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Optimal quadratic quantization for numerics: the Gaussian case

Author

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  • Pagès Gilles

    (Laboratoire de Probabilités et Modèles Aléatoires, CNRS UMR 7599, Université Paris 6, case 188, 4, pl. Jussieu, F-75252 Paris Cedex 5. E-mail: gpa@ccr.jussieu.fr)

  • Printems Jacques

    (INRIA, MathFi project and Centre de Mathématiques, CNRS UMR 8050, Université Paris 12, 61, av. du Général de Gaulle, F-94010 Créteil. E-mail: printems@univ-paris12.fr)

Abstract

Optimal quantization has been recently revisited in multi-dimensional numerical integration, multi-asset American option pricing, control theory and nonlinear filtering theory. In this paper, we enlighten some numerical procedures in order to get some accurate optimal quadratic quantization of the Gaussian distribution in one and higher dimensions. We study in particular Newton method in the deterministic case (dimension d = 1) and stochastic gradient in higher dimensional case (d ≥ 2). Some heuristics are provided which concern the step in the stochastic gradient method. Finally numerical examples borrowed from mathematical finance are used to test the accuracy of our Gaussian optimal quantizers.

Suggested Citation

  • Pagès Gilles & Printems Jacques, 2003. "Optimal quadratic quantization for numerics: the Gaussian case," Monte Carlo Methods and Applications, De Gruyter, vol. 9(2), pages 135-165, April.
    • Handle: RePEc:bpj:mcmeap:v:9:y:2003:i:2:p:135-165:n:2
    DOI: 10.1515/156939603322663321
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    References listed on IDEAS

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    1. V. Bally & G. Pagès & J. Printems, 2003. "First‐Order Schemes in the Numerical Quantization Method," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 1-16, January.
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    Cited by:

    1. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2020. "New Weak Error bounds and expansions for Optimal Quantization," Post-Print hal-02361644, HAL.
    2. 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.
    3. Damir Filipovic & Damien Ackerer & Sergio Pulido, 2018. "The Jacobi Stochastic Volatility Model," Post-Print hal-01338330, HAL.
    4. Bally, Vlad & Pagès, Gilles, 2003. "Error analysis of the optimal quantization algorithm for obstacle problems," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 1-40, July.
    5. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2019. "New Weak Error bounds and expansions for Optimal Quantization," Working Papers hal-02361644, HAL.
    6. Giorgia Callegaro & Abass Sagna, 2009. "An application to credit risk of a hybrid Monte Carlo-Optimal quantization method," Papers 0907.0645, arXiv.org.
    7. Moritz Duembgen & L. C. G. Rogers, 2014. "Estimate nothing," Quantitative Finance, Taylor & Francis Journals, vol. 14(12), pages 2065-2072, December.
    8. L C G Rogers & Pawel Zaczkowski, 2013. "Monte Carlo approximation to optimal investment," Papers 1305.3433, arXiv.org.
    9. repec:hal:wpaper:hal-00400666 is not listed on IDEAS
    10. 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.
    11. Anne Laure Bronstein & Gilles Pages & Benedikt Wilbertz, 2010. "How to speed up the quantization tree algorithm with an application to swing options," Quantitative Finance, Taylor & Francis Journals, vol. 10(9), pages 995-1007.
    12. Doan, Viet_Dung & Gaikwad, Abhijeet & Bossy, Mireille & Baude, Françoise & Stokes-Rees, Ian, 2010. "Parallel pricing algorithms for multi-dimensional Bermudan/American options using Monte Carlo methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(3), pages 568-577.

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