IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2304.03042.html
   My bibliography  Save this paper

Rough volatility, path-dependent PDEs and weak rates of convergence

Author

Listed:
  • Ofelia Bonesini
  • Antoine Jacquier
  • Alexandre Pannier

Abstract

In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional It\^o formula developed by [Viens, F., & Zhang, J. (2019). A martingale approach for fractional Brownian motions and related path dependent PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of convergence for discretised stochastic integrals of smooth functions of a Riemann-Liouville fractional Brownian motion with Hurst parameter $H \in (0,1/2)$. These integrals approximate log-stock prices in rough volatility models. We obtain weak error rates of order 1 if the test function is quadratic and of order $H+1/2$ for smooth test functions.

Suggested Citation

  • Ofelia Bonesini & Antoine Jacquier & Alexandre Pannier, 2023. "Rough volatility, path-dependent PDEs and weak rates of convergence," Papers 2304.03042, arXiv.org.
    • Handle: RePEc:arx:papers:2304.03042
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2304.03042
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Abi Jaber, Eduardo & El Euch, Omar, 2019. "Markovian structure of the Volterra Heston model," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 63-72.
    2. Richard, Alexandre & Tan, Xiaolu & Yang, Fan, 2021. "Discrete-time simulation of Stochastic Volterra equations," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 109-138.
    3. Wu, Peng & Muzy, Jean-François & Bacry, Emmanuel, 2022. "From rough to multifractal volatility: The log S-fBM model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    4. Masaaki Fukasawa & Blanka Horvath & Peter Tankov, 2021. "Hedging under rough volatility," Papers 2105.04073, arXiv.org.
    5. Ofelia Bonesini & Giorgia Callegaro & Antoine Jacquier, 2021. "Functional quantization of rough volatility and applications to volatility derivatives," Papers 2104.04233, arXiv.org, revised Mar 2024.
    6. Antoine Jacquier & Aitor Muguruza & Alexandre Pannier, 2021. "Rough multifactor volatility for SPX and VIX options," Papers 2112.14310, arXiv.org, revised Nov 2023.
    7. Christian Bayer & Masaaki Fukasawa & Shonosuke Nakahara, 2022. "On the weak convergence rate in the discretization of rough volatility models," Papers 2203.02943, arXiv.org.
    8. Paul Gassiat, 2022. "Weak error rates of numerical schemes for rough volatility," Papers 2203.09298, arXiv.org, revised Feb 2023.
    9. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2015. "Hybrid scheme for Brownian semistationary processes," Papers 1507.03004, arXiv.org, revised May 2017.
    10. Peter K. Friz & William Salkeld & Thomas Wagenhofer, 2022. "Weak error estimates for rough volatility models," Papers 2212.01591, arXiv.org.
    11. Anine E. Bolko & Kim Christensen & Mikko S. Pakkanen & Bezirgen Veliyev, 2020. "A GMM approach to estimate the roughness of stochastic volatility," Papers 2010.04610, arXiv.org, revised Apr 2022.
    12. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2017. "Hybrid scheme for Brownian semistationary processes," Finance and Stochastics, Springer, vol. 21(4), pages 931-965, October.
    13. Blanka Horvath & Josef Teichmann & Zan Zuric, 2021. "Deep Hedging under Rough Volatility," Papers 2102.01962, arXiv.org.
    14. Antoine Jacquier & Mugad Oumgari, 2019. "Deep Curve-dependent PDEs for affine rough volatility," Papers 1906.02551, arXiv.org, revised Jan 2023.
    15. Viktor Bezborodov & Luca Persio & Yuliya Mishura, 2019. "Option Pricing with Fractional Stochastic Volatility and Discontinuous Payoff Function of Polynomial Growth," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 331-366, March.
    16. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    17. Blanka Horvath & Josef Teichmann & Zan Zuric, 2021. "Deep Hedging under Rough Volatility," Swiss Finance Institute Research Paper Series 21-88, Swiss Finance Institute.
    18. Eduardo Abi Jaber & Omar El Euch, 2019. "Markovian structure of the Volterra Heston model," Post-Print hal-01716696, HAL.
    19. Peng Wu & Jean-Franc{c}ois Muzy & Emmanuel Bacry, 2022. "From Rough to Multifractal volatility: the log S-fBM model," Papers 2201.09516, arXiv.org, revised Jul 2022.
    20. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    21. Blanka Horvath & Josef Teichmann & Žan Žurič, 2021. "Deep Hedging under Rough Volatility," Risks, MDPI, vol. 9(7), pages 1-20, July.
    22. Bruno Dupire, 2019. "Functional Itô calculus," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 721-729, May.
    23. Dupret, Jean-Loup & Barbarin, Jérôme & Hainaut , Donatien, 2022. "Impact of rough stochastic volatility models on long-term life insurance pricing," LIDAM Reprints ISBA 2022022, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    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. Alexandre Pannier, 2023. "Path-dependent PDEs for volatility derivatives," Papers 2311.08289, arXiv.org, revised Jan 2024.
    2. Antoine Jacquier & Mugad Oumgari, 2023. "Interest rate convexity in a Gaussian framework," Papers 2307.14218, arXiv.org, revised Mar 2024.
    3. Peter Bank & Christian Bayer & Peter K. Friz & Luca Pelizzari, 2023. "Rough PDEs for local stochastic volatility models," Papers 2307.09216, arXiv.org.

