Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Friesen, Martin Gerhold, Stefan Wiedermann, Kristof |
| author_facet | Friesen, Martin Gerhold, Stefan Wiedermann, Kristof |
| contents | We study small-time central limit theorems for stochastic Volterra integral equations with Hölder continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional distributions, a functional CLT, and limit theorems for smooth transformations of the process, which covers a large class of Volterra kernels that includes rough models based on Riemann-Liouville kernels with short- and long-range dependencies. To illustrate our results, we derive asymptotic pricing formulae for digital calls on the realized variance in three different regimes. The latter provides a robust and model-independent pricing method for small maturities in rough volatility models. Finally, for the case of completely monotone kernels, we introduce a flexible framework of Hilbert space-valued Markovian lifts and derive analogous limit theorems for such lifts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15971 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts Friesen, Martin Gerhold, Stefan Wiedermann, Kristof Probability Mathematical Finance 60F05, 60F17, 60H15, 60H20, 60G15, 60G22, 91G20 We study small-time central limit theorems for stochastic Volterra integral equations with Hölder continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional distributions, a functional CLT, and limit theorems for smooth transformations of the process, which covers a large class of Volterra kernels that includes rough models based on Riemann-Liouville kernels with short- and long-range dependencies. To illustrate our results, we derive asymptotic pricing formulae for digital calls on the realized variance in three different regimes. The latter provides a robust and model-independent pricing method for small maturities in rough volatility models. Finally, for the case of completely monotone kernels, we introduce a flexible framework of Hilbert space-valued Markovian lifts and derive analogous limit theorems for such lifts. |
| title | Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts |
| topic | Probability Mathematical Finance 60F05, 60F17, 60H15, 60H20, 60G15, 60G22, 91G20 |
| url | https://arxiv.org/abs/2412.15971 |