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Multilevel Monte Carlo methods for positivity-preserving approximations of the Heston 3/2-model

Xiaojuan Wu, Siqing Gan

TL;DR

The paper tackles efficient evaluation of expectations for the Heston 3/2-model with super-linear coefficients by developing an explicit, positivity-preserving split-step Milstein discretization. Coupled with the multilevel Monte Carlo framework, it proves order-one mean-square convergence under non-globally Lipschitz conditions and establishes the optimal MLMC complexity O(ε^{−2}). Theoretical results are supported by numerical experiments demonstrating correct convergence rates, variance decay, and computational scaling. This work enables stable and fast pricing and risk assessment in models with positivity constraints and non-globally Lipschitz dynamics.

Abstract

This article is concerned with the multilevel Monte Carlo (MLMC) methods for approximating expectations of some functions of the solution to the Heston 3/2-model from mathematical finance, which takes values in $(0, \infty)$ and possesses superlinearly growing drift and diffusion coefficients. To discretize the SDE model, a new Milstein-type scheme is proposed to produce independent sample paths. The proposed scheme can be explicitly solved and is positivity-preserving unconditionally, i.e., for any time step-size $h>0$. This positivity-preserving property for large discretization time steps is particularly desirable in the MLMC setting. Furthermore, a mean-square convergence rate of order one is proved in the non-globally Lipschitz regime, which is not trivial, as the diffusion coefficient grows super-linearly. The obtained order-one convergence in turn promises the desired relevant variance of the multilevel estimator and justifies the optimal complexity $\mathcal{O}(ε^{-2})$ for the MLMC approach, where $ε> 0$ is the required target accuracy. Numerical experiments are finally reported to confirm the theoretical findings.

Multilevel Monte Carlo methods for positivity-preserving approximations of the Heston 3/2-model

TL;DR

The paper tackles efficient evaluation of expectations for the Heston 3/2-model with super-linear coefficients by developing an explicit, positivity-preserving split-step Milstein discretization. Coupled with the multilevel Monte Carlo framework, it proves order-one mean-square convergence under non-globally Lipschitz conditions and establishes the optimal MLMC complexity O(ε^{−2}). Theoretical results are supported by numerical experiments demonstrating correct convergence rates, variance decay, and computational scaling. This work enables stable and fast pricing and risk assessment in models with positivity constraints and non-globally Lipschitz dynamics.

Abstract

This article is concerned with the multilevel Monte Carlo (MLMC) methods for approximating expectations of some functions of the solution to the Heston 3/2-model from mathematical finance, which takes values in and possesses superlinearly growing drift and diffusion coefficients. To discretize the SDE model, a new Milstein-type scheme is proposed to produce independent sample paths. The proposed scheme can be explicitly solved and is positivity-preserving unconditionally, i.e., for any time step-size . This positivity-preserving property for large discretization time steps is particularly desirable in the MLMC setting. Furthermore, a mean-square convergence rate of order one is proved in the non-globally Lipschitz regime, which is not trivial, as the diffusion coefficient grows super-linearly. The obtained order-one convergence in turn promises the desired relevant variance of the multilevel estimator and justifies the optimal complexity for the MLMC approach, where is the required target accuracy. Numerical experiments are finally reported to confirm the theoretical findings.
Paper Structure (8 sections, 8 theorems, 69 equations, 3 figures)

This paper contains 8 sections, 8 theorems, 69 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The proposed time discretization scheme
  3. MLMC and its optimal complexity
  4. Basic elements of MLMC
  5. Order-one mean-square convergence of the scheme
  6. Optimal complexity of MLMC
  7. Numerical experiments
  8. Conclusion

Key Result

Proposition 2.1

Let $\mu , \alpha , \beta > 0$, $X_0 >0$. Then the Heston $3/2$-model Milstein-eq:SODE-eq:f-g-notation has a unique global positive solution. Moreover, for $1 \leq p \leq 1 + \tfrac{2 \alpha}{ \beta^2 }$ we have

Figures (3)

  • Figure 1: Mean-square convergence rates of the proposed time-stepping scheme.
  • Figure 2: Convergence of variance
  • Figure 3: Computational costs

Theorems & Definitions (8)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Corollary 3.6