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The quintic Ornstein-Uhlenbeck volatility model that jointly calibrates SPX & VIX smiles

Author

Listed:
  • Eduardo Abi Jaber

    (X - École polytechnique)

  • Camille Illand

    (AXA Investment Managers, Multi Asset Client Solutions, Quantitative Research - AXA)

  • Shaun Xiaoyuan Li

    (UP1 - Université Paris 1 Panthéon-Sorbonne, AXA Investment Managers, Multi Asset Client Solutions, Quantitative Research - AXA)

Abstract

The quintic Ornstein-Uhlenbeck volatility model is a stochastic volatility model where the volatility process is a polynomial function of degree five of a single Ornstein-Uhlenbeck process with fast mean reversion and large vol-of-vol. The model is able to achieve remarkable joint fits of the SPX-VIX smiles with only 6 effective parameters and an input curve that allows to match certain term structures. Even better, the model remains very simple and tractable for pricing and calibration: the VIX squared is again polynomial in the Ornstein-Uhlenbeck process, leading to efficient VIX derivative pricing by a simple integration against a Gaussian density; simulation of the volatility process is exact; and pricing SPX products can be done efficiently and accurately by standard Monte Carlo techniques with suitable antithetic and control variates.

Suggested Citation

  • Eduardo Abi Jaber & Camille Illand & Shaun Xiaoyuan Li, 2023. "The quintic Ornstein-Uhlenbeck volatility model that jointly calibrates SPX & VIX smiles," Working Papers hal-03909334, HAL.
    • Handle: RePEc:hal:wpaper:hal-03909334
    Note: View the original document on HAL open archive server: https://hal.science/hal-03909334v2
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    References listed on IDEAS

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    1. Stéphane Goutte & Amine Ismail & Huyên Pham, 2017. "Regime-switching stochastic volatility model: estimation and calibration to VIX options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 24(1), pages 38-75, January.
    2. Thomas Kokholm & Martin Stisen, 2015. "Joint pricing of VIX and SPX options with stochastic volatility and jump models," Journal of Risk Finance, Emerald Group Publishing, vol. 16(1), pages 27-48, January.
    3. Eduardo Abi Jaber & Camille Illand & Shaun Xiaoyuan Li, 2022. "Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints," Working Papers hal-03902513, HAL.
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    9. Eduardo Abi Jaber & Camille Illand & Shaun & Li, 2022. "Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints," Papers 2212.08297, arXiv.org.
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    Cited by:

    1. Eduardo Abi Jaber & Nathan De Carvalho, 2023. "Reconciling rough volatility with jumps," Papers 2303.07222, arXiv.org.
    2. Eduardo Abi Jaber & Shaun & Li, 2024. "Volatility models in practice: Rough, Path-dependent or Markovian?," Papers 2401.03345, arXiv.org.

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    Keywords

    SPX and VIX modeling; Stochastic volatility; Pricing; Calibration;
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