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Recursive Marginal Quantization of Higher-Order Schemes

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  • T. A. McWalter
  • R. Rudd
  • J. Kienitz
  • E. Platen

Abstract

Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive Marginal Quantization of the Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This method involves recursively quantizing the conditional marginals of the discrete-time Euler approximation of the underlying process. By generalizing this approach, we show that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak order 2.0 scheme. As part of this generalization a simple matrix formulation is presented, allowing efficient implementation. We further extend the applicability of recursive marginal quantization by showing how absorption and reflection at the zero boundary may be incorporated, when this is necessary. To illustrate the improved accuracy of the higher order schemes, various computations are performed using geometric Brownian motion and its generalization, the constant elasticity of variance model. For both processes, we show numerical evidence of improved weak order convergence and we compare the marginal distributions implied by the three schemes to the known analytical distributions. By pricing European, Bermudan and Barrier options, further evidence of improved accuracy of the higher order schemes is demonstrated.

Suggested Citation

  • T. A. McWalter & R. Rudd & J. Kienitz & E. Platen, 2017. "Recursive Marginal Quantization of Higher-Order Schemes," Papers 1701.02681, arXiv.org.
  • Handle: RePEc:arx:papers:1701.02681
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    References listed on IDEAS

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    1. Hsu, Y.L. & Lin, T.I. & Lee, C.F., 2008. "Constant elasticity of variance (CEV) option pricing model: Integration and detailed derivation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(1), pages 60-71.
    2. repec:bla:jfinan:v:44:y:1989:i:1:p:211-19 is not listed on IDEAS
    3. Sagna, Abass, 2011. "Pricing of barrier options by marginal functional quantization," Monte Carlo Methods and Applications, De Gruyter, vol. 17(4), pages 371-398, December.
    4. Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
    5. Gilles Pagès & Abass Sagna, 2015. "Recursive Marginal Quantization of the Euler Scheme of a Diffusion Process," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(5), pages 463-498, November.
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    Citations

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    Cited by:

    1. 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.
    2. Bonollo, Michele & Di Persio, Luca & Oliva, Immacolata, 2020. "A quantization approach to the counterparty credit exposure estimation," International Review of Economics & Finance, Elsevier, vol. 70(C), pages 335-356.
    3. Gilles Pagès & Thibaut Montes & Vincent Lemaire, 2020. "Stationary Heston model: Calibration and Pricing of exotics using Product Recursive Quantization," Working Papers hal-02434232, HAL.
    4. Ralph Rudd & Thomas A. McWalter & Joerg Kienitz & Eckhard Platen, 2017. "Fast Quantization of Stochastic Volatility Models," Papers 1704.06388, arXiv.org.
    5. Damien Ackerer & Damir Filipović, 2020. "Option pricing with orthogonal polynomial expansions," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 47-84, January.
    6. Giorgia Callegaro & Lucio Fiorin & Andrea Pallavicini, 2021. "Quantization goes polynomial," Quantitative Finance, Taylor & Francis Journals, vol. 21(3), pages 361-376, March.
    7. Lucio Fiorin & Wim Schoutens, 2020. "Conic quantization: stochastic volatility and market implied liquidity," Quantitative Finance, Taylor & Francis Journals, vol. 20(4), pages 531-542, April.
    8. Giorgia Callegaro & Alessandro Gnoatto & Martino Grasselli, 2021. "A Fully Quantization-based Scheme for FBSDEs," Papers 2105.09276, arXiv.org.
    9. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2020. "New Weak Error bounds and expansions for Optimal Quantization," Post-Print hal-02361644, HAL.
    10. Ralph Rudd & Thomas A. McWalter & Joerg Kienitz & Eckhard Platen, 2018. "Quantization Under the Real-world Measure: Fast and Accurate Valuation of Long-dated Contracts," Papers 1801.07044, arXiv.org, revised Jan 2018.
    11. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2019. "New Weak Error bounds and expansions for Optimal Quantization," Working Papers hal-02361644, HAL.
    12. Callegaro, Giorgia & Gnoatto, Alessandro & Grasselli, Martino, 2023. "A fully quantization-based scheme for FBSDEs," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    13. Vincent Lemaire & Thibaut Montes & Gilles Pag`es, 2020. "Stationary Heston model: Calibration and Pricing of exotics using Product Recursive Quantization," Papers 2001.03101, arXiv.org, revised Jul 2020.
    14. Damien Ackerer & Damir Filipovic, 2017. "Option Pricing with Orthogonal Polynomial Expansions," Papers 1711.09193, arXiv.org, revised May 2019.
    15. Ralph Rudd & Thomas A. McWalter & Joerg Kienitz & Eckhard Platen, 2020. "Robust Product Markovian Quantization," Papers 2006.15823, arXiv.org.
    16. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2022. "Stationary Heston model: Calibration and Pricing of exotics using Product Recursive Quantization," Post-Print hal-02434232, HAL.

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