IDEAS home Printed from https://ideas.repec.org/p/aiz/louvad/2022001.html
   My bibliography  Save this paper

A subdiffusive stochastic volatility jump model

Author

Listed:
  • Dupret, Jean-Loup

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Hainaut, Donatien

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

Abstract

Subdiffusions appear as good candidates for modeling illiquidity in financial markets. Existing subdiffusive models of asset prices are indeed able to capture the motionless periods in the quotes of thinly-traded assets. However, they fail at reproducing the jumps and the time-varying volatility observed in the price of these assets. The aim of this work is hence to propose a new model of subdiffusive asset prices reproducing the main characteristics exhibited in illiquid markets. This is done by considering a stochastic volatility jump model, time-changed by an inverse subordinator. We derive the forward fractional partial differential equations (PDE) governing the probability density function of the introduced model and we prove that it leads to an arbitrage-free and incomplete market. By proposing a new procedure for estimating the model parameters and using a series expansion for solving numerically the obtained fractional PDE, we are able to price various financial derivatives on illiquid assets and to propose a corresponding hedging strategy. This way, we show that the introduced subdiffusive stochastic volatility jump model yields consistent and more reliable results in illiquid markets.

Suggested Citation

  • Dupret, Jean-Loup & Hainaut, Donatien, 2022. "A subdiffusive stochastic volatility jump model," LIDAM Discussion Papers ISBA 2022001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2022001
    as

