High order approximations and simulation schemes for the log-Heston process

ArXiv ID: 2407.17151 “View on arXiv”

Authors: Unknown

Abstract

We present weak approximations schemes of any order for the Heston model that are obtained by using the method developed by Alfonsi and Bally (2021). This method consists in combining approximation schemes calculated on different random grids to increase the order of convergence. We apply this method with either the Ninomiya-Victoir scheme (2008) or a second-order scheme that samples exactly the volatility component, and we show rigorously that we can achieve then any order of convergence. We give numerical illustrations on financial examples that validate the theoretical order of convergence. We also present promising numerical results for the multifactor/rough Heston model and hint at applications to other models, including the Bates model and the double Heston model.

Keywords: Heston model, Ninomiya-Victoir scheme, High-order approximation, Rough Heston model, Bates model, Equity Derivatives

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats
  • Why: The paper is intensely theoretical, focusing on high-order weak approximations, random grid methods, and rigorous convergence proofs for the log-Heston SDE, demonstrating advanced stochastic calculus and functional analysis. However, its empirical validation is limited to numerical illustrations of theoretical convergence order, with no backtesting or real-world trading strategies presented.
  flowchart TD
    A["Research Goal"] --> B["Methodology: Combined Approximation Schemes"]
    B --> C["Base Schemes: Ninomiya-Victoir &<br>Exact Volatility Sampling"]
    C --> D["Simulation on Random Grids"]
    D --> E["Weak Approximation of<br>Log-Heston Process"]
    E --> F["Outcomes"]
    
    subgraph F ["Key Findings"]
        F1["Proved rigorous convergence<br>to any desired order"]
        F2["Numerical validation of<br>theoretical convergence rates"]
        F3["Extensions to Rough Heston,<br>Bates, and Double Heston models"]
    end