    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. Alexandre Pannier, 2023. "Path-dependent PDEs for volatility derivatives," Papers 2311.08289, arXiv.org, revised Jan 2024.
    2. Etienne Chevalier & Sergio Pulido & Elizabeth Zúñiga, 2021. "American options in the Volterra Heston model," Working Papers hal-03178306, HAL.
    3. 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.
    4. Paul Gassiat, 2022. "Weak error rates of numerical schemes for rough volatility," Papers 2203.09298, arXiv.org, revised Feb 2023.
    5. Siow Woon Jeng & Adem Kiliçman, 2021. "On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model," Mathematics, MDPI, vol. 9(22), pages 1-32, November.
    6. Antoine Jacquier & Zan Zuric, 2023. "Random neural networks for rough volatility," Papers 2305.01035, arXiv.org.
    7. Mathieu Rosenbaum & Jianfei Zhang, 2021. "Deep calibration of the quadratic rough Heston model," Papers 2107.01611, arXiv.org, revised May 2022.
    8. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Working Papers hal-02946146, HAL.
    9. 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.
    10. Alexandre Carbonneau & Fr'ed'eric Godin, 2021. "Deep equal risk pricing of financial derivatives with non-translation invariant risk measures," Papers 2107.11340, arXiv.org.
    11. Mathieu Rosenbaum & Jianfei Zhang, 2022. "On the universality of the volatility formation process: when machine learning and rough volatility agree," Papers 2206.14114, arXiv.org.
    12. Alexandre Pannier & Cristopher Salvi, 2024. "A path-dependent PDE solver based on signature kernels," Papers 2403.11738, arXiv.org.
    13. Christa Cuchiero & Sara Svaluto-Ferro, 2021. "Infinite-dimensional polynomial processes," Finance and Stochastics, Springer, vol. 25(2), pages 383-426, April.
    14. Peter K. Friz & Paul Gassiat & Paolo Pigato, 2022. "Short-dated smile under rough volatility: asymptotics and numerics," Quantitative Finance, Taylor & Francis Journals, vol. 22(3), pages 463-480, March.
    15. Florian Bourgey & Stefano De Marco & Peter K. Friz & Paolo Pigato, 2023. "Local volatility under rough volatility," Mathematical Finance, Wiley Blackwell, vol. 33(4), pages 1119-1145, October.
    16. Christa Cuchiero & Sara Svaluto-Ferro, 2019. "Infinite dimensional polynomial processes," Papers 1911.02614, arXiv.org.
    17. Blanka Horvath & Josef Teichmann & Zan Zuric, 2021. "Deep Hedging under Rough Volatility," Papers 2102.01962, arXiv.org.
    18. 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.
    19. Qinwen Zhu & Gr'egoire Loeper & Wen Chen & Nicolas Langren'e, 2020. "Markovian approximation of the rough Bergomi model for Monte Carlo option pricing," Papers 2007.02113, arXiv.org.
    20. Qinwen Zhu & Gregoire Loeper & Wen Chen & Nicolas Langrené, 2021. "Markovian approximation of the rough Bergomi model for Monte Carlo option pricing," Post-Print hal-02910724, HAL.

    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:2304.03042. 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.