Numerical analysis on locally risk-minimizing strategies for Barndorff-Nielsen and Shephard models

ArXiv ID: 2505.00255 “View on arXiv”

Authors: Takuji Arai

Abstract

We develop a numerical method for locally risk-minimizing (LRM) strategies for Barndorff-Nielsen and Shephard (BNS) models. Arai et al. (2017) derived a mathematical expression for LRM strategies in BNS models using Malliavin calculus for Lévy processes and presented some numerical results only for the case where the asset price process is a martingale. Subsequently, Arai and Imai (2024) developed the first Monte Carlo (MC) method available for non-martingale BNS models with infinite active jumps. Here, we modify the expression obtained by Arai et al. (2017) into a numerically tractable form, and, using the MC method developed by Arai and Imai (2024), propose a numerical method of LRM strategies available for non-martingale BNS models with infinite active jumps. In the final part of this paper, we will conduct some numerical experiments.

Keywords: BNS Model, Local Risk-Minimization, Malliavin Calculus, Monte Carlo Method, Lévy Processes, Equities

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is extremely dense in advanced mathematics, utilizing Malliavin calculus for Lévy processes and complex stochastic calculus derivations for a non-martingale model. While it mentions numerical experiments, the focus is heavily on modifying theoretical expressions into a tractable form using Monte Carlo methods, with no presented code, specific datasets, or empirical performance metrics in the provided excerpt.

  flowchart TD
    A["Research Goal: Develop Numerical Method<br>for LRM Strategies in BNS Models"] --> B["Key Methodology Step: Modify Arai et al.<br>Expression via Malliavin Calculus"]
    B --> C{"Model Type?"}
    C -- Martingale --> D["Direct Computation<br>from Arai et al. (2017)"]
    C -- Non-Martingale w/ Infinite Jumps --> E["Monte Carlo Method<br>Arai & Imai (2024)"]
    D & E --> F["Computational Process:<br>Simulate Lévy Processes &<br>Estimate LRM Strategies"]
    F --> G["Key Outcomes:<br>Numerical Experiments &<br>Analysis of Strategies"]