IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-02879356.html
   My bibliography  Save this paper

Regime-switching stochastic volatility model: estimation and calibration to VIX options

Author

Listed:
  • Stéphane Goutte

    (LED - Laboratoire d'Economie Dionysien - UP8 - Université Paris 8 Vincennes-Saint-Denis)

  • Amine Ismail

    (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

  • Huyen Pham

    (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

Abstract

We develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynamics with the instantaneous variance driven by a Cox–Ingersoll–Ross process and each parameter modulated by a hidden Markov chain. We propose an extension of the EM algorithm through the Baum–Welch implementation to estimate our model and filter the hidden state of the Markov chain while using the VIX index to invert the latent volatility state. Using Monte Carlo simulations, we test the convergence of our algorithm and compare it with an approximate likelihood procedure where the volatility state is replaced by the VIX index. We found that our method is more accurate than the approximate procedure. Then, we apply Fourier methods to derive a semi-analytical expression of S&P500 and VIX option prices, which we calibrate to market data. We show that the model is sufficiently rich to encapsulate important features of the joint dynamics of the stock and the volatility and to consistently fit option market prices.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Stéphane Goutte & Amine Ismail & Huyen Pham, 2017. "Regime-switching stochastic volatility model: estimation and calibration to VIX options," Post-Print hal-02879356, HAL.
  • Handle: RePEc:hal:journl:hal-02879356
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Jan Baldeaux & Alexander Badran, 2014. "Consistent Modelling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(4), pages 299-312, September.
    2. Peter Christoffersen & Kris Jacobs & Karim Mimouni, 2010. "Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns, and Option Prices," The Review of Financial Studies, Society for Financial Studies, vol. 23(8), pages 3141-3189, August.
    3. 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.
    4. Anders B. Trolle & Eduardo S. Schwartz, 2009. "A General Stochastic Volatility Model for the Pricing of Interest Rate Derivatives," The Review of Financial Studies, Society for Financial Studies, vol. 22(5), pages 2007-2057, May.
    5. Christian Bayer & Jim Gatheral & Morten Karlsmark, 2013. "Fast Ninomiya--Victoir calibration of the double-mean-reverting model," Quantitative Finance, Taylor & Francis Journals, vol. 13(11), pages 1813-1829, November.
    6. Viktor Todorov, 2010. "Variance Risk-Premium Dynamics: The Role of Jumps," The Review of Financial Studies, Society for Financial Studies, vol. 23(1), pages 345-383, January.
    7. Mencía, Javier & Sentana, Enrique, 2013. "Valuation of VIX derivatives," Journal of Financial Economics, Elsevier, vol. 108(2), pages 367-391.
    8. Wong, Hoi Ying & Lo, Yu Wai, 2009. "Option pricing with mean reversion and stochastic volatility," European Journal of Operational Research, Elsevier, vol. 197(1), pages 179-187, August.
    9. Robert Elliott & Tak Kuen Siu & Leunglung Chan, 2007. "Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(1), pages 41-62.
    10. Durham, Garland B., 2007. "SV mixture models with application to S&P 500 index returns," Journal of Financial Economics, Elsevier, vol. 85(3), pages 822-856, September.
    11. Duan, Jin-Chuan & Yeh, Chung-Ying, 2010. "Jump and volatility risk premiums implied by VIX," Journal of Economic Dynamics and Control, Elsevier, vol. 34(11), pages 2232-2244, November.
    12. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    13. Andrew Papanicolaou & Ronnie Sircar, 2014. "A regime-switching Heston model for VIX and S&P 500 implied volatilities," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1811-1827, October.
    14. Hillebrand, Eric, 2005. "Neglecting parameter changes in GARCH models," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 121-138.
