data/Inputs:<br>Jump-diffusion parameters,<br>investment constraints,<br>discrete rebalancing intervals"]
    C --> D["Computational Process:<br>Approximate convolution integral<br>using monotone integration scheme<br>(Quadrature or FFT with Toeplitz structure)"]
    D --> E["Key Findings/Outcomes:<br>• Order of magnitude faster<br>• Proven stability & convergence<br>• Matches benchmark solutions<br>• Converges to impulse control VIE"]