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General Error Estimates for the Longstaff–Schwartz Least-Squares Monte Carlo Algorithm

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  • Daniel Z. Zanger

    (California Science and Technology University, Milpitas, California 95035)

Abstract

We establish error estimates for the Longstaff–Schwartz algorithm, employing just a single set of independent Monte Carlo sample paths that is reused for all exercise time steps. We obtain, within the context of financial derivative payoff functions bounded according to the uniform norm, new bounds on the stochastic part of the error of this algorithm for an approximation architecture that may be any arbitrary set of L 2 functions of finite Vapnik–Chervonenkis (VC) dimension whenever the algorithm’s least-squares regression optimization step is solved either exactly or approximately. Moreover, we show how to extend these estimates to the case of payoff functions bounded only in L p , p a real number greater than 2 < p < ∞ . We also establish new overall error bounds for the Longstaff–Schwartz algorithm, including estimates on the approximation error also for unconstrained linear, finite-dimensional polynomial approximation. Our results here extend those in the literature by not imposing any uniform boundedness condition on the approximation architectures, allowing each of them to be any set of L 2 functions of finite VC dimension and by establishing error estimates as well in the case of ɛ-additive approximate least-squares optimization, ɛ greater than or equal to 0.

Suggested Citation

  • Daniel Z. Zanger, 2020. "General Error Estimates for the Longstaff–Schwartz Least-Squares Monte Carlo Algorithm," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 923-946, August.
    • Handle: RePEc:inm:ormoor:v:45:y:2020:i:3:p:923-946
    DOI: 10.1287/moor.2019.1017
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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. Daniel Z. Zanger, 2018. "Convergence Of A Least†Squares Monte Carlo Algorithm For American Option Pricing With Dependent Sample Data," Mathematical Finance, Wiley Blackwell, vol. 28(1), pages 447-479, January.
    3. Daniel Zanger, 2009. "Convergence of a Least-Squares Monte Carlo Algorithm for Bounded Approximating Sets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(2), pages 123-150.
    4. Daniel Zanger, 2013. "Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing," Finance and Stochastics, Springer, vol. 17(3), pages 503-534, July.
    5. L.A. Abbas-Turki & S. Vialle & Bernard Lapeyre & P. Mercier, 2014. "Pricing derivatives on graphics processing units using Monte Carlo simulation," Post-Print hal-01667067, HAL.
    6. Lars Stentoft, 2004. "Convergence of the Least Squares Monte Carlo Approach to American Option Valuation," Management Science, INFORMS, vol. 50(9), pages 1193-1203, September.
    7. Philip Protter & Emmanuelle Clément & Damien Lamberton, 2002. "An analysis of a least squares regression method for American option pricing," Finance and Stochastics, Springer, vol. 6(4), pages 449-471.
    8. Denis Belomestny, 2011. "Pricing Bermudan options by nonparametric regression: optimal rates of convergence for lower estimates," Finance and Stochastics, Springer, vol. 15(4), pages 655-683, December.
    9. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    10. Lars Stentoft, 2004. "Assessing the Least Squares Monte-Carlo Approach to American Option Valuation," Review of Derivatives Research, Springer, vol. 7(2), pages 129-168, August.
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