Simulation of square-root processes made simple: applications to the Heston model

ArXiv ID: 2412.11264 “View on arXiv”

Authors: Unknown

Abstract

We introduce a simple, efficient and accurate nonnegative preserving numerical scheme for simulating the square-root process. The novel idea is to simulate the integrated square-root process first instead of the square-root process itself. Numerical experiments on realistic parameter sets, applied for the integrated process and the Heston model, display high precision with a very low number of time steps. As a bonus, our scheme yields the exact limiting Inverse Gaussian distributions of the integrated square-root process with only one single time-step in two scenarios: (i) for high mean-reversion and volatility-of-volatility regimes, regardless of maturity; and (ii) for long maturities, independent of the other parameters.

Keywords: square-root process, Heston model, numerical simulation, integrated process, Inverse Gaussian, Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper presents advanced mathematical concepts like stochastic differential equations, affine structures, and explicit simulation schemes, but also includes numerical experiments and validation on realistic parameter sets with precise performance metrics.

  flowchart TD
    A["Research Goal: Simulate square-root process efficiently & accurately"] --> B["Key Methodology: Simulate integrated process first"]
    B --> C["Data/Inputs: Heston model parameters<br>realistic numerical setups"]
    C --> D["Computational Process: Apply new scheme<br>to integrated square-root process"]
    D --> E{"Key Outcomes"}
    E --> F["High precision with low time steps"]
    E --> G["Exact limiting Inverse Gaussian distributions<br>obtained in specific regimes"]
    E --> H["Nonnegativity preservation ensured"]