Numerical analysis of a particle system for the calibrated Heston-type local stochastic volatility model

ArXiv ID: 2504.14343 “View on arXiv”

Authors: Unknown

Abstract

We analyse a Monte Carlo particle method for the simulation of the calibrated Heston-type local stochastic volatility (H-LSV) model. The common application of a kernel estimator for a conditional expectation in the calibration condition results in a McKean-Vlasov (MV) stochastic differential equation (SDE) with non-standard coefficients. The primary challenges lie in certain mean-field terms in the drift and diffusion coefficients and the $1/2$-Hölder regularity of the diffusion coefficient. We establish the well-posedness of this equation for a fixed but arbitrarily small bandwidth of the kernel estimator. Moreover, we prove a strong propagation of chaos result, ensuring convergence of the particle system under a condition on the Feller ratio and up to a critical time. For the numerical simulation, we employ an Euler-Maruyama scheme for the log-spot process and a full truncation Euler scheme for the CIR volatility process. Under certain conditions on the inputs and the Feller ratio, we prove strong convergence of the Euler-Maruyama scheme with rate $1/2$ in time, up to a logarithmic factor. Numerical experiments illustrate the convergence of the discretisation scheme and validate the propagation of chaos in practice.

Keywords: Heston Model, Local Stochastic Volatility, Monte Carlo Simulation, McKean-Vlasov SDE, Propagation of Chaos, Derivatives (Options)

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, proving existence, uniqueness, and propagation of chaos for a McKean-Vlasov SDE with irregular coefficients using advanced stochastic analysis, yet it only presents minimal numerical experiments to illustrate the convergence of the scheme, lacking backtests or practical trading implementation.

  flowchart TD
    A["<b>Research Goal:</b><br>Analyse Monte Carlo particle method<br>for calibrated H-LSV model"] --> B["<b>Key Methodology:</b><br>1. McKean-Vlasov SDE Analysis<br>2. Propagation of Chaos<br>3. Euler-Maruyama Scheme"]
    B --> C["<b>Mathematical Framework:</b><br>Conditional Expectation via Kernel Estimator<br>Mean-field terms in SDE coefficients"]
    C --> D["<b>Computational Process:</b><br>Particle System Simulation<br>Log-spot (Euler-Maruyama)<br>CIR Volatility (Full Truncation)<br>Condition: Feller Ratio"]
    D --> E["<b>Key Findings/Outcomes:</b><br>1. Well-posedness for fixed bandwidth<br>2. Strong Propagation of Chaos (up to critical time)<br>3. Strong convergence rate 1/2 (log factor)<br>4. Numerical validation of convergence"]