VIX options in the SABR model
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908504876384256 |
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| author | Pirjol, Dan Zhu, Lingjiong |
| author_facet | Pirjol, Dan Zhu, Lingjiong |
| contents | We study the pricing of VIX options in the SABR model $dS_t = σ_t S_t^βdB_t, dσ_t = ωσ_t dZ_t$ where $B_t,Z_t$ are standard Brownian motions correlated with correlation $ρ<0$ and $0 \leq β< 1$. VIX is expressed as a risk-neutral conditional expectation of an integral over the volatility process $v_t = S_t^{β-1} σ_t$. We show that $v_t$ is the unique solution to a one-dimensional diffusion process. Using the Feller test, we show that $v_t$ explodes in finite time with non-zero probability. As a consequence, VIX futures and VIX call prices are infinite, and VIX put prices are zero for any maturity. As a remedy, we propose a capped volatility process by capping the drift and diffusion terms in the $v_{t}$ process such that it becomes non-explosive and well-behaved, and study the short-maturity asymptotics for the pricing of VIX options. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06398 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | VIX options in the SABR model Pirjol, Dan Zhu, Lingjiong Pricing of Securities We study the pricing of VIX options in the SABR model $dS_t = σ_t S_t^βdB_t, dσ_t = ωσ_t dZ_t$ where $B_t,Z_t$ are standard Brownian motions correlated with correlation $ρ<0$ and $0 \leq β< 1$. VIX is expressed as a risk-neutral conditional expectation of an integral over the volatility process $v_t = S_t^{β-1} σ_t$. We show that $v_t$ is the unique solution to a one-dimensional diffusion process. Using the Feller test, we show that $v_t$ explodes in finite time with non-zero probability. As a consequence, VIX futures and VIX call prices are infinite, and VIX put prices are zero for any maturity. As a remedy, we propose a capped volatility process by capping the drift and diffusion terms in the $v_{t}$ process such that it becomes non-explosive and well-behaved, and study the short-maturity asymptotics for the pricing of VIX options. |
| title | VIX options in the SABR model |
| topic | Pricing of Securities |
| url | https://arxiv.org/abs/2501.06398 |