Path-dependent PDEs for volatility derivatives
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911064716738560 |
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| author | Pannier, Alexandre |
| author_facet | Pannier, Alexandre |
| contents | We regard options on VIX and Realised Variance as solutions to path-dependent partial differential equations (PDEs) in a continuous stochastic volatility model. The modeling assumption specifies that the instantaneous variance is a $C^3$ function of a multidimensional Gaussian Volterra process; this includes a large class of models suggested for the purpose of VIX option pricing, either rough, or not, or mixed. We unveil the path-dependence of those volatility derivatives and, under a regularity hypothesis on the payoff function, we prove the well-posedness of the associated PDE. The latter is of heat type, because of the Gaussian assumption, and the terminal condition is also path-dependent. Furthermore, formulae for the greeks are provided, the implied volatility is shown to satisfy a quasi-linear path-dependent PDE and, in Markovian models, finite-dimensional pricing PDEs are obtained for VIX options. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_08289 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Path-dependent PDEs for volatility derivatives Pannier, Alexandre Probability Analysis of PDEs Mathematical Finance 60G22, 35K10, 91G20 We regard options on VIX and Realised Variance as solutions to path-dependent partial differential equations (PDEs) in a continuous stochastic volatility model. The modeling assumption specifies that the instantaneous variance is a $C^3$ function of a multidimensional Gaussian Volterra process; this includes a large class of models suggested for the purpose of VIX option pricing, either rough, or not, or mixed. We unveil the path-dependence of those volatility derivatives and, under a regularity hypothesis on the payoff function, we prove the well-posedness of the associated PDE. The latter is of heat type, because of the Gaussian assumption, and the terminal condition is also path-dependent. Furthermore, formulae for the greeks are provided, the implied volatility is shown to satisfy a quasi-linear path-dependent PDE and, in Markovian models, finite-dimensional pricing PDEs are obtained for VIX options. |
| title | Path-dependent PDEs for volatility derivatives |
| topic | Probability Analysis of PDEs Mathematical Finance 60G22, 35K10, 91G20 |
| url | https://arxiv.org/abs/2311.08289 |