IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2203.02323.html
   My bibliography  Save this paper

Solution of integrals with fractional Brownian motion for different Hurst indices

Author

Listed:
  • Fei Gao
  • Shuaiqiang Liu
  • Cornelis W. Oosterlee
  • Nico M. Temme

Abstract

In this paper, we will evaluate integrals that define the conditional expectation, variance and characteristic function of stochastic processes with respect to fractional Brownian motion (fBm) for all relevant Hurst indices, i.e. $H \in (0,1)$. The fractional Ornstein-Uhlenbeck (fOU) process, for example, gives rise to highly nontrivial integration formulas that need careful analysis when considering the whole range of Hurst indices. We will show that the classical technique of analytic continuation, from complex analysis, provides a way of extending the domain of validity of an integral, from $H\in(1/2,1)$, to the larger domain, $H\in(0,1)$. Numerical experiments for different Hurst indices confirm the robustness and efficiency of the integral formulations presented here. Moreover, we provide accurate and highly efficient financial option pricing results for processes that are related to the fOU process, with the help of Fourier cosine expansions.

Suggested Citation

  • Fei Gao & Shuaiqiang Liu & Cornelis W. Oosterlee & Nico M. Temme, 2022. "Solution of integrals with fractional Brownian motion for different Hurst indices," Papers 2203.02323, arXiv.org, revised Mar 2022.
    • Handle: RePEc:arx:papers:2203.02323
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2203.02323
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Fred Espen Benth & Jūratė Šaltytė Benth, 2012. "Modeling and Pricing in Financial Markets for Weather Derivatives," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8457, August.
    2. Dorje Brody & Joanna Syroka & Mihail Zervos, 2002. "Dynamical pricing of weather derivatives," Quantitative Finance, Taylor & Francis Journals, vol. 2(3), pages 189-198.
    3. Power, Gabriel J. & Turvey, Calum G., 2010. "Long-range dependence in the volatility of commodity futures prices: Wavelet-based evidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(1), pages 79-90.
    4. Boonstra, Boris C. & Oosterlee, Cornelis W., 2021. "Valuation of electricity storage contracts using the COS method," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    5. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    6. Fred Espen Benth, 2003. "On arbitrage-free pricing of weather derivatives based on fractional Brownian motion," Applied Mathematical Finance, Taylor & Francis Journals, vol. 10(4), pages 303-324.
    7. Akinlar, M.A. & Inc, Mustafa & Gómez-Aguilar, J.F. & Boutarfa, B., 2020. "Solutions of a disease model with fractional white noise," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    8. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    9. 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.
    10. Fred Espen Benth & Jūratė Šaltytė Benth, 2012. "Financial markets for weather," World Scientific Book Chapters, in: Modeling and Pricing in Financial Markets for Weather Derivatives, chapter 1, pages 1-13, World Scientific Publishing Co. Pte. Ltd..
    11. Giulia Livieri & Saad Mouti & Andrea Pallavicini & Mathieu Rosenbaum, 2018. "Rough volatility: Evidence from option prices," IISE Transactions, Taylor & Francis Journals, vol. 50(9), pages 767-776, September.
    12. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
    13. Xiao, Wei-Lin & Zhang, Wei-Guo & Zhang, Xi-Li & Wang, Ying-Luo, 2010. "Pricing currency options in a fractional Brownian motion with jumps," Economic Modelling, Elsevier, vol. 27(5), pages 935-942, September.
    Full references (including those not matched with items on IDEAS)

    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. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Papers 2210.12393, arXiv.org.
    2. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Working Papers hal-03827332, HAL.
    3. Fabio Baschetti & Giacomo Bormetti & Silvia Romagnoli & Pietro Rossi, 2020. "The SINC way: A fast and accurate approach to Fourier pricing," Papers 2009.00557, arXiv.org, revised May 2021.
    4. Fred Espen Benth & Asma Khedher & Michèle Vanmaele, 2020. "Pricing of Commodity Derivatives on Processes with Memory," Risks, MDPI, vol. 8(1), pages 1-32, January.
    5. 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.
    6. Calypso Herrera & Florian Krach & Pierre Ruyssen & Josef Teichmann, 2021. "Optimal Stopping via Randomized Neural Networks," Papers 2104.13669, arXiv.org, revised Dec 2023.
    7. Mehrdoust, Farshid & Noorani, Idin & Hamdi, Abdelouahed, 2023. "Two-factor Heston model equipped with regime-switching: American option pricing and model calibration by Levenberg–Marquardt optimization algorithm," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 660-678.
    8. Marc Mukendi Mpanda & Safari Mukeru & Mmboniseni Mulaudzi, 2020. "Generalisation of Fractional-Cox-Ingersoll-Ross Process," Papers 2008.07798, arXiv.org, revised Jul 2022.
    9. Ying Jiao & Chunhua Ma & Simone Scotti & Chao Zhou, 2021. "The Alpha‐Heston stochastic volatility model," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 943-978, July.
    10. Zhang, Sumei & Gao, Xiong, 2019. "An asymptotic expansion method for geometric Asian options pricing under the double Heston model," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 1-9.
    11. Hideharu Funahashi & Masaaki Kijima, 2017. "Does the Hurst index matter for option prices under fractional volatility?," Annals of Finance, Springer, vol. 13(1), pages 55-74, February.
    12. 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.
    13. 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.
    14. Kononovicius, Aleksejus & Kazakevičius, Rytis & Kaulakys, Bronislovas, 2022. "Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processes," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    15. Brandi, Giuseppe & Di Matteo, T., 2022. "Multiscaling and rough volatility: An empirical investigation," International Review of Financial Analysis, Elsevier, vol. 84(C).
    16. 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.
    17. Aleksejus Kononovicius & Rytis Kazakeviv{c}ius & Bronislovas Kaulakys, 2022. "Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processes," Papers 2205.07563, arXiv.org, revised Jul 2022.
    18. Matthieu Garcin, 2019. "Fractal analysis of the multifractality of foreign exchange rates [Analyse fractale de la multifractalité des taux de change]," Working Papers hal-02283915, HAL.
    19. Cui, Zhenyu & Lars Kirkby, J. & Nguyen, Duy, 2019. "A general framework for time-changed Markov processes and applications," European Journal of Operational Research, Elsevier, vol. 273(2), pages 785-800.
    20. Giuseppe Brandi & T. Di Matteo, 2022. "Multiscaling and rough volatility: an empirical investigation," Papers 2201.10466, arXiv.org.

    More about this item

    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:2203.02323. 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.