IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1909.02972.html
   My bibliography  Save this paper

Portfolio optimisation under rough Heston models

Author

Listed:
  • Benjamin James Duthie

Abstract

This thesis investigates Merton's portfolio problem under two different rough Heston models, which have a non-Markovian structure. The motivation behind this choice of problem is due to the recent discovery and success of rough volatility processes. The optimisation problem is solved from two different approaches: firstly by considering an auxiliary random process, which solves the optimisation problem with the martingale optimality principle, and secondly, by a finite dimensional approximation of the volatility process which casts the problem into its classical stochastic control framework. In addition, we show how classical results from Merton's portfolio optimisation problem can be used to help motivate the construction of the solution in both cases. The optimal strategy under both approaches is then derived in a semi-closed form, and comparisons between the results made. The approaches discussed in this thesis, combined with the historical works on the distortion transformation, provide a strong foundation to build models capable of handling increasing complexity demanded by the ever growing financial market.

Suggested Citation

  • Benjamin James Duthie, 2019. "Portfolio optimisation under rough Heston models," Papers 1909.02972, arXiv.org.
    • Handle: RePEc:arx:papers:1909.02972
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1909.02972
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jim Gatheral & Martin Keller-Ressel, 2018. "Affine forward variance models," Papers 1801.06416, arXiv.org, revised Oct 2018.
    2. Peter Carr & Liuren Wu, 2003. "What Type of Process Underlies Options? A Simple Robust Test," Journal of Finance, American Finance Association, vol. 58(6), pages 2581-2610, December.
    3. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    4. Thaleia Zariphopoulou, 2001. "A solution approach to valuation with unhedgeable risks," Finance and Stochastics, Springer, vol. 5(1), pages 61-82.
    5. Bingyan Han & Hoi Ying Wong, 2019. "Merton's portfolio problem under Volterra Heston model," Papers 1905.05371, arXiv.org, revised Nov 2019.
    6. Martin Keller-Ressel & Martin Larsson & Sergio Pulido, 2018. "Affine Rough Models," Papers 1812.08486, arXiv.org.
    7. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    8. ,, 2004. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 20(2), pages 427-429, April.
    9. 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.
    10. Harms, Philipp & Stefanovits, David, 2019. "Affine representations of fractional processes with applications in mathematical finance," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1185-1228.
    11. ,, 2004. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 20(1), pages 223-229, February.
    12. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    13. Masaaki Fukasawa & Tetsuya Takabatake & Rebecca Westphal, 2019. "Is Volatility Rough ?," Papers 1905.04852, arXiv.org, revised May 2019.
    14. Nicole Bauerle & Sascha Desmettre, 2018. "Portfolio Optimization in Fractional and Rough Heston Models," Papers 1809.10716, arXiv.org, revised May 2019.
    15. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    16. F. Comte & L. Coutin & E. Renault, 2012. "Affine fractional stochastic volatility models," Annals of Finance, Springer, vol. 8(2), pages 337-378, May.
    17. Jean‐Pierre Fouque & Ruimeng Hu, 2019. "Optimal portfolio under fractional stochastic environment," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 697-734, July.
    18. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    19. Bingyan Han & Hoi Ying Wong, 2019. "Mean-variance portfolio selection under Volterra Heston model," Papers 1904.12442, arXiv.org, revised Jan 2020.
    20. Philippe Carmona & Laure Coutin & G. Montseny, 2000. "Approximation of Some Gaussian Processes," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 161-171, January.
    21. Josselin Garnier & Knut Solna, 2017. "Option Pricing under Fast-varying and Rough Stochastic Volatility," Papers 1707.00610, arXiv.org, revised Apr 2018.
    22. Tehranchi, Michael, 2004. "Explicit solutions of some utility maximization problems in incomplete markets," Stochastic Processes and their Applications, Elsevier, vol. 114(1), pages 109-125, November.
    23. Josselin Garnier & Knut Solna, 2016. "Option pricing under fast-varying long-memory stochastic volatility," Papers 1604.00105, arXiv.org, revised Apr 2018.
    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. Etienne Chevalier & Sergio Pulido & Elizabeth Zúñiga, 2021. "American options in the Volterra Heston model," Working Papers hal-03178306, HAL.
    2. Etienne Chevalier & Sergio Pulido & Elizabeth Zúñiga, 2022. "American options in the Volterra Heston model," Post-Print hal-03178306, HAL.
    3. Etienne Chevalier & Sergio Pulido & Elizabeth Z'u~niga, 2021. "American options in the Volterra Heston model," Papers 2103.11734, arXiv.org, revised May 2022.
    4. Bingyan Han & Hoi Ying Wong, 2019. "Time-inconsistency with rough volatility," Papers 1907.11378, arXiv.org, revised Dec 2021.
    5. Giulia Di Nunno & Anton Yurchenko-Tytarenko, 2022. "Sandwiched Volterra Volatility model: Markovian approximations and hedging," Papers 2209.13054, arXiv.org.
    6. Christian Bayer & Peter K. Friz & Paul Gassiat & Jorg Martin & Benjamin Stemper, 2020. "A regularity structure for rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 782-832, July.
    7. 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.
    8. Bingyan Han & Hoi Ying Wong, 2019. "Mean-variance portfolio selection under Volterra Heston model," Papers 1904.12442, arXiv.org, revised Jan 2020.
    9. 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.
    10. 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.
    11. Giulia Di Nunno & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2022. "Option pricing in Sandwiched Volterra Volatility model," Papers 2209.10688, arXiv.org, revised Dec 2023.
    12. Yicun Li & Yuanyang Teng, 2022. "Estimation of the Hurst Parameter in Spot Volatility," Mathematics, MDPI, vol. 10(10), pages 1-26, May.
    13. Anine E. Bolko & Kim Christensen & Mikko S. Pakkanen & Bezirgen Veliyev, 2020. "Roughness in spot variance? A GMM approach for estimation of fractional log-normal stochastic volatility models using realized measures," CREATES Research Papers 2020-12, Department of Economics and Business Economics, Aarhus University.
    14. Han, Bingyan & Wong, Hoi Ying, 2021. "Merton’s portfolio problem under Volterra Heston model," Finance Research Letters, Elsevier, vol. 39(C).
    15. Jacquier, Antoine & Pannier, Alexandre, 2022. "Large and moderate deviations for stochastic Volterra systems," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 142-187.
    16. Ackermann, Julia & Kruse, Thomas & Overbeck, Ludger, 2022. "Inhomogeneous affine Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 250-279.
    17. Bingyan Han & Hoi Ying Wong, 2019. "Merton's portfolio problem under Volterra Heston model," Papers 1905.05371, arXiv.org, revised Nov 2019.
    18. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Working Papers hal-02946146, HAL.
    19. 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.
    20. 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.

    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:1909.02972. 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.