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Decomposition Formula For Rough Volterra Stochastic Volatility Models

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  • RAÚL MERINO

    (Universitat de Barcelona, Facultat de Matemàtiques, Gran Via 585, 08007 Barcelona, Spain¶VidaCaixa S.A., Investment Risk Management Department, C/Juan Gris, 2-8, Barcelona 08014, Spain)

  • JAN POSPÍŠIL

    (#x2020;Department of Mathematics, University of West Bohemia, Univerzitní 2732/8, 301 00 Plzeň, Czech Republic)

  • TOMÁŠ SOBOTKA

    (#x2020;Department of Mathematics, University of West Bohemia, Univerzitní 2732/8, 301 00 Plzeň, Czech Republic§Ernst & Young, s.r.o., Na Florenci 2116/15, 110 00 Praha, Czech Republic)

  • TOMMI SOTTINEN

    (#x2021;Department of Mathematics and Statistics, University of Vaasa, P. O. Box 700, FIN-65101 Vaasa, Finland)

  • JOSEP VIVES

    (Universitat de Barcelona, Facultat de Matemàtiques, Gran Via 585, 08007 Barcelona, Spain)

Abstract

The research presented in this paper provides an alternative option pricing approach for a class of rough fractional stochastic volatility models. These models are increasingly popular between academics and practitioners due to their surprising consistency with financial markets. However, they bring several challenges alongside. Most noticeably, even simple nonlinear financial derivatives as vanilla European options are typically priced by means of Monte–Carlo (MC) simulations which are more computationally demanding than similar MC schemes for standard stochastic volatility models. In this paper, we provide a proof of the prediction law for general Gaussian Volterra processes. The prediction law is then utilized to obtain an adapted projection of the future squared volatility — a cornerstone of the proposed pricing approximation. Firstly, a decomposition formula for European option prices under general Volterra volatility models is introduced. Then we focus on particular models with rough fractional volatility and we derive an explicit semi-closed approximation formula. Numerical properties of the approximation for a popular model — the rBergomi model — are studied and we propose a hybrid calibration scheme which combines the approximation formula alongside MC simulations. This scheme can significantly speed up the calibration to financial markets as illustrated on a set of AAPL options.

Suggested Citation

  • Raúl Merino & Jan Pospíšil & Tomáš Sobotka & Tommi Sottinen & Josep Vives, 2021. "Decomposition Formula For Rough Volterra Stochastic Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 24(02), pages 1-47, March.
    • Handle: RePEc:wsi:ijtafx:v:24:y:2021:i:02:n:s0219024921500084
    DOI: 10.1142/S0219024921500084
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    Cited by:

    1. Jan Matas & Jan Posp'iv{s}il, 2021. "On simulation of rough Volterra stochastic volatility models," Papers 2108.01999, arXiv.org, revised Aug 2022.
    2. Alòs, Elisa & Antonelli, Fabio & Ramponi, Alessandro & Scarlatti, Sergio, 2023. "CVA in fractional and rough volatility models," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    3. Ömer ÖNALAN, 2022. "Joint Modelling of S&P500 and VIX Indices with Rough Fractional Ornstein-Uhlenbeck Volatility Model," Journal for Economic Forecasting, Institute for Economic Forecasting, vol. 0(1), pages 68-84, April.
    4. Jan Matas & Jan Pospíšil, 2023. "Robustness and sensitivity analyses of rough Volterra stochastic volatility models," Annals of Finance, Springer, vol. 19(4), pages 523-543, December.
    5. Jan Matas & Jan Posp'iv{s}il, 2021. "Robustness and sensitivity analyses for rough Volterra stochastic volatility models," Papers 2107.12462, arXiv.org, revised Jun 2023.

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