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

2025-04-19

AI Summary

This work addresses the well-posedness and numerical simulation challenges arising in the calibrated Heston-type local-stochastic volatility (H-LSV) model, whose mean-field SDE features a nonstandard McKean–Vlasov term induced by a kernel estimator and a diffusion coefficient with only 1/2-Hölder regularity. We propose a Monte Carlo particle scheme coupling the Euler–Maruyama discretization for the log-price process with the full truncation Euler scheme for the CIR-type volatility process. For the first time, we establish the well-posedness theory for H-LSV-type mean-field SDEs under small-bandwidth kernel estimation. Under the Feller condition, we prove strong propagation of chaos and strong convergence of order 1/2 over a critical time horizon. Both theoretical analysis and numerical experiments validate the scheme’s effectiveness and confirm the predicted convergence rate. This provides a rigorous, implementable numerical framework for calibration and pricing under high-dimensional nonlinear stochastic volatility models.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Economics, Online Markets and Human Computation: Humans versus LLMs for data annotation and labeling
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 $1/2$-H""{o}lder 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.

Problem

• Simulating Heston-type local stochastic volatility model with particle method
• Addressing McKean-Vlasov SDE challenges in model calibration
• Proving convergence of Euler-Maruyama scheme for log-spot process

Innovation

• Monte Carlo particle method for H-LSV model
• McKean-Vlasov SDE with non-standard coefficients
• Euler-Maruyama and full truncation Euler schemes