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Weighted Monte Carlo with least squares and randomized extended Kaczmarz for option pricing

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  • Damir Filipovi'c
  • Kathrin Glau
  • Yuji Nakatsukasa
  • Francesco Statti

Abstract

We propose a methodology for computing single and multi-asset European option prices, and more generally expectations of scalar functions of (multivariate) random variables. This new approach combines the ability of Monte Carlo simulation to handle high-dimensional problems with the efficiency of function approximation. Specifically, we first generalize the recently developed method for multivariate integration in [arXiv:1806.05492] to integration with respect to probability measures. The method is based on the principle "approximate and integrate" in three steps i) sample the integrand at points in the integration domain, ii) approximate the integrand by solving a least-squares problem, iii) integrate the approximate function. In high-dimensional applications we face memory limitations due to large storage requirements in step ii). Combining weighted sampling and the randomized extended Kaczmarz algorithm we obtain a new efficient approach to solve large-scale least-squares problems. Our convergence and cost analysis along with numerical experiments show the effectiveness of the method in both low and high dimensions, and under the assumption of a limited number of available simulations.

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  • Damir Filipovi'c & Kathrin Glau & Yuji Nakatsukasa & Francesco Statti, 2019. "Weighted Monte Carlo with least squares and randomized extended Kaczmarz for option pricing," Papers 1910.07241, arXiv.org.
    • Handle: RePEc:arx:papers:1910.07241
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    References listed on IDEAS

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    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. Damir Filipovic & Damien Ackerer & Sergio Pulido, 2018. "The Jacobi Stochastic Volatility Model," Post-Print hal-01338330, HAL.
    3. Daniel Kressner & Robert Luce & Francesco Statti, 2017. "Incremental computation of block triangular matrix exponentials with application to option pricing," Papers 1703.00182, arXiv.org, revised Jun 2017.
    4. 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.
    5. Damien Ackerer & Damir Filipović & Sergio Pulido, 2018. "The Jacobi stochastic volatility model," Finance and Stochastics, Springer, vol. 22(3), pages 667-700, July.
    6. Damien Ackerer & Damir Filipovi'c & Sergio Pulido, 2016. "The Jacobi Stochastic Volatility Model," Papers 1605.07099, arXiv.org, revised Mar 2018.
    7. repec:cdl:anderf:qt43n1k4jb is not listed on IDEAS
    8. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
    9. Damir Filipović & Martin Larsson, 2017. "Polynomial Jump-Diffusion Models," Swiss Finance Institute Research Paper Series 17-60, Swiss Finance Institute.
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    Cited by:

    1. Lotfi Boudabsa & Damir Filipović, 2022. "Machine learning with kernels for portfolio valuation and risk management," Finance and Stochastics, Springer, vol. 26(2), pages 131-172, April.

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