    15. Gabriel G. Drimus, 2012. "Options on realized variance by transform methods: a non-affine stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 12(11), pages 1679-1694, November.
    16. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    17. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    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. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Working Papers hal-03827332, HAL.
    2. Léo Parent, 2022. "The EWMA Heston model," Post-Print hal-04431111, HAL.
    3. Florian Bourgey & Stefano De Marco & Emmanuel Gobet, 2022. "Weak approximations and VIX option price expansions in forward variance curve models," Papers 2202.10413, arXiv.org, revised May 2022.
    4. Zhiqiang Zhou & Wei Xu & Alexey Rubtsov, 2024. "Joint calibration of S&P 500 and VIX options under local stochastic volatility models," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 29(1), pages 273-310, January.
    5. 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.
    6. Liu, Yue & Sun, Huaping & Zhang, Jijian & Taghizadeh-Hesary, Farhad, 2020. "Detection of volatility regime-switching for crude oil price modeling and forecasting," Resources Policy, Elsevier, vol. 69(C).
    7. F. Leung & M. Law & S. K. Djeng, 2024. "Deterministic modelling of implied volatility in cryptocurrency options with underlying multiple resolution momentum indicator and non-linear machine learning regression algorithm," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 10(1), pages 1-25, December.
    8. Eduardo Abi Jaber & Camille Illand & Shaun Xiaoyuan Li, 2022. "Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints," Working Papers hal-03902513, HAL.
    9. Antoine Jacquier & Aitor Muguruza & Alexandre Pannier, 2021. "Rough multifactor volatility for SPX and VIX options," Papers 2112.14310, arXiv.org, revised Nov 2023.
    10. Eduardo Abi Jaber & Camille Illand & Shaun Xiaoyuan Li, 2023. "The quintic Ornstein-Uhlenbeck volatility model that jointly calibrates SPX & VIX smiles," Post-Print hal-03909334, HAL.
    11. Eduardo Abi Jaber & Camille Illand & Shaun & Li, 2022. "The quintic Ornstein-Uhlenbeck volatility model that jointly calibrates SPX & VIX smiles," Papers 2212.10917, arXiv.org, revised May 2023.
    12. Abid, Ilyes & Dhaoui, Abderrazak & Goutte, Stéphane & Guesmi, Khaled, 2019. "Contagion and bond pricing: The case of the ASEAN region," Research in International Business and Finance, Elsevier, vol. 47(C), pages 371-385.
    13. Farshid Mehrdoust & Idin Noorani, 2019. "Pricing S&P500 barrier put option of American type under Heston–CIR model with regime-switching," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 6(02), pages 1-17, June.
    14. Julien Guyon, 2020. "Inversion of convex ordering in the VIX market," Quantitative Finance, Taylor & Francis Journals, vol. 20(10), pages 1597-1623, October.
    15. Jaegi Jeon & Geonwoo Kim & Jeonggyu Huh, 2019. "Consistent and Efficient Pricing of SPX and VIX Options under Multiscale Stochastic Volatility," Papers 1909.10187, arXiv.org.
    16. repec:hal:wpaper:hal-03909334 is not listed on IDEAS
    17. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Papers 2210.12393, arXiv.org.
    18. Noorani, Idin & Mehrdoust, Farshid & Nasroallah, Abdelaziz, 2021. "A generalized antithetic variates Monte-Carlo simulation method for pricing of Asian option in a Markov regime-switching model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 1-15.
    19. Lech A. Grzelak, 2022. "On Randomization of Affine Diffusion Processes with Application to Pricing of Options on VIX and S&P 500," Papers 2208.12518, arXiv.org.
    20. Ivan Guo & Gregoire Loeper & Jan Obloj & Shiyi Wang, 2020. "Joint Modelling and Calibration of SPX and VIX by Optimal Transport," Papers 2004.02198, arXiv.org, revised Sep 2021.
    21. Mehrdoust, Farshid & Noorani, Idin & Kanniainen, Juho, 2024. "Valuation of option price in commodity markets described by a Markov-switching model: A case study of WTI crude oil market," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 215(C), pages 228-269.
    22. Jim Gatheral & Paul Jusselin & Mathieu Rosenbaum, 2020. "The quadratic rough Heston model and the joint S&P 500/VIX smile calibration problem," Papers 2001.01789, arXiv.org.
