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The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation

Author

Listed:
  • ANTHONIE W. VAN DER STOEP

    (Derivatives Research and Validation Group, Rabobank, Graadt van Roggenweg 400, 3531 AH, Utrecht, The Netherlands;
    CWI — National Research Institute for Mathematics and Computer Science, Science Park 123, 1098 XG, Amsterdam, The Netherlands)

  • LECH A. GRZELAK

    (Derivatives Research and Validation Group, Rabobank, Graadt van Roggenweg 400, 3531 AH, Utrecht, The Netherlands;
    CWI — National Research Institute for Mathematics and Computer Science, Science Park 123, 1098 XG, Amsterdam, The Netherlands)

  • CORNELIS W. OOSTERLEE

    (Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands;
    CWI — National Research Institute for Mathematics and Computer Science, Science Park 123, 1098 XG, Amsterdam, The Netherlands)

Abstract

In this paper we propose an efficient Monte Carlo scheme for simulating the stochastic volatility model of Heston (1993) enhanced by a nonparametric local volatility component. This hybrid model combines the main advantages of the Heston model and the local volatility model introduced by Dupire (1994) and Derman & Kani (1998). In particular, the additional local volatility component acts as a "compensator" that bridges the mismatch between the nonperfectly calibrated Heston model and the market quotes for European-type options. By means of numerical experiments we show that our scheme enables a consistent and fast pricing of products that are sensitive to the forward volatility skew. Detailed error analysis is also provided.

Suggested Citation

  • Anthonie W. Van Der Stoep & Lech A. Grzelak & Cornelis W. Oosterlee, 2014. "The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(07), pages 1-30.
    • Handle: RePEc:wsi:ijtafx:v:17:y:2014:i:07:n:s0219024914500459
    DOI: 10.1142/S0219024914500459
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    Citations

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    Cited by:

    1. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    2. Akihiro Kaneko, 2023. "Multi-stage Euler-Maruyama methods for backward stochastic differential equations driven by continuous-time Markov chains," Papers 2311.08826, arXiv.org, revised Nov 2023.
    3. Andrei Cozma & Christoph Reisinger, 2015. "Exponential integrability properties of Euler discretization schemes for the Cox-Ingersoll-Ross process," Papers 1601.00919, arXiv.org.
    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. Maarten Wyns & Karel in 't Hout, 2016. "An adjoint method for the exact calibration of Stochastic Local Volatility models," Papers 1609.00232, arXiv.org.
    6. Misha Beek & Michel Mandjes & Peter Spreij & Erik Winands, 2020. "Regime switching affine processes with applications to finance," Finance and Stochastics, Springer, vol. 24(2), pages 309-333, April.
    7. Cui, Zhenyu & Kirkby, J. Lars & Nguyen, Duy, 2021. "Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations," European Journal of Operational Research, Elsevier, vol. 290(3), pages 1046-1062.
    8. Maarten Wyns & Jacques Du Toit, 2016. "A Finite Volume - Alternating Direction Implicit Approach for the Calibration of Stochastic Local Volatility Models," Papers 1611.02961, arXiv.org.
    9. Andrei Cozma & Matthieu Mariapragassam & Christoph Reisinger, 2015. "Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets," Papers 1501.06084, arXiv.org, revised Oct 2016.
    10. Kaustav Das & Nicolas Langren'e, 2020. "Explicit approximations of option prices via Malliavin calculus in a general stochastic volatility framework," Papers 2006.01542, arXiv.org, revised Jan 2024.
    11. Andrei Cozma & Christoph Reisinger, 2017. "Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models," Papers 1706.07375, arXiv.org, revised Oct 2018.
    12. Ferreiro-Ferreiro, Ana María & García-Rodríguez, José A. & Souto, Luis & Vázquez, Carlos, 2020. "A new calibration of the Heston Stochastic Local Volatility Model and its parallel implementation on GPUs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 467-486.
    13. Martino Grasselli & Andrea Mazzoran & Andrea Pallavicini, 2024. "A general framework for a joint calibration of VIX and VXX options," Annals of Operations Research, Springer, vol. 336(1), pages 3-26, May.
    14. Sergey Nasekin & Wolfgang Karl Hardle, 2020. "Model-driven statistical arbitrage on LETF option markets," Papers 2009.09713, arXiv.org.
    15. Jingtang Ma & Wensheng Yang & Zhenyu Cui, 2021. "Semimartingale and continuous-time Markov chain approximation for rough stochastic local volatility models," Papers 2110.08320, arXiv.org, revised Oct 2021.
    16. Julio Guerrero & Giuseppe Orlando, 2022. "Stochastic Local Volatility models and the Wei-Norman factorization method," Papers 2201.11241, arXiv.org.
    17. Ma, Jingtang & Yang, Wensheng & Cui, Zhenyu, 2021. "CTMC integral equation method for American options under stochastic local volatility models," Journal of Economic Dynamics and Control, Elsevier, vol. 128(C).
    18. Kaustav Das & Nicolas Langren'e, 2018. "Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility," Papers 1812.07803, arXiv.org, revised Oct 2021.
    19. Luca De Gennaro Aquino & Carole Bernard, 2019. "Semi-analytical prices for lookback and barrier options under the Heston model," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 715-741, December.
    20. Daniel Guterding & Wolfram Boenkost, 2018. "The Heston stochastic volatility model with piecewise constant parameters - efficient calibration and pricing of window barrier options," Papers 1805.04704, arXiv.org, revised Jan 2019.

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