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Product Markovian Quantization of a Diffusion Process with Applications to Finance

Author

Listed:
  • Lucio Fiorin

    (University of Padova)

  • Gilles Pagès

    (Sorbonne Université (formerly UPMC), UMR 8001)

  • Abass Sagna

    (Université d’Evry Val-d’Essonne, UMR CNRS 8071)

Abstract

We introduce a new methodology for the quantization of the Euler scheme for a d-dimensional diffusion process. This method is based on a Markovian and componentwise product quantization and allows us, from a numerical point of view, to speak of fast online quantization in a dimension greater than one since the product quantization of the Euler scheme of the diffusion process and its companion weights and transition probabilities may be computed quite instantaneously. We show that the resulting quantization process is a Markov chain, then we compute the associated weights and transition probabilities from (semi-) closed formulas. From the analytical point of view, we show that the induced quantization errors at the k-th discretization step is a cumulative of the marginal quantization error up to that time. Numerical experiments are performed for the pricing of a Basket call option in a correlated Black Scholes framework, for the pricing of a European call option in a Heston model and for the approximation of the solution of backward stochastic differential equations in order to show the performances of the method.

Suggested Citation

  • Lucio Fiorin & Gilles Pagès & Abass Sagna, 2019. "Product Markovian Quantization of a Diffusion Process with Applications to Finance," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1087-1118, December.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:4:d:10.1007_s11009-018-9652-1
    DOI: 10.1007/s11009-018-9652-1
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    References listed on IDEAS

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    1. Peter Christoffersen & Steven Heston & Kris Jacobs, 2009. "The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work So Well," Management Science, INFORMS, vol. 55(12), pages 1914-1932, December.
    2. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    3. Pagès, Gilles & Sagna, Abass, 2018. "Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 847-883.
    4. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    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.
    6. 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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    8. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    9. Bender, Christian & Denk, Robert, 2007. "A forward scheme for backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1793-1812, December.
    10. Crisan, D. & Manolarakis, K. & Touzi, N., 2010. "On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1133-1158, July.
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    Cited by:

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