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Full and fast calibration of the Heston stochastic volatility model

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  • Yiran Cui
  • Sebastian del Ba~no Rollin
  • Guido Germano

Abstract

This paper presents an algorithm for a complete and efficient calibration of the Heston stochastic volatility model. We express the calibration as a nonlinear least squares problem. We exploit a suitable representation of the Heston characteristic function and modify it to avoid discontinuities caused by branch switchings of complex functions. Using this representation, we obtain the analytical gradient of the price of a vanilla option with respect to the model parameters, which is the key element of all variants of the objective function. The interdependency between the components of the gradient enables an efficient implementation which is around ten times faster than a numerical gradient. We choose the Levenberg-Marquardt method to calibrate the model and do not observe multiple local minima reported in previous research. Two-dimensional sections show that the objective function is shaped as a narrow valley with a flat bottom. Our method is the fastest calibration of the Heston model developed so far and meets the speed requirement of practical trading.

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  • Yiran Cui & Sebastian del Ba~no Rollin & Guido Germano, 2015. "Full and fast calibration of the Heston stochastic volatility model," Papers 1511.08718, arXiv.org, revised May 2016.
    • Handle: RePEc:arx:papers:1511.08718
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    Cited by:

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    2. Oliver Pfante & Nils Bertschinger, 2019. "Information-Theoretic Analysis Of Stochastic Volatility Models," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 22(01), pages 1-21, February.
    3. Giorgia Callegaro & Lucio Fiorin & Martino Grasselli, 2019. "Quantization meets Fourier: a new technology for pricing options," Annals of Operations Research, Springer, vol. 282(1), pages 59-86, November.
    4. Yu Feng & Ralph Rudd & Christopher Baker & Qaphela Mashalaba & Melusi Mavuso & Erik Schlögl, 2021. "Quantifying the Model Risk Inherent in the Calibration and Recalibration of Option Pricing Models," Risks, MDPI, vol. 9(1), pages 1-20, January.
    5. Fan, Qingqian & Feng, Sixian, 2022. "An empirical study on the characterization of implied volatility and pricing in the Chinese option market," Finance Research Letters, Elsevier, vol. 49(C).
    6. Pierre-Edouard Arrouy & Alexandre Boumezoued & Bernard Lapeyre & Sophian Mehalla, 2022. "Economic Scenario Generators: a risk management tool for insurance," Post-Print hal-03671943, HAL.
    7. Gudmundsson, Hilmar & Vyncke, David, 2019. "On the calibration of the 3/2 model," European Journal of Operational Research, Elsevier, vol. 276(3), pages 1178-1192.
    8. 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.
    9. Sascha Desmettre, 2018. "Change of Measure in the Heston Model given a violated Feller Condition," Papers 1809.10955, arXiv.org, revised Oct 2019.
    10. Kontosakos, Vasileios E. & Mendonca, Keegan & Pantelous, Athanasios A. & Zuev, Konstantin M., 2021. "Pricing discretely-monitored double barrier options with small probabilities of execution," European Journal of Operational Research, Elsevier, vol. 290(1), pages 313-330.
    11. Recchioni, Maria Cristina & Iori, Giulia & Tedeschi, Gabriele & Ouellette, Michelle S., 2021. "The complete Gaussian kernel in the multi-factor Heston model: Option pricing and implied volatility applications," European Journal of Operational Research, Elsevier, vol. 293(1), pages 336-360.
    12. Oliver Pfante & Nils Bertschinger, 2019. "Volatility Inference And Return Dependencies In Stochastic Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-44, May.
    13. Pascal François & Lars Stentoft, 2021. "Smile‐implied hedging with volatility risk," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(8), pages 1220-1240, August.
    14. Shuaiqiang Liu & Anastasia Borovykh & Lech A. Grzelak & Cornelis W. Oosterlee, 2019. "A neural network-based framework for financial model calibration," Papers 1904.10523, arXiv.org.
    15. Eudald Romo & Luis Ortiz-Gracia, 2021. "SWIFT calibration of the Heston model," Papers 2103.01570, arXiv.org.
    16. Hervé Andres & Pierre-Edouard Arrouy & Paul Bonnefoy & Alexandre Boumezoued & Sophian Mehalla, 2020. "Fast calibration of the LIBOR Market Model with Stochastic Volatility based on analytical gradient," Working Papers hal-02875623, HAL.
    17. Eudald Romo & Luis Ortiz-Gracia, 2021. "SWIFT Calibration of the Heston Model," Mathematics, MDPI, vol. 9(5), pages 1-20, March.
    18. Gaetano Bua & Daniele Marazzina, 2021. "On the application of Wishart process to the pricing of equity derivatives: the multi-asset case," Computational Management Science, Springer, vol. 18(2), pages 149-176, June.
    19. Feng, Chengxiao & Tan, Jie & Jiang, Zhenyu & Chen, Shuang, 2020. "A generalized European option pricing model with risk management," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    20. Herv'e Andres & Pierre-Edouard Arrouy & Paul Bonnefoy & Alexandre Boumezoued & Sophian Mehalla, 2020. "Fast calibration of the LIBOR Market Model with Stochastic Volatility based on analytical gradient," Papers 2006.13521, arXiv.org.
    21. Pierre-Edouard Arrouy & Alexandre Boumezoued & Bernard Lapeyre & Sophian Mehalla, 2022. "Economic Scenario Generators: a risk management tool for insurance," Working Papers hal-03671943, HAL.

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    • G3 - Financial Economics - - Corporate Finance and Governance

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