IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1805.12587.html
   My bibliography  Save this paper

Fast Hybrid Schemes for Fractional Riccati Equations (Rough is not so Tough)

Author

Listed:
  • Callegaro Giorgia
  • Grasselli Martino
  • Pag`es Gilles

Abstract

We solve a family of fractional Riccati differential equations with constant (possibly complex) coefficients. These equations arise, e.g., in fractional Heston stochastic volatility models, that have received great attention in the recent financial literature thanks to their ability to reproduce a rough volatility behavior. We first consider the case of a zero initial value corresponding to the characteristic function of the log-price. Then we investigate the case of a general starting value associated to a transform also involving the volatility process. The solution to the fractional Riccati equation takes the form of power series, whose convergence domain is typically finite. This naturally suggests a hybrid numerical algorithm to explicitly obtain the solution also beyond the convergence domain of the power series representation. Our numerical tests show that the hybrid algorithm turns out to be extremely fast and stable. When applied to option pricing, our method largely outperforms the only available alternative in the literature, based on the Adams method.

Suggested Citation

  • Callegaro Giorgia & Grasselli Martino & Pag`es Gilles, 2018. "Fast Hybrid Schemes for Fractional Riccati Equations (Rough is not so Tough)," Papers 1805.12587, arXiv.org, revised Feb 2020.
    • Handle: RePEc:arx:papers:1805.12587
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1805.12587
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Eduardo Abi Jaber & Nathan de Carvalho, 2024. "Reconciling rough volatility with jumps," Post-Print hal-04295416, HAL.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jiangze Du & Shaojie Lai & Kin Keung Lai & Shifei Zhou, 2021. "A novel term structure stochastic model with adaptive correlation for trend analysis," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 26(4), pages 5485-5498, October.
    2. Omar Euch & Masaaki Fukasawa & Mathieu Rosenbaum, 2018. "The microstructural foundations of leverage effect and rough volatility," Finance and Stochastics, Springer, vol. 22(2), pages 241-280, April.
    3. Raul Merino & Jan Posp'iv{s}il & Tom'av{s} Sobotka & Tommi Sottinen & Josep Vives, 2019. "Decomposition formula for rough Volterra stochastic volatility models," Papers 1906.07101, arXiv.org, revised Aug 2019.
    4. Christensen, Kim & Thyrsgaard, Martin & Veliyev, Bezirgen, 2019. "The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing," Journal of Econometrics, Elsevier, vol. 212(2), pages 556-583.
    5. Giorgia Callegaro & Martino Grasselli & Gilles Paèes, 2021. "Fast Hybrid Schemes for Fractional Riccati Equations (Rough Is Not So Tough)," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 221-254, February.
    6. M.E. Mancino & S. Scotti & G. Toscano, 2020. "Is the Variance Swap Rate Affine in the Spot Variance? Evidence from S&P500 Data," Applied Mathematical Finance, Taylor & Francis Journals, vol. 27(4), pages 288-316, July.
    7. Jingtang Ma & Wensheng Yang & Zhenyu Cui, 2021. "Semimartingale and continuous-time Markov chain approximation for rough stochastic local volatility models," Papers 2110.08320, arXiv.org, revised Oct 2021.
    8. Elisa Alòs & Jorge A. León, 2021. "An Intuitive Introduction to Fractional and Rough Volatilities," Mathematics, MDPI, vol. 9(9), pages 1-22, April.
    9. Bolko, Anine E. & Christensen, Kim & Pakkanen, Mikko S. & Veliyev, Bezirgen, 2023. "A GMM approach to estimate the roughness of stochastic volatility," Journal of Econometrics, Elsevier, vol. 235(2), pages 745-778.
    10. 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.
    11. Giulia Di Nunno & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2022. "Option pricing in Sandwiched Volterra Volatility model," Papers 2209.10688, arXiv.org, revised Jul 2024.
    12. Ariel Neufeld, 2018. "Buy-And-Hold Property For Fully Incomplete Markets When Super-Replicating Markovian Claims," Journal of Enterprising Culture (JEC), World Scientific Publishing Co. Pte. Ltd., vol. 21(08), pages 1-12, December.
    13. Jan Matas & Jan Posp'iv{s}il, 2021. "On simulation of rough Volterra stochastic volatility models," Papers 2108.01999, arXiv.org, revised Aug 2022.
    14. Marc Lagunas-Merino & Salvador Ortiz-Latorre, 2020. "A decomposition formula for fractional Heston jump diffusion models," Papers 2007.14328, arXiv.org.
    15. Siow Woon Jeng & Adem Kilicman, 2020. "Series Expansion and Fourth-Order Global Padé Approximation for a Rough Heston Solution," Mathematics, MDPI, vol. 8(11), pages 1-26, November.
    16. Josselin Garnier & Knut Sølna, 2018. "Option pricing under fast-varying and rough stochastic volatility," Annals of Finance, Springer, vol. 14(4), pages 489-516, November.
    17. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Working Papers hal-02946146, HAL.
    18. Peter Carr & Liuren Wu, 2014. "Static Hedging of Standard Options," Journal of Financial Econometrics, Oxford University Press, vol. 12(1), pages 3-46.
    19. Söderlind, Paul, 2009. "The C-CAPM without ex post data," Journal of Macroeconomics, Elsevier, vol. 31(4), pages 721-729, December.
    20. Damir Filipovi'c & Martin Larsson, 2017. "Polynomial Jump-Diffusion Models," Papers 1711.08043, arXiv.org, revised Jul 2019.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1805.12587. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.