Multilevel Monte Carlo methods for positivity-preserving approximations of the Heston 3/2-model
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910361425281024 |
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| author | Wu, Xiaojuan Gan, Siqing |
| author_facet | Wu, Xiaojuan Gan, Siqing |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05837 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multilevel Monte Carlo methods for positivity-preserving approximations of the Heston 3/2-model Wu, Xiaojuan Gan, Siqing Numerical Analysis 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. |
| title | Multilevel Monte Carlo methods for positivity-preserving approximations of the Heston 3/2-model |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2403.05837 |