    23. Jiling Cao & Xinfeng Ruan & Shu Su & Wenjun Zhang, 2020. "Pricing VIX derivatives with infinite‐activity jumps," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 40(3), pages 329-354, March.
    24. Eduardo Abi Jaber & Camille Illand & Shaun & Li, 2022. "Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints," Papers 2212.08297, 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. Stéphane Goutte & Amine Ismail & Huyên Pham, 2017. "Regime-switching Stochastic Volatility Model : Estimation and Calibration to VIX options," Working Papers hal-01212018, HAL.
    2. Andrew Papanicolaou, 2018. "Consistent Time-Homogeneous Modeling of SPX and VIX Derivatives," Papers 1812.05859, arXiv.org, revised Mar 2022.
    3. Andrea Barletta & Elisa Nicolato & Stefano Pagliarani, 2019. "The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 928-966, July.
    4. Andrew Papanicolaou & Ronnie Sircar, 2014. "A regime-switching Heston model for VIX and S&P 500 implied volatilities," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1811-1827, October.
    5. Bardgett, Chris & Gourier, Elise & Leippold, Markus, 2019. "Inferring volatility dynamics and risk premia from the S&P 500 and VIX markets," Journal of Financial Economics, Elsevier, vol. 131(3), pages 593-618.
    6. Andrew Papanicolaou, 2022. "Consistent time‐homogeneous modeling of SPX and VIX derivatives," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 907-940, July.
    7. Andrea Barletta & Paolo Santucci de Magistris & Francesco Violante, 2016. "Retrieving Risk-Neutral Densities Embedded in VIX Options: a Non-Structural Approach," CREATES Research Papers 2016-20, Department of Economics and Business Economics, Aarhus University.
    8. Kaeck, Andreas & Seeger, Norman J., 2020. "VIX derivatives, hedging and vol-of-vol risk," European Journal of Operational Research, Elsevier, vol. 283(2), pages 767-782.
    9. Barletta, Andrea & Santucci de Magistris, Paolo & Violante, Francesco, 2019. "A non-structural investigation of VIX risk neutral density," Journal of Banking & Finance, Elsevier, vol. 99(C), pages 1-20.
    10. Pacati, Claudio & Pompa, Gabriele & Renò, Roberto, 2018. "Smiling twice: The Heston++ model," Journal of Banking & Finance, Elsevier, vol. 96(C), pages 185-206.
    11. Chenxu Li, 2014. "Closed-Form Expansion, Conditional Expectation, and Option Valuation," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 487-516, May.
    12. Cao, Jiling & Kim, Jeong-Hoon & Liu, Wenqiang & Zhang, Wenjun, 2023. "Rescaling the double-mean-reverting 4/2 stochastic volatility model for derivative pricing," Finance Research Letters, Elsevier, vol. 58(PB).
    13. Liexin Cheng & Xue Cheng & Xianhua Peng, 2024. "Joint Calibration to SPX and VIX Derivative Markets with Composite Change of Time Models," Papers 2404.16295, arXiv.org, revised Aug 2024.
    14. Xinyu WU & Hailin ZHOU, 2016. "GARCH DIFFUSION MODEL, iVIX, AND VOLATILITY RISK PREMIUM," ECONOMIC COMPUTATION AND ECONOMIC CYBERNETICS STUDIES AND RESEARCH, Faculty of Economic Cybernetics, Statistics and Informatics, vol. 50(1), pages 327-342.
    15. Alexander Badran & Beniamin Goldys, 2015. "A Market Model for VIX Futures," Papers 1504.00428, arXiv.org.
    16. Song, Zhaogang & Xiu, Dacheng, 2016. "A tale of two option markets: Pricing kernels and volatility risk," Journal of Econometrics, Elsevier, vol. 190(1), pages 176-196.
    17. Alexandre Pannier, 2023. "Path-dependent PDEs for volatility derivatives," Papers 2311.08289, arXiv.org, revised Jan 2024.
    18. Andrew Papanicolaou, 2021. "Extreme-Strike Comparisons and Structural Bounds for SPX and VIX Options," Papers 2101.00299, arXiv.org, revised Mar 2021.
    19. Florian Bourgey & Stefano De Marco & Emmanuel Gobet, 2022. "Weak approximations and VIX option price expansions in forward variance curve models," Papers 2202.10413, arXiv.org, revised May 2022.
    20. Lech A. Grzelak, 2022. "On Randomization of Affine Diffusion Processes with Application to Pricing of Options on VIX and S&P 500," Papers 2208.12518, 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:hal:journl:hal-02879356. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.