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The characteristic function of Gaussian stochastic volatility models: an analytic expression

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  • Eduardo Abi Jaber

    (Université Paris 1 Panthéon-Sorbonne)

Abstract

Stochastic volatility models based on Gaussian processes, like fractional Brownian motion, are able to reproduce important stylised facts of financial markets such as rich autocorrelation structures, persistence and roughness of sample paths. This is made possible by virtue of the flexibility introduced in the choice of the covariance function of the Gaussian process. The price to pay is that in general, such models are no longer Markovian nor semimartingales, which limits their practical use. We derive, in two different ways, an explicit analytic expression for the joint characteristic function of the log-price and its integrated variance in general Gaussian stochastic volatility models. That analytic expression can be approximated by closed-form matrix expressions. This opens the door to fast approximation of the joint density and pricing of derivatives on both the stock and its realised variance by using Fourier inversion techniques. In the context of rough volatility modelling, our results apply to the (rough) fractional Stein–Stein model and provide the first analytic formulas for option pricing known to date, generalising that of Stein–Stein, Schöbel–Zhu and a special case of Heston.

Suggested Citation

  • Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Finance and Stochastics, Springer, vol. 26(4), pages 733-769, October.
  • Handle: RePEc:spr:finsto:v:26:y:2022:i:4:d:10.1007_s00780-022-00489-4
    DOI: 10.1007/s00780-022-00489-4
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    Citations

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    Cited by:

    1. Luca Galimberti & Anastasis Kratsios & Giulia Livieri, 2022. "Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis," Papers 2210.13300, arXiv.org, revised May 2023.
    2. Eduardo Abi Jaber & Nathan De Carvalho, 2023. "Reconciling rough volatility with jumps," Papers 2303.07222, arXiv.org.
    3. Huy N. Chau & Duy Nguyen & Thai Nguyen, 2024. "On short-time behavior of implied volatility in a market model with indexes," Papers 2402.16509, arXiv.org, revised Apr 2024.
    4. 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.
    5. Eduardo Abi Jaber & Louis-Amand G'erard, 2024. "Signature volatility models: pricing and hedging with Fourier," Papers 2402.01820, arXiv.org.
    6. Eduardo Abi Jaber & Eyal Neuman & Moritz Voss, 2023. "Equilibrium in Functional Stochastic Games with Mean-Field Interaction," Working Papers hal-04119787, HAL.
    7. Eduardo Abi Jaber & Eyal Neuman, 2022. "Optimal Liquidation with Signals: the General Propagator Case," Working Papers hal-03835948, HAL.
    8. Eduardo Abi Jaber & Eyal Neuman, 2022. "Optimal Liquidation with Signals: the General Propagator Case," Papers 2211.00447, arXiv.org.
    9. Eduardo Abi Jaber & Eyal Neuman & Moritz Vo{ss}, 2023. "Equilibrium in Functional Stochastic Games with Mean-Field Interaction," Papers 2306.05433, arXiv.org, revised Feb 2024.
    10. Peter K. Friz & William Salkeld & Thomas Wagenhofer, 2022. "Weak error estimates for rough volatility models," Papers 2212.01591, arXiv.org.
    11. 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.
    12. Carsten Chong & Marc Hoffmann & Yanghui Liu & Mathieu Rosenbaum & Gr'egoire Szymanski, 2022. "Statistical inference for rough volatility: Minimax Theory," Papers 2210.01214, arXiv.org, revised Feb 2024.

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    More about this item

    Keywords

    Stochastic volatility models; Non-Markovian models; Fast pricing;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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