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

Christoph Reisinger, Maria Olympia Tsianni

TL;DR

This work provides a rigorous treatment of simulating a calibrated Heston-type local-stochastic volatility model via a kernel-based particle method that yields a McKean–Vlasov SDE with irregular diffusion. The authors establish well-posedness of the regularised MV-SDE for fixed kernel bandwidth, prove a strong propagation of chaos up to a critical time, and prove strong convergence of the Euler–Maruyama scheme for the particle system, with rate 1/2 in time up to a logarithmic factor under appropriate Feller and Hölder conditions. The full-truncation Euler scheme for the CIR volatility is employed to preserve non-negativity and achieve strong convergence in L^p for the volatility component. Numerical experiments validate discretisation convergence and demonstrate practical propagation of chaos, including parametric regimes where the Feller condition is stressed, showing the method's reliability for calibration and pricing tasks in finance.

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.

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

TL;DR

This work provides a rigorous treatment of simulating a calibrated Heston-type local-stochastic volatility model via a kernel-based particle method that yields a McKean–Vlasov SDE with irregular diffusion. The authors establish well-posedness of the regularised MV-SDE for fixed kernel bandwidth, prove a strong propagation of chaos up to a critical time, and prove strong convergence of the Euler–Maruyama scheme for the particle system, with rate 1/2 in time up to a logarithmic factor under appropriate Feller and Hölder conditions. The full-truncation Euler scheme for the CIR volatility is employed to preserve non-negativity and achieve strong convergence in L^p for the volatility component. Numerical experiments validate discretisation convergence and demonstrate practical propagation of chaos, including parametric regimes where the Feller condition is stressed, showing the method's reliability for calibration and pricing tasks in finance.

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 -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 in time, up to a logarithmic factor. Numerical experiments illustrate the convergence of the discretisation scheme and validate the propagation of chaos in practice.

Paper Structure

This paper contains 14 sections, 10 theorems, 103 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Structure of paper
  3. Notation and Preliminaries
  4. Problem formulation
  5. Particle method
  6. Properties of the Cox--Ingersoll--Ross process
  7. Well-posedness of the regularised system
  8. Propagation of chaos
  9. Time-discretisation scheme
  10. The full truncation Euler (FTE) scheme
  11. Strong convergence of Euler--Maruyama (EM) scheme
  12. Proof of well-posedness of the regularised equations
  13. Numerical experiments
  14. Proof of Proposition \ref{['existencetheorem']}

Key Result

Lemma 1

Let $V^i$ denote the CIR volatility process as in volparticles. It holds that

Figures (12)

  • Figure 1: $\kappa_1 =1.5$
  • Figure 2: $\kappa_2 =6$
  • Figure 3: $\kappa_3 = 18$
  • Figure 4: $\xi_4 = 0.1$
  • Figure 5: $\xi_5 = 0.5$
  • ...and 7 more figures

Theorems & Definitions (22)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1: Well-posedness
  • Theorem 2
  • proof
  • Lemma 3
  • ...and 12 more