    Download full text from publisher

    File URL: https://dial.uclouvain.be/pr/boreal/en/object/boreal%3A257566/datastream/PDF_01/view
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Dupret, Jean-Loup & Hainaut, Donatien, 2021. "Portfolio insurance under rough volatility and Volterra processes," LIDAM Discussion Papers ISBA 2021026, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Hainaut, Donatien, 2021. "Moment generating function of non-Markov self-excited claims processes," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 406-424.
    3. Michael S. Johannes & Nicholas G. Polson & Jonathan R. Stroud, 2009. "Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices," Review of Financial Studies, Society for Financial Studies, vol. 22(7), pages 2559-2599, July.
    4. Claudio Albanese & Ken Jackson & Petter Wiberg, 2004. "A new Fourier transform algorithm for value-at-risk," Quantitative Finance, Taylor & Francis Journals, vol. 4(3), pages 328-338.
    5. Jean-Philippe Bouchaud & Yuval Gefen & Marc Potters & Matthieu Wyart, 2003. "Fluctuations and response in financial markets: the subtle nature of `random' price changes," Papers cond-mat/0307332, arXiv.org, revised Aug 2003.
    6. Christophe Andrieu & Arnaud Doucet & Roman Holenstein, 2010. "Particle Markov chain Monte Carlo methods," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(3), pages 269-342, June.
    7. Adrian Dragulescu & Victor Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 443-453.
    8. Gu, Hui & Liang, Jin-Rong & Zhang, Yun-Xiu, 2012. "Time-changed geometric fractional Brownian motion and option pricing with transaction costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(15), pages 3971-3977.
    9. Guo, Zhidong & Yuan, Hongjun, 2014. "Pricing European option under the time-changed mixed Brownian-fractional Brownian model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 406(C), pages 73-79.
    10. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
    11. Hainaut, Donatien, 2020. "Fractional Hawkes processes," LIDAM Reprints ISBA 2020009, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    12. Hainaut, Donatien, 2021. "A fractional multi-states model for insurance," LIDAM Reprints ISBA 2021014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    13. Dupret, Jean-Loup & Hainaut, Donatien, 2021. "Portfolio insurance under rough volatility and Volterra processes," LIDAM Reprints ISBA 2021051, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    14. Bollerslev, Tim & Gibson, Michael & Zhou, Hao, 2011. "Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities," Journal of Econometrics, Elsevier, vol. 160(1), pages 235-245, January.
    15. Enrico Scalas, 2006. "Five Years of Continuous-time Random Walks in Econophysics," Lecture Notes in Economics and Mathematical Systems, in: Akira Namatame & Taisei Kaizouji & Yuuji Aruka (ed.), The Complex Networks of Economic Interactions, pages 3-16, Springer.
    16. Hainaut, Donatien, 2021. "A fractional multi-states model for insurance," LIDAM Discussion Papers ISBA 2021019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    17. Hainaut, Donatien, 2021. "Moment generating function of non-Markov self-excited claims processes," LIDAM Discussion Papers ISBA 2021028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    18. Hainaut, Donatien, 2021. "A fractional multi-states model for insurance," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 120-132.
    19. Pauline Barrieu & Nicole El Karoui, 2002. "Optimal design of derivatives in illiquid markets," Quantitative Finance, Taylor & Francis Journals, vol. 2(3), pages 181-188.
    20. Hainaut, Donatien, 2021. "Moment generating function of non-Markov self-excited claims processes," LIDAM Reprints ISBA 2021046, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    21. Hainaut, Donatien, 2020. "Fractional Hawkes processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 549(C).
    22. Wang, Jun & Liang, Jin-Rong & Lv, Long-Jin & Qiu, Wei-Yuan & Ren, Fu-Yao, 2012. "Continuous time Black–Scholes equation with transaction costs in subdiffusive fractional Brownian motion regime," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 750-759.
    23. Magdziarz, Marcin, 2009. "Stochastic representation of subdiffusion processes with time-dependent drift," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3238-3252, October.
    24. Janczura, Joanna & Wyłomańska, Agnieszka, 2009. "Subdynamics of financial data from fractional Fokker-Planck equation," MPRA Paper 30649, University Library of Munich, Germany.
    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. Hainaut, Donatien, 2022. "Multivariate claim processes with rough intensities: Properties and estimation," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 269-287.
    2. Hainaut, Donatien, 2022. "Pricing of spread and exchange options in a rough jump-diffusion market," LIDAM Discussion Papers ISBA 2022012, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Dupret, Jean-Loup & Hainaut, Donatien, 2023. "A fractional Hawkes process for illiquidity modeling," LIDAM Discussion Papers ISBA 2023001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Hainaut, Donatien, 2022. "Multivariate rough claim processes: properties and estimation," LIDAM Discussion Papers ISBA 2022002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Foad Shokrollahi & Marcin Marcin Magdziarz, 2020. "Equity warrant pricing under subdiffusive fractional Brownian motion of the short rate," Papers 2007.12228, arXiv.org, revised Nov 2020.
    6. Foad Shokrollahi, 2016. "Subdiffusive fractional Brownian motion regime for pricing currency options under transaction costs," Papers 1612.06665, arXiv.org, revised Aug 2017.
    7. Ketelbuters, John-John & Hainaut, Donatien, 2022. "CDS pricing with fractional Hawkes processes," European Journal of Operational Research, Elsevier, vol. 297(3), pages 1139-1150.
    8. Kumar, A. & Wyłomańska, A. & Połoczański, R. & Sundar, S., 2017. "Fractional Brownian motion time-changed by gamma and inverse gamma process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 648-667.
    9. Ballestra, Luca Vincenzo & Pacelli, Graziella & Radi, Davide, 2016. "A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 240-248.
    10. Leunga Njike, Charles Guy & Hainaut, Donatien, 2024. "Affine Heston model style with self-exciting jumps and long memory," LIDAM Discussion Papers ISBA 2024001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Hainaut, Donatien & Chen, Maggie & Scalas, Enrico, 2023. "The rough Hawkes process," LIDAM Discussion Papers ISBA 2023007, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    12. Foad Shokrollahi & Adem Kılıçman & Marcin Magdziarz, 2016. "Pricing European options and currency options by time changed mixed fractional Brownian motion with transaction costs," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(01), pages 1-22, March.
    13. Hainaut, Donatien, 2023. "A mutually exciting rough jump diffusion for financial modelling," LIDAM Discussion Papers ISBA 2023011, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    14. Brix, Anne Floor & Lunde, Asger & Wei, Wei, 2018. "A generalized Schwartz model for energy spot prices — Estimation using a particle MCMC method," Energy Economics, Elsevier, vol. 72(C), pages 560-582.
    15. Maneesoonthorn, Worapree & Martin, Gael M. & Forbes, Catherine S. & Grose, Simone D., 2012. "Probabilistic forecasts of volatility and its risk premia," Journal of Econometrics, Elsevier, vol. 171(2), pages 217-236.
    16. Hainaut, Donatien, 2021. "A fractional multi-states model for insurance," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 120-132.
    17. Guglielmo Maria Caporale & Luis A. Gil-Alana & Miguel Martin-Valmayor, 2020. "Persistence in the Realized Betas: Some Evidence for the Spanish Stock Market," CESifo Working Paper Series 8171, CESifo.
    18. Bence Toth & Imon Palit & Fabrizio Lillo & J. Doyne Farmer, 2011. "Why is order flow so persistent?," Papers 1108.1632, arXiv.org, revised Nov 2014.
    19. Tim Bollerslev & Natalia Sizova & George Tauchen, 2011. "Volatility in Equilibrium: Asymmetries and Dynamic Dependencies," Review of Finance, European Finance Association, vol. 16(1), pages 31-80.
    20. Guay, François & Schwenkler, Gustavo, 2021. "Efficient estimation and filtering for multivariate jump–diffusions," Journal of Econometrics, Elsevier, vol. 223(1), pages 251-275.

    More about this item

    Keywords

    Illiquidity modeling ; subdiffusion ; fractional Fokker-Planck equations ; stochastic volatility jump model ; option pricing;
    All these keywords.

    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:aiz:louvad:2022001. 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: Nadja Peiffer (email available below). General contact details of provider: https://edirc.repec.org/data/isuclbe.html .

    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.