Rough differential equations for volatility
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arXiv
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| Format: | Preprint |
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2024
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| author | Bonesini, Ofelia Ferrucci, Emilio Gasteratos, Ioannis Jacquier, Antoine |
| author_facet | Bonesini, Ofelia Ferrucci, Emilio Gasteratos, Ioannis Jacquier, Antoine |
| contents | We introduce a canonical way of performing the joint lift of a Brownian motion $W$ and a low-regularity adapted stochastic rough path $\mathbf{X}$, extending [Diehl, Oberhauser and Riedel (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations]. Applying this construction to the case where $\mathbf{X}$ is the canonical lift of a one-dimensional fractional Brownian motion (possibly correlated with $W$) completes the partial rough path of [Fukasawa and Takano (2024). A partial rough path space for rough volatility]. We use this to model rough volatility with the versatile toolkit of rough differential equations (RDEs), namely by taking the price and volatility processes to be the solution to a single RDE. We argue that our framework is already interesting when $W$ and $X$ are independent, as correlation between the price and volatility can be introduced in the dynamics. The lead-lag scheme of [Flint, Hambly, and Lyons (2016). Discretely sampled signals and the rough Hoff process] is extended to our fractional setting as an approximation theory for the rough path in the correlated case. Continuity of the solution map transforms this into a numerical scheme for RDEs. We numerically test this framework and use it to calibrate a simple new rough volatility model to market data. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_21192 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rough differential equations for volatility Bonesini, Ofelia Ferrucci, Emilio Gasteratos, Ioannis Jacquier, Antoine Mathematical Finance Probability 60L20, 60L90, 60G22, 65C30, 91G20, 91G60 We introduce a canonical way of performing the joint lift of a Brownian motion $W$ and a low-regularity adapted stochastic rough path $\mathbf{X}$, extending [Diehl, Oberhauser and Riedel (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations]. Applying this construction to the case where $\mathbf{X}$ is the canonical lift of a one-dimensional fractional Brownian motion (possibly correlated with $W$) completes the partial rough path of [Fukasawa and Takano (2024). A partial rough path space for rough volatility]. We use this to model rough volatility with the versatile toolkit of rough differential equations (RDEs), namely by taking the price and volatility processes to be the solution to a single RDE. We argue that our framework is already interesting when $W$ and $X$ are independent, as correlation between the price and volatility can be introduced in the dynamics. The lead-lag scheme of [Flint, Hambly, and Lyons (2016). Discretely sampled signals and the rough Hoff process] is extended to our fractional setting as an approximation theory for the rough path in the correlated case. Continuity of the solution map transforms this into a numerical scheme for RDEs. We numerically test this framework and use it to calibrate a simple new rough volatility model to market data. |
| title | Rough differential equations for volatility |
| topic | Mathematical Finance Probability 60L20, 60L90, 60G22, 65C30, 91G20, 91G60 |
| url | https://arxiv.org/abs/2412.21192 |