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Robust Product Markovian Quantization

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  • Ralph Rudd
  • Thomas A. McWalter
  • Joerg Kienitz
  • Eckhard Platen

Abstract

Recursive marginal quantization (RMQ) allows the construction of optimal discrete grids for approximating solutions to stochastic differential equations in d-dimensions. Product Markovian quantization (PMQ) reduces this problem to d one-dimensional quantization problems by recursively constructing product quantizers, as opposed to a truly optimal quantizer. However, the standard Newton-Raphson method used in the PMQ algorithm suffers from numerical instabilities, inhibiting widespread adoption, especially for use in calibration. By directly specifying the random variable to be quantized at each time step, we show that PMQ, and RMQ in one dimension, can be expressed as standard vector quantization. This reformulation allows the application of the accelerated Lloyd's algorithm in an adaptive and robust procedure. Furthermore, in the case of stochastic volatility models, we extend the PMQ algorithm by using higher-order updates for the volatility or variance process. We illustrate the technique for European options, using the Heston model, and more exotic products, using the SABR model.

Suggested Citation

  • Ralph Rudd & Thomas A. McWalter & Joerg Kienitz & Eckhard Platen, 2020. "Robust Product Markovian Quantization," Papers 2006.15823, arXiv.org.
    • Handle: RePEc:arx:papers:2006.15823
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. T. A. McWalter & R. Rudd & J. Kienitz & E. Platen, 2018. "Recursive marginal quantization of higher-order schemes," Quantitative Finance, Taylor & Francis Journals, vol. 18(4), pages 693-706, April.
    4. 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.
    5. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    6. Ralph Rudd & Thomas A. McWalter & Joerg Kienitz & Eckhard Platen, 2017. "Fast Quantization of Stochastic Volatility Models," Papers 1704.06388, arXiv.org.
    7. Giorgia Callegaro & Lucio Fiorin & Martino Grasselli, 2017. "Pricing via recursive quantization in stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 17(6), pages 855-872, June.
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