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Convergence Of A Least†Squares Monte Carlo Algorithm For American Option Pricing With Dependent Sample Data

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  • Daniel Z. Zanger

Abstract

We analyze the convergence of the Longstaff–Schwartz algorithm relying on only a single set of independent Monte Carlo sample paths that is repeatedly reused for all exercise time†steps. We prove new estimates on the stochastic component of the error of this algorithm whenever the approximation architecture is any uniformly bounded set of L2 functions of finite Vapnik–Chervonenkis dimension (VC†dimension), but in particular need not necessarily be either convex or closed. We also establish new overall error estimates, incorporating bounds on the approximation error as well, for certain nonlinear, nonconvex sets of neural networks.

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  • 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.
    • Handle: RePEc:bla:mathfi:v:28:y:2018:i:1:p:447-479
    DOI: 10.1111/mafi.12125
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    Cited by:

    1. 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.
    2. Chen Liu & Henry Schellhorn & Qidi Peng, 2019. "American Option Pricing With Regression: Convergence Analysis," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(08), pages 1-31, December.
    3. Ravi Kashyap, 2022. "Options as Silver Bullets: Valuation of Term Loans, Inventory Management, Emissions Trading and Insurance Risk Mitigation using Option Theory," Annals of Operations Research, Springer, vol. 315(2), pages 1175-1215, August.
    4. Christian Bayer & Ra'ul Tempone & Soren Wolfers, 2018. "Pricing American Options by Exercise Rate Optimization," Papers 1809.07300, arXiv.org, revised Aug 2019.
    5. Sérgio C. Bezerra & Alberto Ohashi & Francesco Russo & Francys Souza, 2020. "Discrete-type Approximations for Non-Markovian Optimal Stopping Problems: Part II," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1221-1255, September.
    6. Denis Belomestny & Maxim Kaledin & John Schoenmakers, 2020. "Semitractability of optimal stopping problems via a weighted stochastic mesh algorithm," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1591-1616, October.
    7. Hampus Engsner, 2021. "Least Squares Monte Carlo applied to Dynamic Monetary Utility Functions," Papers 2101.10947, arXiv.org, revised Apr 2021.
    8. Jo~ao F. Doriguello & Alessandro Luongo & Jinge Bao & Patrick Rebentrost & Miklos Santha, 2021. "Quantum algorithm for stochastic optimal stopping problems with applications in finance," Papers 2111.15332, arXiv.org, revised Jul 2023.

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