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On calibration of stochastic and fractional stochastic volatility models

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  • Mrázek, Milan
  • Pospíšil, Jan
  • Sobotka, Tomáš

Abstract

In this paper we study optimization techniques for calibration of stochastic volatility models to real market data. Several optimization techniques are compared and used in order to solve the nonlinear least squares problem arising in the minimization of the difference between the observed market prices and the model prices. To compare several approaches we use a popular stochastic volatility model firstly introduced by Heston (1993) and a more complex model with jumps in the underlying and approximative fractional volatility. Calibration procedures are performed on two main data sets that involve traded DAX index options. We show how well both models can be fitted to a given option price surface. The routines alongside models are also compared in terms of out-of-sample errors. For the calibration tasks without having a good knowledge of the market (e.g. a suitable initial model parameters) we suggest an approach of combining local and global optimizers. This way we are able to retrieve superior error measures for all considered tasks and models.

Suggested Citation

  • Mrázek, Milan & Pospíšil, Jan & Sobotka, Tomáš, 2016. "On calibration of stochastic and fractional stochastic volatility models," European Journal of Operational Research, Elsevier, vol. 254(3), pages 1036-1046.
    • Handle: RePEc:eee:ejores:v:254:y:2016:i:3:p:1036-1046
    DOI: 10.1016/j.ejor.2016.04.033
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    2. 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.
    3. R. Merino & J. Pospíšil & T. Sobotka & J. Vives, 2018. "Decomposition Formula For Jump Diffusion Models," Journal of Enterprising Culture (JEC), World Scientific Publishing Co. Pte. Ltd., vol. 21(08), pages 1-36, December.
    4. Ben-zhang Yang & Xinjiang He & Nan-jing Huang, 2019. "Equilibrium price and optimal insider trading strategy under stochastic liquidity with long memory," Papers 1901.00345, arXiv.org, revised Jan 2019.
    5. Moawia Alghalith & Christos Floros & Konstantinos Gkillas, 2020. "Estimating Stochastic Volatility under the Assumption of Stochastic Volatility of Volatility," Risks, MDPI, vol. 8(2), pages 1-15, April.
    6. Jan Matas & Jan Pospíšil, 2023. "Robustness and sensitivity analyses of rough Volterra stochastic volatility models," Annals of Finance, Springer, vol. 19(4), pages 523-543, December.
    7. Raul Merino & Jan Posp'iv{s}il & Tom'av{s} Sobotka & Josep Vives, 2019. "Decomposition formula for jump diffusion models," Papers 1906.06930, arXiv.org.
    8. Han, Jinhui & Li, Xiaolong & Ma, Guiyuan & Kennedy, Adrian Patrick, 2023. "Strategic trading with information acquisition and long-memory stochastic liquidity," European Journal of Operational Research, Elsevier, vol. 308(1), pages 480-495.
    9. Ewald, Christian & Zou, Yihan, 2021. "Analytic formulas for futures and options for a linear quadratic jump diffusion model with seasonal stochastic volatility and convenience yield: Do fish jump?," European Journal of Operational Research, Elsevier, vol. 294(2), pages 801-815.
    10. Cui, Zhenyu & Lars Kirkby, J. & Nguyen, Duy, 2017. "A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps," European Journal of Operational Research, Elsevier, vol. 262(1), pages 381-400.
    11. Ying Chang & Yiming Wang & Sumei Zhang, 2021. "Option Pricing under Double Heston Jump-Diffusion Model with Approximative Fractional Stochastic Volatility," Mathematics, MDPI, vol. 9(2), pages 1-10, January.
    12. Jan Posp'iv{s}il & Tom'av{s} Sobotka & Philipp Ziegler, 2019. "Robustness and sensitivity analyses for stochastic volatility models under uncertain data structure," Papers 1912.06709, arXiv.org.
    13. Omid Jenabi & Nazar Dahmardeh Ghale No, 2018. "Option Pricing in Stochastic Volatility Models Driven by Fractional Jump-Diffusion Processes," International Journal of Finance, Insurance and Risk Management, International Journal of Finance, Insurance and Risk Management, vol. 8(1), pages 1374-1374.

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