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The stochastic collocation Monte Carlo sampler: highly efficient sampling from ‘expensive’ distributions

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  • L. A. Grzelak
  • J. A. S. Witteveen
  • M. Suárez-Taboada
  • C. W. Oosterlee

Abstract

In this article, we propose an efficient approach for inverting computationally expensive cumulative distribution functions. A collocation method, called the Stochastic Collocation Monte Carlo sampler (SCMC sampler), within a polynomial chaos expansion framework, allows us the generation of any number of Monte Carlo samples based on only a few inversions of the original distribution plus independent samples from a standard normal variable. We will show that with this path-independent collocation approach the exact simulation of the Heston stochastic volatility model, as proposed in Broadie and Kaya [Oper. Res., 2006, 54, 217–231], can be performed efficiently and accurately. We also show how to efficiently generate samples from the squared Bessel process and perform the exact simulation of the SABR model.

Suggested Citation

  • L. A. Grzelak & J. A. S. Witteveen & M. Suárez-Taboada & C. W. Oosterlee, 2019. "The stochastic collocation Monte Carlo sampler: highly efficient sampling from ‘expensive’ distributions," Quantitative Finance, Taylor & Francis Journals, vol. 19(2), pages 339-356, February.
    • Handle: RePEc:taf:quantf:v:19:y:2019:i:2:p:339-356
    DOI: 10.1080/14697688.2018.1459807
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    Citations

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    Cited by:

    1. K. B. Gubbels & J. Y. Ypma & C. W. Oosterlee, 2023. "Principal Component Copulas for Capital Modelling," Papers 2312.13195, arXiv.org.
    2. Pingping Zeng & Ziqing Xu & Pingping Jiang & Yue Kuen Kwok, 2023. "Analytical solvability and exact simulation in models with affine stochastic volatility and Lévy jumps," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 842-890, July.
    3. Leonardo Perotti & Lech A. Grzelak, 2021. "Fast Sampling from Time-Integrated Bridges using Deep Learning," Papers 2111.13901, arXiv.org.
    4. Shuaiqiang Liu & Lech A. Grzelak & Cornelis W. Oosterlee, 2022. "The Seven-League Scheme: Deep Learning for Large Time Step Monte Carlo Simulations of Stochastic Differential Equations," Risks, MDPI, vol. 10(3), pages 1-27, February.
    5. Lech A. Grzelak & Juliusz Jablecki & Dariusz Gatarek, 2022. "Efficient Pricing and Calibration of High-Dimensional Basket Options," Papers 2206.09877, arXiv.org.
    6. Leonardo Perotti & Lech A. Grzelak, 2022. "On Pricing of Discrete Asian and Lookback Options under the Heston Model," Papers 2211.03638, arXiv.org, revised Feb 2024.
    7. Grzelak, Lech A., 2022. "Sparse grid method for highly efficient computation of exposures for xVA," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    8. 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.
    9. Shuaiqiang Liu & Graziana Colonna & Lech A. Grzelak & Cornelis W. Oosterlee, 2023. "GPU acceleration of the Seven-League Scheme for large time step simulations of stochastic differential equations," Papers 2302.05170, arXiv.org.
    10. Lech A. Grzelak, 2022. "Randomization of Short-Rate Models, Analytic Pricing and Flexibility in Controlling Implied Volatilities," Papers 2211.05014, arXiv.org.
    11. George Hong, 2020. "Skewing Quanto with Simplicity," Papers 2009.02566, arXiv.org.

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