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Fast Quantization of Stochastic Volatility Models

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

Abstract

Recursive Marginal Quantization (RMQ) allows fast approximation of solutions to stochastic differential equations in one-dimension. When applied to two factor models, RMQ is inefficient due to the fact that the optimization problem is usually performed using stochastic methods, e.g., Lloyd's algorithm or Competitive Learning Vector Quantization. In this paper, a new algorithm is proposed that allows RMQ to be applied to two-factor stochastic volatility models, which retains the efficiency of gradient-descent techniques. By margining over potential realizations of the volatility process, a significant decrease in computational effort is achieved when compared to current quantization methods. Additionally, techniques for modelling the correct zero-boundary behaviour are used to allow the new algorithm to be applied to cases where the previous methods would fail. The proposed technique is illustrated for European options on the Heston and Stein-Stein models, while a more thorough application is considered in the case of the popular SABR model, where various exotic options are also priced.

Suggested Citation

  • Ralph Rudd & Thomas A. McWalter & Joerg Kienitz & Eckhard Platen, 2017. "Fast Quantization of Stochastic Volatility Models," Papers 1704.06388, arXiv.org.
    • Handle: RePEc:arx:papers:1704.06388
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    References listed on IDEAS

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    1. Bin Chen & Cornelis W. Oosterlee & Hans Van Der Weide, 2012. "A Low-Bias Simulation Scheme For The Sabr Stochastic Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(02), pages 1-37.
    2. 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.
    3. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    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. 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.
    6. 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.
    7. 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.
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    Citations

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    Cited by:

    1. Gilles Pagès & Thibaut Montes & Vincent Lemaire, 2020. "Stationary Heston model: Calibration and Pricing of exotics using Product Recursive Quantization," Working Papers hal-02434232, HAL.
    2. 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.
    3. 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.
    4. Ralph Rudd & Thomas A. McWalter & Joerg Kienitz & Eckhard Platen, 2020. "Robust Product Markovian Quantization," Papers 2006.15823, arXiv.org.
    5. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2022. "Stationary Heston model: Calibration and Pricing of exotics using Product Recursive Quantization," Post-Print hal-02434232, HAL.
    6. 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.

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